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About Google Book Search Google’s mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world’s books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at[http: //books . google .com/| Clarendon Drees Series RECENT RESEARCHES ELECTRICITY AND MAGNETISM J. J. THOMSON Ss Bondon HENRY FROWDE Oxrorn University Press WAREHOUSE. Amun Corner, E.C. Qew York MACMILLAN & CO. 112 FOURTH AVENUE THE BEM PUBLIC Liz 34¢0 Orford PRINTED AT THE CLARENDON PRESS Wy MORACE MaMT, PRINTER TO THE UNIVERSITY vi PREFACE. Electricity and diminishes the value of the mental training afforded by the study of that science, Th the first place, though no instrument of research is more powerful than Mathematical Analysis, which indeed is indispens- able in many departments of Electricity, yet analysis works to the best advantage when employed in developing the suggestions afforded by other and more physical methods, One example of such a method, and one which is very closely connected with the initiation and development of Maxwell's Theory, is that of the *tubes of force’ used by Faraday, Faraday interpreted all the laws of Electrostatics in terms of his tubes, which served him in place of the symbols of the mathematician, while in his hands the laws according to which these tubes acted on each other served instead of the differential equations satisfied by such symbols, The method of the tubes is distinetly physical, that of the symbols and differential equations is analytical, The physical method has all the advantages in vividness which arise from the use of concrete quantitics instead of abstract symbols to represent the atate of the electric field; it is more easily wielded, and is thus more suitable for obtaining rapidly the main features of any problem ; when, however, the problem has to be worked out in all its details, the analytical method is necessary. Tn a research in any of tho various ficlds of electricity we shall ‘be acting in accordance with Bacon's dictum that the beat results are obtained when a resoarch begins with Physics and ends with Mathematics, if we use the physical theory to, 80 to speak, make a general survey of the country, and when thix has been done use the analytical method to lay down firm roads along the line indicated by the survey. ‘The use of a physical theory will help to correct the tendency —which I think all who bave had occasion to examine in Mathe- matical Physics will admit is by no means uncommon—to look on analytical processes aa the modern equivalents of the Philosopher's Machine in the Grund Academy of Lagado, and to regard as the normal process of investigation in this subject the manipulation of a large number of aymbols in the hope that every now and then some valuable result may happen to drop out, = i Fe currents, both for experimental and commorcial purposes, much more general than when Maxwell's Treatise was written; and though the principles which govern the action of these currents are clearly Inid down by Maxwell, they axe not developed to the extent which the present importance of the subject: demands. Chapter IV contains an investigation of the theory of such currents when the conductors in which they flow are cylin- drical or spherical, while in Chapter V an account of Horta’s experiments on Electromagnetic Waves is given, This Chapter also contains some investigations on the Electromagnetic Theory of Light, especially on the seattering of light by small metallic particles; on reflection from metals; and on the rotation of the plane of polarization by reflection from a magnet. I regret that it was only when this volume was passing through the pross that I became acquainted with a valuable paper by Drude (Wiede- mann’s Annalen, 46, p. 353, 1892) on this subject. Chapter VI mainly consists of an account of Lord Rayleigh’s investigations on the laws according to which alternating currents distribute themselves among a network of conductors ; while the last Chapter contains a discussion of the equations which hold when a dielectric is moving in magnetic field, and some problems on the distribution of currents in rotating conductors. T have not said anything about recent researches on Magnetic Tnduetion, as a complete account of these in an easily accessible form is contained in Professor Ewing's ‘Treatise on Magnetic Induction in Iron and other Metals.” I have again to thank Mr. Chree, Fellow of King’s College, Cambridge, for many most valuable suggestions, aa well as for @ very careful revision of the proofs, PREFACE. J. J, THOMSON. ” ” 43-47 Rid se: AE 84, Metallic and electrolytic condedilanns «oe ws ee 60 CHAPTER IT. VASHAGK OV XLECTRICITY THROUGH GASKS. 35, Introduction 4, “ . 53 36, Goss the molecules of n gas be electrified! » 53 87. Hot gases... eS rs hd 54 Woo Blectrio propertiea of flames. ._ 57 89. Effect of ultra-violet light on the discharge 57 40. Electrification by ultra-violet light .. .. 50 41. Disintegration of the negative electrode... 60 42. Discharge of electricity from illuminated metals « 60 43. Discharge of electricity by glowing bodies .. 62 44. Volta-potential 20. eee 63 45. Electrification by wun-light wesw 66 46. ‘Electric Strongth’ of a gas. = 68 47. Effect of the nature of the Gictesiea 5 on the pak Tength » 69 48. Effect of curvature of the electrodes on the spark length... 69 49. Baille’s experimonts on the connection betweon epee differ ence and spark length .. 70 60. Liebig’s on the same subject .. oe 72 By Potential difference oxpremed in terms of spark length « .. 74 62-53. Minimum potential difference required to produce » spark 74 54-61. Discharge when the field is not uniform .. .. .. 77-84 62-65, Poace’s experiments on the connection between pressure and spark potential 4 « om BH89 66-68. Critical prowure we = 89-90 69-71. oe difference required to space though » various 90-92 72-76, 1 Methods of producing cleotrodelom diary oo 92-96 77. Appearance of such discharges... 2 » 96 78-80, Critical pressure for such discharges .... 96-97 81. Difficulty of getting the discharge to odere. fom. gua to metal “ 82-86, Fligh cnndackieity af rarefied gasce c 87. Discharge through a mixture of gases... 88-93, inpiea OF a niagaet pu thie: electrodslens dlacberges 94. Appearancs of discharge when electrodes are used ., 95, Crookes’ theory of the dark space 4. = a = ne | —_— xii CONTENTS, cS . Bea ste bain 6 eaoiealy ec: 6p. pioens esse NE infinite plata ,, ce 296. Ce of lat etwen two infiteplel pte » « BIE 237. Correction for thicknoss of plate... ee 238. Caso of one cube inside another = 4. se vse BRD 239-240. Cube over an infinite plate .. «as 225-227 241, Case iB piles: with (qamr)clig, whens tba 2H bs lore as 287 242. Correction when guurd-ring is not at the same potential as the plte ow Sai teia fo ok BA Case of condanser swith jruard-elag when the’ sRVin deep . 232 244, Correction when guard-ring is not at the ap as the plate a » 296 245. Application of elliptic fictions | bw poblene in in 1 oloctroatation 236 246. Cupacity of a pile of plates. « «Ge 247. Capacity of a system of radial plates 2... ws BAD 248, Finite plate at right angles to two infinite ones eae 249. Two sets of parallel plates 4. stew e BME 250. Two sets of radial plates... a oe AE 251. Finite strip pluced parallel to two infinite ste et 252. Two sets of parallel plates... ae sa. aaa 253. ‘Two sets of radial plates... eee BAD 254, Limitation of problems solved. «ee BBO CHAPTER Iy, ELECTHICAL WAVES AND OSCILLATIONS. 255. Seope of the chapter acre we wee ae SBD 256, General equations... oe wk Ge BEE +267, Alternating currents in two siwseoalotb a: = we Dee 268. Case when rato of alternation is very rapid .. ~ 269 + 269-260. Periodic currents along cylindrical contotors. 262 261, Valuo of Beseel’s functions for very ae or very small values of the variable. ” a ee Becaigsaoncfcloctiowsveeslong Wire Gs S08 * 263-264. Slowly alternating currents... +. 270-273 265. Expansion of aJ,(2)/J,(@) 0 eee TE 266, Moderately rapid alternating currents 4.0) 276 267. Very rapidly alternating currents .. Sie aa 278 268, Currents confined toathinskin 4 2380 269. Mognetic force in dielectric 2. 4 282 +270. Transmission of disturbances along wires. 283 i ast Pa 319-320, Effect of radial currents in the sphere .. 4. 982-889 $21. Currenta induced in a sphere by the annihilation of a uniform magnatic field se ee. 322, Magnotie effects of thees currents whan the sphere’ ia not made of irons. 4 ee ewe wets BBO $23. When the aphere is made of iron .. eeee SBF OHAPTER V. XLEOTROMAGNETIC WAVES. 824. Herts’s experiments... 925-327. Hertz's vibrator. 828, The resonator ow sa Ate ae 820, Effect of altering the position of the air gop 930-331, Explanation of those effects... ws 332. Resonance... ny 389-985, Rato of decay of the beatin, ss 936-339. Reflection of waves from a metal plate .. 40-342, Sarasin’s and De la Rive’s genet 343. Parnbolic mirrors... ~ 844-346, Electric screening .. inde Betraction, of electromagustic waves 348. Angle of polarization 0 ws 349-350. Theory of reflection of téctroimaggnidios A waves. by # dielectric .. = 407-411 351. Reflection of these waves ses ro #rabeninaion through a thin motal plate... a 44 852-854. Rofloction cf light from mntalg |... 417-419 855. Table of refractive indices of metals ss swe 420 956, Inndequacy of the theary of metallic reflection... «. 421 357. Magnetic properties of fron for light waves... nn. 422 358, Transmission of light through thin films... 423 359-360, Reflection of electromaguotic waves from a erating 425-428 861-368. Scattering of thcao waves by a wire. - 428-496 869. Scattering of light by metal sphores =... - 370. Lamb's theorem “ o 488 871. Expressions for magnetic iRiries and leche polartestion = 440 372. Polarization in plano wave exprossed in torma of spherical hharmonicg ee on 442 373-376. Aéstiacing of aiplantisvatelby.8 mpaitrol6{ any bine 443-445 977. Beattering by a small sphere 4. sew sn AT $78. Direction in which the scattered light vanishes as) oni san ASD | = A: NOTES ON ELECTRICITY AND MAGNETISM. CHAPTER L Im Chapters IV and VI, substitute ‘effective resistance’ for ‘impedance’ wherever the latter occurs. Thomson : Notes on Electricity. of this kind does more than serve as s vehicle for the clear ex- pression of well-known results, it often renders important services by suggesting the possibility of the existence of new phenomena. The descriptive hypothesis, that of displacement in a dieleo- tric, used by Maxwell to illustrate his mathematical theory, seems to have been found by many readers neither so simple nor so easy of comprehension as the old fluid theory; indeed this seems to have been one of the chief reasons why his views did not sooner meet with the general acceptance they have since received. As many students find the conception of ‘displacement’ difficult, I venture to give an alternative method of regarding the pro- cesses occurring in the electric field, which I have often found useful and which is, from a mathematical point of view, equiva- lent to Maxwell's Theory. B on his view indicated by the existence of eapacity in dielectrics, Although the language which Faraday usod about lines of force leaves the impression that he usually regarded them as chains of polarized particles in the dielectric, yet there seem to be indications that he occasionally regarded them from another aspect ; i.e. assomething having an existence apart from the molecules of the dielectric, though these were polarized by the tubes when they passed through tho dielectric, Thus, for example, in § 1616 of the Leperimental seoms to regard these tubes as stretching across a vacuum. this latter view of the tubes of electrostatic induction which 1 shall adopt, wo shall regard them as having their seat im the ether, the polarization of the particles which accompanies their passage through a dielectric being a secondary phenomenan, We shall for the sake of brevity call such tubes Faraday Tubes, In addition to the tubes which stretch from positive to nega- tive electricity, we suppose that there are, in the ether, multitudes of tubes of similar constitution but which form discrete closed curves instead of having freo ends; we shall call such tubes *elosed” tubes. The difference between the two kinds of tubes is similar to that between a vortex filament with its ends on the free surface of a liquid and one forming a closed vortex ring inside it. These closed tubes which are supposed to be present in the ether whether electric forces exist or not, impart n fibrous structure to the ether. In his theory of electric and magnetic phenomena Faraday Gibdestno\lof tubes of maguotio an wll, ne\of cloctiostatin induction, we shall find however that if we keep to the con- | ception of tubes of electrostatic induction we can explain the phenomena of the magnetic field as due to the motion of such | tubes. 4 & tensity at any place being tees f talon at that plac bet of Un exe at \ 4) In this chapter we shall endeavour to show that the various field may all be ae junt an it is the object of the Kinetic Theory of Gases to ey the properties of # gas as due to the motion of its molecules, ‘These tubes also resemble the molecules of a gas in another re~ spect, as we regard them as incapable of destruction or creation. _ 5,] It may be asked at the outset, why we have taken tho tubes: of electrostatic induction as our molecules, ao to speak, rather thanthe tubes of magnotic induction ? Tho answer to this question is, that the evidence afforded by the phenomena which sccom~ pany the passnge of electricity through liquids and gases shows that molecular structure has an exceedingly close connection with tubes of electrostatic induction, much closer than we have any reason to believe it has with tubes of magnetic induction. ‘The choice of the tubes of electrostatic induction as our molecules woome thus to be the one which affords us the greatest facilities for explaining those clectrical phenomena in which matter as ‘Well as tho other is involved, 6.) Let us consider for a moment on this view the origin of tho energy in tho electrostatic and electromagnetic fields. We suppose that associated with the Faraday tubos there is a dis- ‘tribution of velocity of the ether both in the tubes themselves a — ae apnea ae presipeeplars ee with the quantities which Maxwell denotes by the sam their physical interpretation howover is different. 9.] Wo shall now investigate the rate of change of tl nente of the polarization in a dielectric. Since the Far insuch a medium can neither be ereated nor destroyed, a in the number passing through any fixed area must bo motion or deformation of the tubes. We shall suppo first place, that the tubes at one place are all moving w same velocity. Ta Sage rormser real fn rv of these tubes ot any point, then the change in f, the of tubes passing at the point «, y, =, through unit area at right angles to the axis of #, will be duo to three causes, The firstof these is the motion of the tubes from another part of tho field up to tho area under consideration; the second is the spronding out or concentration of the tubes due to their relative motion; and the third is the alteration in the direction of the tubes duo to the same cause. ‘Let #,f be the change in / due to the first cause, then in consequence of the motion of the tubes, the tubes which at the a! = | ‘ELRCTRIO DISPLACEMENT AND [o. If p is the donsity of the froo olectricity, thon ninco by the definition of Art. 8 the surface integral of the normal taken over any closed surface must be equal to the quantity of oloctricity inside that surface, it follows that df dg dh pa ey, hence equation (1) may be written Gtup = X(ug—w)~ £(uh—nh) 8 yp = (og) 2 (ug of), @) Sette = geluf—ul)— (oho) If p,q, r are the components of the current parallel to x, y, = respectively, a, 8, y the components of the magnetic force in the same directions, then we know Similarly tag _%, (a) Hence, if we regard the current as made up of the convection current whose components are up, Vp, Wp respectively, and the polarization current whose components are x, of py we see by comparing equations (2) and (3) that we may regard the moving Faraday tubes as giving rise to a magnetic force whose components a, 8, y are given by the equation a= 47(vh—wy), B= 4x (uf —uh), (4) y= 4a(ug—w). Thus a Faraday tube when in motion produces a magnetic force at right angles both to itself and to its direction of motion, whose magnitude is proportional to the component of the velocity at right angles to the direction of the tube. The magnetic force = bk 10 BLECTRIC DISPLACEMENT AND tr 11.] The electromotive intensities parallel to x, y, + due to the motion of the tubes are the differential coefficients of the kinetic the following ions for X, Y, Z the components of the electromotive intensity, X = wb—w, Y = wn (6) 2 =va-uh. ‘Thus the direction of the electromotive intensity due to the motion of the tubes is at right angles both to the magnetic induction and to the direction of motion of the tubes, From equations (6) we get aZ aY_ da, da_. pdb, de Gy da" yt Pde" (G, * a) dv dw, _ du du +0(5 tae) ody ~ ode But since the equation da db , de uta ten” holds, as we shall subsequently show, on the view we have taken of the magnetic force as well as on the ordinary view, we have dZ aY_ da, da, da du | dw. du du dy de = dat Vay Pde + O(a + de) ody The right-hand sido of this investigation is by the reasoning grein iol Ast. 0 equal’ to a, the rato of diminution in 4hat number of lines of magnetic induction passing through unit area at right angles to the axis of x: hence we have dz_a¥__ da dy ds ae dX dz db Similarly ped miler he (7) ay _dX__d& de ~ dy at" Now by Stokes’ theorem JS (Xda + Ydy + Zds) angles tothe tube and | i ie (2) a momentum at right ax Fig. L the induetion, (3) an electromotive intensity at iit Gs doclion gh bempar pir se re this always tends to make the tube set itself at right to the direction in which it is moving, Thus in an ii feodieu du whldh tere is 20 five dotGlcty wad coun ‘no electromotive intensities except those which arise from tl motion of the tubes, the tubes set themselves at right ‘to the direction of motion. 18.] We have hitherto only considered the case when the at any one place in a dielectric are moving with a velocity. We can however without difficulty extend these r= sults to the case when we have differant sets of tubes moving with different velocities, | ‘Let us suppose that we have the tubes f,, 9;, 4), moving with a velocity whose components are w, , v, w,, while the tubes f,, 7s, hy ‘move with the volocitios vy, v,, W,, and 60 on, Then the rate of increase in the number of tubes which pass through unit area at right angles to tho axis of x is, by the samo reasoning as before, S2(0-v)—E2(u/—nl)—2(up) x= i yi 13] FARADAY TUBES OF FORCE. 13 Hence we see as before that the tubes may be regarded as producing a magnetic force whose components a, B, y are given by the equations a= 4x2 (vh—ug), \ (8) A= 423(uf—uh), y= 403(ug—v). The Kinetic energy per unit volume, 7, due to the motion of these tubes is given by the equation a =a lttety), or P = 24p[{¥(oh—uy)}*4 (2(uf—uh)}*+ (2(ug—of)}*) Thus d7/du,, the momentum per unit volume parallel to due to the tube with suffix 1, is equal to Amp (or 2(ug—uf)—h,3(uf—uh)}, = gc—h,b, where a, 6, c are the components of the magnetic induction. Thus U, V, W, the components of the momentum per unit volume parallel to the axes of 2, y, z respectively, are given by the equations U =r9—b3h, V=ath-c3f, } (9) W=bsf—adg. Thus when we have a number of tubes moving about in the electric field the resultant momentum at any point is per- -pendicular both to the resultant magnetic induction and to the resultant polarization, and is equal to the product of these two quantities into the sine of the angle between them. The electromotive intensities X, Y, Z parallel to the axes of @, y, 2 respectively are equal to the mean values of dT7/df, dT/dg, dT/dh, hence we have X =ba-ct, Y= ciao} (10) Z=at—ba; where a bar placed over any quantity indicates that the mean ; value of that quantity is to be taken. 56 hans bcLiw, just ide ‘urubh Mie Home ase current, iWin shall sur toca haa Laas ee the sphere is e and that it is moving with velocity w pan to the axis of s. Faraday tubes start from the sphere u carried along with it as it moves through the dielectric ; sines ‘ses Gabeb ara’ nioving thiay''will, elie! have! sceatipa ta magnetic field. We shall suppose that the system has settled down into a steady state, so that the sphere and its tubes ary all moving with the same velocity w. Let f, g, hk be the com- ponents of tho polarization at any point, a, 8, 7 those of the magnetic force, The expressions for X, Y, Z, the componente of the electromotive intensity, will consist of two parts, ono due to the motion of the Faraday tubes and given by equations (6), the other due to tho distribution of these tubes and derivable from a potential ¥; we thus have, if the magnetic permeability is unity, a wa—a* Y=—wa-% (12) dy Z=-7 By equations (4) If K is the specific inductive capacity of the medium, we have 4x in 4m X= Eh Y=5o faqh ee oe 16.] FARADAY TUBES OF FOROE. 17 Since the magnetic permeability of the dielectric is taken as unity, we may put 1/K = V*, where V is the velocity of light through the dielectric. Making these substitutions for the magnetic force and the electromotive intensity, equations (12) become 4nhV* and since df ‘ dg ah da” dy * dz we get ey ay Paw dy aa + Get ye ga = (13) or putting F Vv * aw” equation (18) becomes ay dy dy dat + ap t age= > a solution of which is A Vo fate y?t+22}t (14) To find A we notice that the normal polarization over any sphere concentric with the moving one must equal e, the charge on the sphere; hence if a is the radius of the moving sphere, St pp Vg 4% = S \er+¥g+inlasme o 18 ELECTRIO DISPLACEMENT AND 6. Substituting for eh 9, h their values, wo find rr itary feveeran ae * sin ode a oF area [ 1 * feinto-+ parma conta The integral, if V>w, is equal to 2{V—w*}t, aan a hence A=eV{Vi—-u*}t, so that (18) Thus The Faraday tubes are radial and the resultant polarization varies inversely as uw + 2 2 r’ fi t+ Tap a ’ where 7 is the distance of the point from the centre, and @ the angle which r makes with the direction of motion of the sphere. ‘We see from this result that the polarization is greatest where 6 = 7/2, least where 6=0; the Faraday tubes thus leave the poles of the sphere and tend to congregate at the equator. This arises from the tendency of these tubes to set themselves at right angles to the direction in which they are moving. The surface density of the electricity on the moving sphere varies in- versely a8 w rn 2 ; fa + Pra * a ’ } 16.) FARADAY TUBES OF FORCE, 19 it is thus a maximum at the equator and 4 minimum at the poles. The components a, 8, y of the magnetic force are-given by the equations a= —4rug =-—¥ — __¥__., | APS} {o+y+paet Vw 5 (18) p= srof = > {V—w'} feryvtpeoaaet y=0. These expressions as well as (15) were obtained by Mr. Heaviside by another method in the Phil. Mag. for April, 1889. Thus the lines of magnetic force are circles with their centres in and their planes at right angles to the axis of z. When w is so small that w'/V* may be neglected, the preceding equations take the simpler forms ea e Oy ae OY fit * [-Gr J "aw aap Sod, ga See, aaa a B= (See J. J. Thomson ‘On tho Electric and Magnetic Effects produced by the Motion of Electrified Bodies’, Phil. Mag. April, 1881.) The moving sphere thus produces the same magnetic field as an element of current at the centre of the sphere parallel to z whose moment is equal to ew. When as a limiting case V = w, that is when the sphere is moving with the velocity of light, we seo from equations (15) and (16) that the polarization and magnetic force vanish except when z= 0 when they are infinite. The equatorial plane is thus the seat of infinite magnetic force and polarization, while the rest of the field is absolutely devoid of either. It ought to be noticed that in this case all the Faraday tubes have arranged themselves so as to be at right angles to the direction in which they are moving. We shall now consider the momentum in the dielectric due i ;* the motion of the Faraday tubes. Since the dielectric is o2 20 ELECTRIC DISPLACEMENT AND 16. non-magnetic the eomponents U, V’, W of this are by equations (9) given by the following expressions : Tash =-L i, The resultant momentum at any point is thus at right angles to the radius and to the magnetic force ; it is therefore in the plane through the radius and the direction of motion and at right angles to the former. The magnitude of the resultant momentum per unit volume at a point at a distance r from the centre of the sphere, and where the radius makes an angle @ with the direction of motion, is ew FR 1 sind 4a Via rt { a 3° 1+ cos?) Vow } Thus the momentum vanishes along the line of motion of the sphere, where the Faraday tubes are moving parallel to themselves, and continually increases towards the equator as the tubes get to point more and more at right angles to their direction of motion. The resultant momentum in the whole of the dielectric is evidently parallel to the direction of motion; its magnitude J is given bs the ay ar ae t= 2 ve CLE s fi costo ty _éu on ee “a Wau |, ua 3? fa +7 costo} or putting Y cog 9 = tan, (Pawyt 16.] FARADAY TUBES OF FORCE. 21 we see that "Toei “afr ak. oosty (1 ~ Fy aint) dys or if tan ai eo ve . vy = Fa [peat Ga) + Hsin 29(1-4 4 029) Thus the momentum of the sphere and dielectric parallel to 2 is mw +I, where m is the mass of the sphere ; so that the effect of the charge will be to increase the apparent mass of the sphere by I/w or by ev OR os ve BE pra OCB ga) +B sins (1 +3 000 28) When the velocity of the sphere is very small compared to that of light, é. wt s=5 (at ip) approximately, and the apparent increase in the mass of the sphere is 2e ‘When in the limit w = V the increase in mass is infinite, thus a charged sphere moving with the velocity of light behaves as if its mass were infinite, its velocity therefore will remain con- stant, in other words it is impossible to increase the velocity of a charged body moving through the dielectric beyond that of light. The kinetic energy per unit volume of the dielectric is alte), and hence by equations (16) and (17) it is equal to w is thus the total kinetic energy in the dielectric is equal to | iu, W; 22 ELEOTRIO DISPLACEMENT AND [16. that is to aegis eo 4a” {Vath ‘We shall now proceed to investigate the mechanical forees acting on the sphere when it is moving parallel to the axis of s in @ uniform magnetic field in which the magnetic force is everywhere parallel to the axis of z and equal to H. If U, V’, W are the components of the momentum, U = ge —hb, V’=ha-fe, W= fb —ga. e=0, b=f, a=atH, fsa-aZaa sin29(1+427c0038)t- In this case where a and @ have the values given in equations (16). The momentum transmitted in unit time across the surface of a sphere concentric with the moving one has for components J wl cos 6 dS, [fuvreosoas, [fom coseas, the integration being extended over the surface of the sphere. Substituting the values of U, V’,W, we see that the first and third of these expressions vanish, while the second reduces to VE 268 ares Tai costo Vina eVHw OY pend) LL ow'trintdt or Leos ‘ al ° fre pa onto! pep oosta? which is equal to _ _ellw¥ {rut _ imu (Vw {V2—w}t Vue (F log (7) 1. wT) fy E log (9) )- or to When /V is very small this expression reduces to teHw, which must be satisfied since the value of Bel [loves Adedyas must, in a medium whose magnetic pormeability is : ee es gia Boe 17.) FARADAY TUBES OF FORCE. 26 _7. For let ay, Bo, yo be any particular values of the components of the magnetic force which satisfy the assigned conditions, then the most general values of these components are expressed by the equations where ¢ is an arbitrary function of x, y, z. Then if SSS (+ B +7?) da dy dz is stationary, SSS (ada+ 38+ yb7) da dy de = 0. (19) Let the variations in a, 8, y be due to the increment of ¢ by an arbitrary funetion 34, then da= ae B= ue » bys ote : Substituting these values for ed 3p, dy, and integrating by parts, equation (19) becomes [fs (adydz+Bdedz+ydzdy) —ff{frofie + ++ + Macdyde =o, and therefore since 3¢ is arbitrary da dp dy_ dat ay tds The values of a, 8, y given by equation (18) cannot therefore be the complete expressions for the magnetic force, and since we regard all magnetic force as due to the motion of Faraday tubes, it follows that the tubes which connect the positive to the nega- tive charges on the plates of the condenser cannot be the only tubes in the field which are in motion; the motion of these tabes must set in motion the closed tubes which, Art. 2, exist in their neighbourhood. The motion of the closed tubes will produce a magnetic field in which the forces can be derived from athe oftheFuninytei When tne mgeet 30 ELECTBIO DISPLACEMENT AND [18 The potential outaide the cylinder, if a is the radius of the cylinder, is equal to a? e088 Tr where r is the distance from the axis of the cylinder of the point at which the potential is reckoned, and @ the azimuth of r measured from the direction of magnetization. Thus inside the cylinder the equipotential surfaces are planes at right angles to the , Fig. 2. sliveotion of magnetization, while outside they are a system of cir- cular cylindora which if prolonged would pass through the axis of the magnot; the axes of all these cylinders are parallel to the axin of 3 lio in the plane of az. The cross sections of the original oylinder and the equipotential surfaces are represented in Wig. 2. Wo «hall supposo that tho Faraday tubes are parallel to the axix of tho oylinder; thon wo may regard the magnotic field as produced by such tubes travelling round the equipotential | | 82 ELECTRIC DISPLACEMENT AND [19 of the tube is proportional to the normal magnetic induetion, #0 that the continuity of the tangential momentum is equivalent to that of the normal component of the magnetic induction. We have thus deduced from this view of the magnetic field the ordinary boundary conditions (1) that the tangential component of the magnetic force is continuous, and (2) that the normal | component of the magnetic induction is continuous. The paths along which the tubes move coincide with the lines of flow produced by moving the cylinder uniformly at right angles to the direction of magnetization through an incom pressible fluid. Induction of Currents due to Changes im the Magnetic Field. 19.] Let Fig. 3 represent a section of the magnetized eylinder and one of its equipotential surfaces, the directions of the mag- netic force round the cylinder being denoted by the dotted lines. We shall call those Faraday tubes which point upwards from the plane of the paper positive, the negative Faraday tubes of course pointing downwards. The positive and the negative tubes circulate round the equi potential surface in the direc- tions marked in the figure. Let A and B represent the cros- sections of the wires of a cir cuit, the wires being at right angles to the plane of the paper. When the magnetic field is steady no current will be produced in this cirenit, be- cause there are as many positive as negative tubes at any point Fig. 8. in the field. Let us now sup- pose that the magnetic field is suddenly destroyed; we may imagine that this is done by placing barriers across the equipotential surfaces in the mag- netized cylinder so as to stop the circulation of the Faraday tubes. The inertia of these tubes will for a short time carry} ll L ce aa . ear which are | right angles to the equipotential surface turn in consequ towards tho cylinder as indicated in Fig. 4, in which the do lines represent lines of magnetic force. 36 ELMOTRIC DISPLACEMENT AND {e3. magnet to maintain the motion of the Faraday tubes in the field, and to point out that the motion of molecular tubes is able to furnish such a mechanism. Steady Owrrent flowing along a Straight Wire, 23.) We shall now proceed to express in terms of the Faraday tubes the phenomena produced by a steady current flowing along an infinitely long straight vertical wire. We shall suppose that the circumstances aro such that there ig no froo electricity on the surface of the wire, so that the Faraday tubes in its neigh- bourhood are parallel to its length. If we take the direction of the current as the positive direction, the positive tubes parallel to the wire will be moving in radially to keep up the eurrent, and this inward radial flow of positive tubes will be accompanied by an outward radial flow of negative tubes, a positive tube when entering the wire displacing negative tubo which moves outward from the wire. This shearing of the positive and negative tubes past each other will give riso to a magnetic force which will be at right angles both to the direction of the tubes and the direction in which they are moving ; thus the magnetic foree is tangential to a circle whose plane is horizontal and whose contre is on the axis of the wire, When the positive tubes enter the wire they shrink to molecular dimensions in the manner to be described in Art. $1. At a distance 7 from the axis of the wire let V be the number of positive tubes passing through unit area of a plane at right angles to the wire, v the velocity of these tubes inwards, let V’ be the number of negative tubes per unit aroa at tho samo point, » their velocity outwards. The algebraical sum of the number of tubes which cross the eirele whose radius is r and whose centre is on the axis of the wire is thus OV 4vN") 270, Whon tho fiold is steady the valuo of this oxprossion must be the same at all distances from the wire, because as many tubes must flow into any region ax flow ont of it, Hence when the field is steady this expression must equal the algebrajcal sum of the number of positive tubes which enter the wire in unit time; this number is however equal fey the Suereok hare ees Mess Bare (N40) 2nrai. D : L ‘momentum will come to A from tho left ‘thus A will be pushod from loft to right or towards g. There will thus be an attraction between tho parallel currents. 24.] It will be noticed that the tubes in the preceding ease move radially in towards the wire, 0 that the energy converted into heat in the cireuit comes from the dielectric way tothe wie ed nt tent gal lng it. This was first pointed out by Poynting in his paper on { Transfer of Energy in the Electromagnetic Field (Phi. Trane. 1984, Part. II. p. 348), ‘ When however the current instead of being constant is alternating very rapidly, tho motion of the tubes in the diclee- tric is mainly longitudinal and not transversal. Wo shall show in Chapter IV that if p is the frequency of the current, o the resistance of the wire, a its radius, and its magnetic permeability, then when 47u:pa/o is a large quantity the eleotro~ motive intensity outside the wire is normal to the wire and therefore radial. ‘Thus in this ease the Faraday tubes will be - | radial, and they will move at right angles to themsel’ to the wire. There is thus a great contrast between this and tho previous one in which tho tubes are per move radially, while in this the tubes are radial and moye longitudinally, “a Diwharge of « Leyden Jar. 25.) We shall now proceed to consider the distribution and motion of the Faraday tubes during the discharge of a We shall take the aym- -~ metrical case in which the outside coatings: of two Leyden jars a and 8 (Fig. 6) are connected by a wire, whilo the in- side coating of a is can A . neeted to ono terminal of an electrical machine, Hy. 6. tho inside coating of B | to the other. When electrical machine is in action tho difference of potential tween the inside coatings of the jars increases until a » 4 — 26.) FARADAY TUBES OF FORCE. 41 mecting the outer coatings of the jars, the Faraday tubes will strike against the circuit on their way to and from the wire. The passage of these tubes across the circuit will, since there is an excess of tubes of one name, produce a current in this cir- cuit, which is the ordinary current in the secondary due to the variation of the intensity of the cur- rent in the primary cir- cuit. Some of the tubes as they rush from the jar to the wire connecting the outside coatings of the jar strike against the second- Fig. 12. ary circuit, break up into two parts, as shown in Fig. 13, the ends of these parts run along this circuit until they meet again, when the tube re- unites and goes off asa single tube. The passage of the tube Ban mary ‘clreult ¥ig. 18. across the secondary circuit is thus equivalent to @ current in the direction of rotation of the hands of a watch; this is opposite to that of the current in the wire connecting the outside coatings of the jars. The circuit by breaking up the ‘y ST pi gaat dee of the tubes and | they are moving, the magnetic force is thus parallel to y- The magnitude of the magnetic force is by eq force. If there is no reflection the electromotive i the magnotic force travel with uniform velocity from the plane of disturbance and always bear a i to each other. By supposing tho number of tubes issuing the plane source per unit time to vary harmonically at tho conception of a divergent wave os « scries of tubes travelling outwards with the velocity of light. ease the places of maximum, zero and minimam on ati illic congo eared ita blacin obec Sac and minimum magnetic force. y= number of tubes, 80 that no more tubes can fall on See ties, wales tis onde aie lees n cular distances. Thus, on this view, shaczlitenssat ira whether on a metal, an electrolyte, or gas, ulways requires existence of free atoms. The production of electrification must be accompanied by chemical dissociation, the disappearance of electrification by chemical combination; in short, on thie view, changes in electrification are always accompanied by changes. ‘This was long thought to be a peculiarity attaching dence to show that it is also true when eleetricity passes gases, ‘Reasous for this ecnalasion ill be gives Ga will be sufficient here to mention one or two of the most ati instances, the details of which will be found in that chap Perrot found that when the electric discharge steam, oxygen camo off in excess at the positive and at the negative electrode, and that the excesses of oxygen ab tl positive and of hydrogen at the negative electrode were the same a8 the quantities of these gases sot free in a water volta- moter placed in series with the discharge through the steam, Grove found that when the discharge passed between a point and a silver plate through a mixture of hydrogen and oxygen, tho plate was oxidised when it was the positive electrode, nob when it was the negative. If the plate was oxidised to begin with, it was reduced by the hydrogen whon it was the negatis electrode, not when it was the positive. These and the oth results mentioned in Chap. IL scem to point unmistakably to the conclusion that the passage of electricity through gases is necessarily attended by chemical decomposition. = & ig. 17. cules, say as, 0b, €F, form a chain, and that the tubes i ‘molecules inetoed of being eusesivaly ; then sines we have ‘The approximate values of Z/{K} for a fow given in the following table :-— - TAR} Silver ee ef ee Lead . ‘ . . or neo Mercury. 87x 10-* | Water with @:3 per cont of H,S0, BL x 1078 Glass at 200°C. . . 2 x10% Since the values of {A} have not beet determined for sib: stances conducting anything like so well as those in the ceding list we cannot determine the value of 7. Cohn and have found however that the specific inductive capacity distilled water is about 76, Cobn and Arons (Wied. Ann. 33, p. 13, 1888), and Cohn (Berl. Ber. p. 1037, 1891) found that the specific inductive capacity of a weak solution differs very — little from that of the solvent, though the difference in the — specific resistance is very great, If we supposo that the K | for water mixed with sulphuric acid is the same as the K for water, we should find 7’ for this electrolyte about 2 x 107% which is about ten thousand times as long as the time of yibri- tion of sodium light; hence this electrolyte when exposed to electrical vibration of this period will behave as if 7’ were in- finite or na if it were an insulator, and so will bo to electrical vibrations as rapid as those of light. We see too that if {A} for the metals wore as gront as {K} for distilled water, the values of 7’ for these substances would not greatly exceed the time of vibrations of the rays in the visible spectrum: this result explains Maxwell's observation, that the opacity of thin metallic films is much Jess than the value calculated on electromagnetic theory, on the assumption that the condueti of the metals for the very rapidly alternating currents wh constitute light is as great as that for steady currents. Galvanic Cell. 83.) The production of a current by a cell is the reverse } to the decomposition of an electrolyte by # current; in the 33] FARADAY TUBES OF FORCE. 49 ease the chemical processes make a long Faraday tube shrink to molecular dimensions, in the former they produce long tube from short molecular tubes. Let a and 6 (Fig. 18) represent two metal plates immersed in an acid which combines chemically with a. Let a be a positive atom in the plate A connected by a A B bmg] cmd SE a) cs Fig, 18, Faraday tube with a negative atom }, then if a enters into chemical combination with a molecule cd of the acid, after the combination a and c will be connected by a Faraday tube, as will also b and d: it will be seen from the second line in the figure that the length of the tube bd has been increased by the chemical action. If now d enters into combination with another molecule ef, the result of this will be still further to increase the length of the tube, and this length will increase as the chemical combination 2 With regard to the first of these differences, we may that though the conductivities of the best conducting m enormously greater than those of electrolytes, thers do to be any abrupt change in the values of the cond ‘we pass from cases where the conduction is c as in fused lead or sodium chlorides, to cases where recognised as being of this nature, as in tellurium or ‘The following table, which contains the relative of a-fow typical substances, is sufficient to show this:-— Silver 63. Mercury Ey Gas Carbon 1x10-%, ‘Tellurium 4x 10-4, Fused Lead Chloride 21074, Fused Sodium Chloride 8-6 x 10-*. ‘With regard to tho second difference between metallic a electrolytic conduction, viz, the effect of temperature on : 52 ELECTRIC DISPLACEMENT AND FARADAY TUBES OF FORCE. Maxwell's experiments on the transparency of their metallic films show, metals show an analogous effect, for their resistance for the light vibrations is enormously greater than their resist- ance to steady currents. The theory of Faraday tubes which we have been considering is, as far as we have taken it, geometrical rather than dy- namical; we have not attempted any theory of the constitution of these tubes, though the analogies which exist between their properties and those of tubes of vortex motion irresistibly sug- gest that we should look to a rotatory motion in the ether for their explanation. Taking however these tubes for granted, they afford, I think, @ convenient means of getting a vivid picture of the processes - ocourring in the electromagnetic field, and are especially suitable for expressing the relations which exist between chemical change and electrical action. tures cannot receive a charge of electricity. ‘This view receives strong support from the results: experiments (Wied. Ann. 19, p, 518, 1883), which confirmed by Sohneke (Wied. Ann. 34, p.925, 188! that not only is there no electricity produced by the of an uneloctrified liquid, bat that the vapour arising olectrified liquid is not electrified. If the molecules of were capable of receiving a charge of electricity und cumstanees wo should expos: them to do so in this ease, ant Ifa saute tot the ama coils cua hundred years ago. ‘Mr. Kinnersley of Philadelphia, in ; dated March 1761, informs bis friend and correspor self and his friend, steam was so far from rising it left its share of common electricity behind.” There does not soem to be any ovidence that an electri body can lose any of its charge by radiation through without convection of electricity by changed particles. Hot Gases. 87.) It is only at moderate temperatares that a co charged to a low potential retains ita eharge when by a gas, for Beequerel (Annales de Chimie et de Physique ad | = 56 ‘THE PASSAGE OF “ara meta anir peat iniae Sab eno ‘to conduct (2) Air, Nitrogen, Carbobie Asta, Steam, Ammonia, Sulphurio Acid gas, Nitric Acid gas, Sulphur (in an atmosphore of nitro- gen), Sulphuretted Hydrogen (in an atmosphere of nitrogen). (2) Iodine, Bromino, Chlorine, Hydriodic Acid gas, Hydro- bromic Acid gas, Hydrochloric Acid gas, Potassium Todide, Sal-Ammoniac, Sodium Chloride, Potassium Chloride. I The conductivities of the two classes of gases differ so greatly, \, both in amount and in the laws they obey, that the mechanism | by which the discharge is effected is probably different in \, two cases, These experiments seem to show that when electricity passed through a gas otherwise than by convection, free atoms, or something chemically equivalent to them, must be present. It should be noticed that on this viow tho moloculos even of a hot gas do not get charged, it is the atome and not the molecules which aro instrumental in carrying the dischargo. Talso examined the conductivities of several metallic vapours, including those of Sodium, Potassium, Thallium, Cadmium, Bismuth, Lead, Aluminium, Magnesium, Tin, Zine, Silver, and Mercury. Of these the vapours of Tin, Mercury, and Thalliom hardly seemed to conduct at all, the vapours of the other metals conducted well, their conductivities being comparable with those of the dissociable gases. The small amount of conductivity which hot gases, which are not decomposed by heat, possess, come to be dus to a con- veetive discharge carried perhaps by dust produced by the de- composition of the electrodes: in some cases perhaps the elec- tricity may be carried by atoms produced by the action of the electrodes on the adjacent gas. | ‘The temperature of the electrodes seems to exert great influence upon the passage of the electricity through the gas into which | the electrodes dip. In the experimonts described above I found it impossible to get electricity to pass through the gas, however hot it might bo, unless the olectrodes were hot enough to glow, A current passing through a hot gas was immediately: pate placing a large piece of cold platinum foil between the « —though a strong up-current of the hot gas was 0 prevent the gas gotting chilled by the cold foil. As soon 39.) ELECTRICITY THROUGH GASES. 5T foil began to glow, the passage of the electricity through the gas was re-established. This is one among the many instances we shall meet with in this chapter of the difficulty which electricity has in passing from a gas to a cold metal. Electric Properties of Flames. 38.] The case in which the passage of electricity through hot gases has been most studied is that of flames; here the con- ditions are far from simple, and the results that have been obtained are too numerous and intricate for us to do more than mention their main features. A full account of the experiments which have been made on this subject will be found in Wiede- mann's Lehre von der Elektricitat, vol. 4, B*. A flame such as the oxy-hydrogen flame conducts electricity, the hotter parts conducting better than the colder: the con- ductivity of the flame is improved by putting volatile salts into it, and the inerease in the conductivity is greater when the salts are placed near the negative electrode than when they are placed near the positive t. The conduction through the flame exhibits polar properties, for if the electrodes are of different sizes the flame conducts better when the larger electrode is negative than when it is positive. If wires made of different metals are connected together and dipped into the flame, there will be an electromotive force round the circuit formed by the flame and the wire; the flame apparently behaving in much the same way as the acid in a one-fluid battery; the electromotive force in some cases amounts to between three and four volts. ‘A current can also be obtained through a bent piece of wire if the ends of the wire are placed in different parts of the flame. Escape of Electricity from a Conductor at Low Potential surrounded by Cold Gas. 89.] Though it seems to be a well-established fact that a conductor at a low potential, surrounded by cold air, may retain its charge for an indefinitely long time, recent researches have * Seo also Giese, Wied. Ann. 88, p. 408, 1889. + For an investigation on the effect of putting volatile salts in flames published subsequently to Wiedemann’s work, soe Arrhenius (Wied. Ann, 42, p. 18, 1891). confined to ultra-violet light, as the | x fall on # cathode also facilitate the discharge. — Great light was thrown on the nature of this effeot inyostigation made by Lenard and Wolf (Wiet, Ann, 37, p. induces a negative one on the illuminated negative electricity escapes, travels up to and noutralises positive electricity which induced it. When the pressure gas surrounding the body is less than 1mm, the eseape of the Geitel, Wied. Ann. 41, p, 166, 1890), Discharge of Electricity caused. by Glowing Bodies. aq 43,] Somewhat similar differences between the discharge of positive nnd negative electricity are observed when the charged pressure, the cold plate discharges positive electricity more easily than negative. Guthrie, who (Phil. Mag. [4] 46, p. 257, 1878) was the first to call attention to phenomena of this kind, Fees ikavianthrets iohars ait oon ota iat Gea tain a charge either of positive or of negative electricity, and that as it cools it acquires the power of retaining a charge before it can retain a positive one. If the sphers _ iin: | differ from that for the dielectric; for the el ) site sign to that which would be left on the conductor place to which it can go, ‘The case is however different when the conductor is expo the action of ultra-violet light, for then, as Lenard experiments prove, one or both of the following effects take place: (1) disintegration of the conductor, (2) | changes in the gas in the neighbourhood of the Cisiy, Tf either of these effects takes place 1 conductor to be electrified, for the electricity of oppo to that left on the conductor may go to the disintegrated n or the gas. The experiments hitherto made leave undecid question which of these bodies serves as the refuge of th tricity discarded from the metal. 3 ‘The researches of Hallwachs and Righi on electrification ultra-violet light can be explained on either hypothesis, aggume that «,, the value of + for the metallic vapour o1 dissociated gas, is greater than «,, the value of « for the solid metal. For when negative electricity —Q eseupes from the metal and positive electricity equal to + Q remains bebind, tho di tion in the part of the potential energy due to the Volta is 1, Q—0,Q or (x;—c,)Q. Thus, since «, is by hypothesis greater _ than o,, the departure of the negative electricity from the metal will bo accompanied by a diminution in tho potential energy, Gaalepill therfore: goon until the. incase ix Shale potential energy due to the now distribution of eloctricity ix sufficient to balance the diminution in the part of the due to the Volta potential. Tho positive electrification of the plate produced by ultra-violet light can thus be aceounted for. ‘Again, if the motal were initially positively olvctrified it would not be so likely to lowe its charge as if it were initinlly charged with negative electricity, for the passage of positive Coie icity from the metal to its vapour or to the dissociated gaa -_! & = a | 68 THE PASSAGE OF t with it. The case is however different whon the metal off as vapour, or when it dissociates the gus in ita ° hood: here the wire and the vapour or gas are in states of aggregation, for which the values of © ars p different, so that there may now be a separation of wire getting the positive and the vapour or gas the negative. Tiv air there io such ans sbbandant depoaitian’ GF lasing glass tube surrounding an incandescent platinum wire that the latter in all probability gives off dust as well as oithor disaoci-_ ating the surrounding gas or giving off platinam vapour ; while | Nahrwold (Wied. Ann. 35, 107, 1888) has preiaanec tion of platinum is so small in hydrogen that very given off us dust in this gas. Let us now consider what will happen in air. When the platinum becomes incandescent there is a separation of the positive remaining on the wire, the negative going to metallic vapour or dissociated gas. Since the wire has positive charge, any lumps that break away from it will be positively clectrified. If the positive electricity given biting lumps to the plate, which in Elster and Geitel's held above the glowing wiro, is greater than the negative | given to it by such vapour or gas as depict it, the charge on the plate will be positive, as in Elstor and Geitel’s experiments. In hydrogen however, where the absent, there is nothing to neutralize the negative electricity on the metallic vapour or dissociated gas, so that the charge on the plate will, as Elster and Geitel found, be negative. > Spark Disowance. Electric Strength of a Gas, 46.) Tn Art. 51 of the first volume of the Electricity and Mage netism Maxwell defines the electric strength of » gas as the greatest electromotive intensity it can sustain without disehange taking place. This definition suggests that the electric strungth | iy a detinite spocific property of a gas, otherwise the introduction of this term would not be of much value, If discharge through @ gas at a definite pressure and temperature always began when the electromotive intensity reached a certain value, then this value, which is what Maxwell calls tho eloctrie strength of the AL : | 70 THE PASSAGE OF [49 Wo shall find too when we consider the relation between spark Jength and potential difference that the distance between the cloctrodes may have an enormous effect on the electromotive intensity required to produco discharge. ‘The ‘electric strength’ as defined by Maxwell seems to upon so many extraneous circumstances that there does nob appear to be any reason for regarding it as an intrinsic property of the gas, Connection between Spark Length and Potential Difference, uhen the Field is approvimately uniform. 49.] This subject has been investigated by » large number of physicists. We have however only space to consider the most recent investigations on this subject. Baille (Annales de OChintie et de Physique, [6] 26, p. 486, 1882) has made an elaborate in- vestigation of the potential difference required to produce in air at atmospheric pressure sparks of varying lengths, between planes, cylinders, and spheres of various diameters. The mothod he used was to charge the conductors between which the passed by a Holtz machine, the potential between the electrodes being measured by an attracted dise electrometer provided with a guard ring; this method is practically the same as that em- ployed by Lord Kelvin (Reprint of Papers on Electrostatics anil Magnetism, p. 247), who in 1860 made the first measurements in absolute units of the electromotive intensity required to pro- duce a spark, For very short sparks between two planes Baile (1.¢., p. 515) found the rosults given in the following table :— Potential Difference and Spark Length ; (temperature 15° to 20° C, presewre 760 mm.) 49] ELECTRICITY THROUGH GASES. Kee In another series of experiments where the sparks were slightly longer, Baile, p. 515, found the following results :— Potential Difference and Spark Length. Fotmtiat | Surface spare | Potent | Surface Bimreoce | Deniey, |] cong, | diserence, | Danity. H 41 817 25-2 08 12-88 123 2 451 17-9 09 13-44 1g os 6-22 16-5 10 14-67 117 eas 732 46 a 15-75 14 05 871 188 a 16-84 1 06 9-84 132 18 17-94 110 a7 11.20 127 “a 19.00 10-8 15 20-16 10-7 For spark lengths between -025 cm. and -5 cm. the following results were obtained, p. 516, in a different series of experi- ments :— Potential Difference and Spark Length. ‘Spark Pomtiat | Surface Spark Potential | Surface Length, ‘Difference, Density. Length. ‘Difference. Density. 025 5.04 18.86 275 | (32.69 946 050 8.68 18.76 +800 35.35 9.87 075 11.87 12-57 H +825 37.88 9.25, 100 14-79 176 || 850 39.95 9.08 125 17-45 11.06 875 | 42.17 8.04 +150 20-29 10-76 400 44-74 8.90 175 92.94 1048 425 «| 47-80 8.86 200 25-51 10.15 1450 49.70 8.79 225 98.17 9.96 475 5218 8.75 250 8047 9.70 +500 5448 867 For longer aparks Baile, 1.c., p. 517, got the numbers given in the two following Tables, which represent the results of different sets of experiments :— Potential Difference and Spark Length. TABLE (1). Spark Potential Surface Spark Potential | Surface Length. ‘Difference, ‘Density. ‘Length. ‘Difference. ‘Density. 40 44:80 8.90 | 60 68.82 847 45 49-63 8-78 65 68-75 842 60 54.86 8.65 70 74.09 842 55 59-09 8.55 75 79.02 8.89 “70 18 “80 35 50.] We may compare with these results those obtained by Liebig (Phil. Mag.-[5], 24, p. 106, 1887), who used a similar — method, but whose electrodes were segments of spheres 9-76 em, — in radius, Licbig’s results are as follows :— _ Potential Difference and Spark Length. | 4 Bark Length Potential Bheatromotivg Spark Potential Klostreanetdre in emtimetres | Difference, Tntensity. Length, ‘Difference. Intensity. 2.630 398-5 62808 127-7 0105 357 310-7 +2800 35-196 126-7 0) 407 BMS 0194 4573 235.7 3920 47.001 199 5.057 ATL 165 0348 (5588 63-708 140 ‘8.803 195-5 6226, 69.980 4 1700 TH05 ‘S219 11 0841 19-548 161-1 8830 13-816 163-0 2576 (102.463 107-0 -— =1000 1.0872 AOT75 to20 204468 B78 erry 117-489 WaT +1560 24775 188-2 ‘The potential difference and electromotive intensity are mea~ sured in electrostatic units, Liebig's results for hydrogen, coal gas and carbonic acid as well as air are exhibited graphically in Fig. 19, where the nearly straight curve represents the relation between potential differ- ence and spark length, and the other the relation between electro- motive intensity and spark length, The absclssae are the spark lengths, the ordinates, the potential difference or electromotive intensity. It will be seen that Lichig’s values for the potential dif — ference required to produce a spark of given length are about & per cont, higher than Buille’s. It.also appears from any of the pre- coding tables that the electromotive intensity required to spark across 1 layer of air varies very greatly with the thickness = — | ce’ THE PASSAGE OF 51,] With regard to the relation between the potential ence V and spark length l, Baille deduced from his the relation P# = 10500 (/-+ 0-08)0. ‘The agreement between the numbers calculated by this formula and those found by experiment is not very close, and Chrystal (Proc. Ray. Soc. Edin. vol. 11. p. 487, 1605) bao sea for spark lengths greater than 2 millimetres the linear relation V = 4-997 + 99-5931 represents Baille's rosults within experimental errors, This linear relation is confirmed by Licbig’s results, as the curves, Fig, 19, are nearly straight when the spark length is greater than one millimetre, Carey Foster and Pryson (Chemical News, 49, p. 114, 1884) found that the linear relation V=a+ 1 was the one which re- presented best the results of their experiments on the discharge through air at atmospheric pressure. 52.] When the spark length in air at atauospherie pressure ix less than about « millimetre, the curve which expresses the rela~ tion between potential difference and spark length geta concave to the axis along which the spark Jengths are measured ; that is, for 4 givon small increase in the spark length the inerease in the corresponding potential difference is greater whon the sparks are short than when they are long. For exceedingly short sparks there seems to be considerable evidence that when the spark longth is reduced to a certain critical value there is a point of inflexion in the potential difference curve, and that when the spark length is reduced below this value the previous concavity is replaced by convexity, the curve for very small spark lengths taking some- what the shape of the one in Fig. 20. This indicates that the potential difference required to produce a spark however short cannot be less than a certain finite value, which for air ab ordinary temperatures ia probably between 300 and 400 volts. If a curve similar to Fig. 20 represents the relation betwoen potential difference and spark length, wo seo that it would be possible under certain conditions to start a spark by pulling two plates maintained ata constant potential difference further apart, and to stop the spark by pushing the plates nearer together. 53.] At atmospheric pressure the spark length at which the potential difference is a minimum must, if such a length exist at 4 e | between the plates whon the distance was reduced to -001 or even to -0004 of an inch, Mr, Pence also found that when be removed the electrodes from the apparatus after sparks had passed between them when they were very close together, the part of the electrodes most affacted by breathing upon them formed an annulus at some little distance from the centre, in- dicating that discharge had taken place most freely at distances which were slightly greater than the shortest distance between slips, while the other pair of electrodes were kept at m greater distance apart by placing between them two or more of the pieces of glass piled one on the top of the other. At atmospheric pressure the spark passed across the short gap rather than the Jong one, but when the pressure was reduced the reverse effect took place, the spark going across the longer air gap before any discharge could be detected across the shorter, and after the spark had first passed across the longer path it required in some cases on additional potential difference of more than 100 volts to make it go across the shorter as well. When in Art. 170 we consider dischange at very low pressures we shall find that in some experiments of Hittorfs a long spark passed much more easily than a very much shorter one betwoen tho same electrodes; in this case however the electrodes were wires, and the field before discharge was not uniform as in the case under consideration, 78 THE PASSAGE OF [sr The two parts into which the table is divided by the horizon line correspond to two different sets of experiments. Paschen’s results (Wied. Ann. 37, p. 79, 1889) are given in { following table :— Potential Difference at first Spark: pressure 756 mm. mean temperature 15° C. SHORT SPARKS. rk Length | Spheres Spheres rea Pomuieten | omrrmdion | Secs mdlan | -26 chs radian 01 3:38 342 361 02 5.04 5-18 5.58 03 6.62, 6.87 6.94 04 8.06 8.22 8.48 05 9.56 9-75 9.86 08 10-81 10-87 11.19 07 11-78 12:14 12.99 08 18-40 18-59 18-77 09 14-89 14.70 14.89 10 15.86 15.97 16-26 Bt 16.79 17.08 17-26 12 18.98 18-42 18-71 u 20.52 20-78 21.26 LONG SPARKS. ‘Spheres eres spheres raking | saneraion | 25 radio, 15-96 16-11 16.45 21-94 22.17 22.59 27.59 27.87 28.18 82.96 38-42 83.60 88.59 39.00 88.65 43.98 44.52 48-28 49.17 49.31 47-64 54.37 54-18 51.56 69-71 59.08 54.67 64.60 68:85 87.27 69.27 67-80 59.95 7851 75.04 68-14 87-76 81.95 66.89 68.65 70-68 74.94 79-42 Here the heavy type again denotes the maximum potenti differences. 54) ELECTRIOITY THROUGH GASES, 79 These results are represented graphically in Fig. 21. They confirm Baille’s conclusion that for a spark of given length the * tain critical diameter, the critical diameter increasing with the length of the spark. eke hasten cee {iv) @ plane and s point, (v) two points. Tt will that the two points, which give the greatest striking for long sparks, give tho least for short sparks. 55,] If the spark length between parallel plates is unity, the spark length corresponding to-various potential ferences for diffarent kinds of electrodes was found Rue and Miller to be as follows (Proc, Roy, Soc, 36, 1883) — Number of calle, each call hisving an EM. F, of 108 volts . . . 1000 8000 6000 9000 13,000 0 Striking distance for point and plane 60 2.09 882 399 353 $80 Swriking distance for two points. . 84 104 465 465 418 | 3.88 This table would appear to indicate that the ratio of the striking distance for pointed electrodes to that of planes attains a maximum. It must however be remembered that when the sparks are long the conditions are not the same in the two cases; — in the case of the plates the discharge takes place abruptly, while when the electrodes are pointed a brush discharge starts long before the spark passes, and materially modifies the con- ditions. 56] Schuster (Phil. Mag. [5] 29, p. 182, 1890) has, hy the aid of Kirchhoff"s solution of the problem of the distribution of elee- tricity over two spheres, calculated from Paille's and Paschen's experiments the maximum electromotive intensity in the field when tho spark passed. The results for Baille’s experiments are given in Tablo 1, for Paschen’s in ‘lable 2. Po can be expressed by an equation of the form R=a+pr%, where a and f are constants and 7 is the radius of the inner eylinder. 58.] The variations in the value of the electromotive intensity is not the value of the not discharge must take place ; this quantity as the measure of the electric strength has d the progress of this subject by withdrawing attention from | most important cause of the discharge to this which is merely secondary. 59,] The following results taken from Paschen’s experiments — show that when the sparks sre not too long the variations in the electromotive intensity are very much greater than the vari- ations in the potential difference; suggesting thet for such sparks the potential difference is the most important com-_ sideration, Reading i ———— L 6 26 Potential Difference: 4 136 138 Maxiivua Totensity | 372 | ira | aon | { Sparklongth -06 em. Maxinam tnveosty | ica | ten | eq |} Seeretengt 24 0m, Rca | at | 192 | 228 | bop hgh tm Temata | a | ne | Stepan tng tt 60.] We can explain by the following geometrical illustration the two effects produced by the irregularity of the ficld—the | diminution in the potential difference, and the increase in the maximum electromotive intensity, When a discharge is passing 4 ‘= = firmed by Baille's and Paschon’s observations. For a theory of the spark discharge the roador is the discussion at tho end of this chapter. 61.) It is sometimes said that the reason a thin layer of , clectrically stronger than a thick one is, that a film of condensed gas is spread over the surface of the electrodes, and that this film is electrically stronger than the free gas. This consideration how- ever, as Chrystal (Proc. Roy. Soc. Edin, 11, 1881-2, p. 487) has pointed out, is quite incapable of explaining the variation in electric strength, for it is evident that if this were all that had to bo taken into account the discharge would pass whenever the electromotive intensity was great onough to break through this film of condensed gas, so that this intensity would be when the spark passed whatever the thickness of the layer of free gas. Connection between Spark Potential and the Presewre of the Gas. 62.] The general nature of this connection is as follows: ic the pressure of the gas diminishes the difference of quired to produce a spark of given length also diminixhes, ‘ont the pressure falls to « critical value depending upon the length of the spark, the nature of the gas, the shape and size of the electrodes and of the vessel in which the gas is contained; at this pressure the potential difference is a minimum, and any Surthor diminution in the pressure is accompanied by an increase in the potential difference, Tho criticnl pressure yaries yory greatly with the length of the spark; in Mr, Peace’s experiments, which we shall consider later, when the spark length was about 1/100 of a millimetre, the critical pressure was that due to bout 250mm, of mercury, while for sparks several millimetres long the critical pressure was less than that due to 1 mm. of mercury, 63.) At pressures considerably greater than the critical pres sure, the curve which represents the relation between potential difference and pressure, the spark length being constant, approx- imates to a straight line, or more accurately to a slightly euryed 4 & = 86 THE PASSAGE OP (6s. perbola. Paschen (Z. ¢. p. 91) made the interesting observation that as long as the product of the density and spark length is constant the sparking potential is for a considerable range of pressure constant for the same gas. This result can also be expressed by saying that the sparking potential for a gas can be expressed in terms of the ratio of the spark length to the moan free path of the molecules of the gas. The curves given in Fig, 24, which roprosent for air, hydrogen and carbonic acid the relation between the spark potential in electrostatic units as ordinates, and the producta of the pressure of the gas in centimetros of mereury and the spark length in centimetres as abscisswe, seem to show that this relation is approximately « linear one. 65.] The preceding experiments were made at pressures much greater than the critical pressure. A series of very interest- ing experiments has lately been made by Mr. Peace in the 1200 oe & = |_ 1200} 1 me | 3 wed f © 00] 4 00} 700} S cco S eo & = me 200] | Mi) d= iw d= soo} |} vol Ee | K f 33 T0100 Bod Hs B00 Air Pressure in titlimetres Fig, 25. Cavendish Laboratory, Cambridge, on the shape of these curves in the neighbourhood of the critical pressure. Tn these experi- ments the potential difference could be determined with great accuracy, as it was produced by « large number of small storage 88 THE PASSAGE OP ‘will be seen to present soveral points of great interest, ate] first place, Fig. 25 shows how much the eritical Electromotive Intensity C.G.8. Units upon the spark length; this will also be seen from the following table — Thus when the epark length was increased twonty-fold the critical pressure was reduced from 250mm. to 35mm. Another vory remarkable feature is the small variation in the minimum potential diffrence required to produce the spark, In the pre- ceding table there is a very considerable range of pressure, but the variation in the potential difference is comparatively small. Mr. Peace too made the interesting observation that he could nob produce # spark however near he put the electrodes together, b il | 90 THE PASSAGE OF ‘Sos pelo ti sicloons, and ike eas pe ere 67.] Although the magnitude of the critical pressure as we have seen, to a very great extent on the distance between the electrodes, the actual existence of a critical pressure does not seem to depend on the presence of electrodes. In Art. 74 @ method is described by which an endless ring discharge can be produced in a bulb containing gas at a low preesure; in this ease the discharge is in the gas throughout the whole of its course, and there are no electrodes. If in such an experiment the bulb is connected to an air pump it will be found that when the pressure of the gas in the bulb is high no discharge at all is visible; as however the pressure is reduced a discharge gradually appears and increases in brightness until the pressure is reduced to a small fraction of a millimetre, when the brightness is a maximum; when the pressure is reduced below this value the discharge has greater difficulty in passing, it gots dimmer and dimmer, and finally stops altogether when tho exhaustion is vory great. This experiment shows that there is a critical preseurs even when there are no electrodes, but that itis very much lower than in an ordinary sized tube when electrodes are used. 68.] De la Rue and Hugo Miller (Proo, Roy. Soc. 35, p. 202, | 1883), using the ordinary discharge with electrodes, found that the critical pressure depends on the diameter of the tube in which the rarefied gas is confined, the critical pressure getting lower as the diameter of the tube is increased, Potential Difference required to produce Sparks Ucrough various Gases. 69.] The potential difference required to send a spark between the same clectrodes, separated by the same distance, depends, as — Faraday found, on the nature of the gas surrounding the elec trodes: thus, for example, the potential difference required to | produce a spark of given length in hydrogen is much lax than in air. Measurements of the potential differences required ! to produce discharge through » series of gases have beon made 92 THE PASSAGE OP ites | { depends on the spark length and the pressure of the gases. If S88 resale by | Mee Pens Hee ee minimum potential difference required to produce a spark varied very little with the sparic length, —wereto hold far olor Asal there would be much more likelihood of this minimum potantial — difference being connected with some physical or chemical proporty of the gas, than tho potential difference required to produce a spark of arbitrary length at a pressuro chosen at random being #0 connected. 71.) If a permanent gas in a closed vessel be heated up to 300°C, the discharge potential does not change (see Cardani, Rend. della R. Ace. dei Lincei, 4, p. 44, 1888; J. J. Thomson, Proc. Camb. Phil. Soc., vol. 6, p. 325, 1889): if however the vessel he open so that the pressure remains constant, there will be a diminution in the discharge potential due to the diminution in density. When the temperature gets so high that chemical changes such as dissociation take place in the gas the discharge potential may fall to zero. A great number of experiments have been made on the relative ‘electric strengths’ of damp and dry air, The only observer who seems to have found any difference is Baille, and in his case the difference was so large as to make it probable that some of the water vapour bad condensed into drops. Phenomena accompanying the Electrio Discharge at Low Pressures. 72.] When the discharge passes between metallic electrodes soaled into # tube filled with gas at a low pressure, the appear- ance it presents is very complicated: many of the effects observed in tho tube are however evidently due to tho action of the electrodes, as tho phenomena at the anode are very different from those at the cathode; it therefore appears desirable to begin the study of the phenomena shown in vacuum tubes by investi- gating the discharge when no electrodes ara present. 73.) If we wish to produce the endless discharge in a closed vessel without electrodes, we must produce in some way or another round a closed curve in the vesscl an electromotive force large enough to break down the insulation of the gas, Since, for discharge to take place, the electromotive foree round « closed curve must be finite, it cannot be produced clectro- a: = = of THE PASSAGE -OF ie jams aro discharged and electrical oscillations set up in tho wire favourable conditions to causes bright discharge to pass through the rarefied gas in the bulb placed inside the coil. We have described in Art. 26 the way in which the Faraday tubes, which before the spark took place were mainly in the glass between the two coatings of the jars, spread through the region outaide the jars, as soon a8 the dischargo passes, keeping their ends on the wire Acs. They will pass in their journey through the bulb in the coil ¢, and if they congregate there in sufficient numbers the electromotive foree will be euflicient to cause & discharge to pass through the gas, Anything which concentrates the Faraday tubes in the bulb will increase the brightness of the discharge through it. 75.] It is necessary to prevent the coil ¢ getting to a high potential before the spark passes, otherwise it may induce a negative electrification on the parts of the inside of the glass bully nearest to it and a positive electrification on the parts more remote: when the potential of the coil suddenly falls in conse= quence of the pnssage of the spark, the positive and negative clectricities will rush together, and in so doing S69 peels tho rarefied gas in the bulb and produce luminosity will spread throughout the bulb and will ‘not he concentrated in a well-dofined ring, aa it is whon it arises from the electromotive force due to the alternating currents passing along the wire acs. This effect may explain the difference in the appearance presented by the discharge in the following experi- ments, where the discharge passes as a bright ring, from that ob- served by Hittorf (Wied. Ann. 21, p. 138, 1884), who obtained the discharge in a tube by twisting round it a wire the two coatings of a Leyden jar: in Hittorf's experiment the luminosity seems to have filled the tube and not to have been concentrated in a bright ring, ‘To prevent these clectrostatic effects, due to causes which operate before the electrical oscilla- tions in the wires begin, the coil ¢ is connected to earth, and as an additional precaution the discharge tube may be separated from the coil by a sereen of blotting paper moistened with dilute acid. The wet blotting paper is a sufficiently good conductor to sereen off any purely tic offect, but not a good enough .&, = = 96 THE PASSAGE OP this can be done by increasing the self-induetion of the It is thus advisable to use rather moro turns im tho coil indicated by the preceding rule. 4 Appearance of the Discharge. 77.) Let us suppose that a bulb fused on to an air pump ts within the coil ¢, and that the jars are kept i while the bulb is being exhausted. When the pressure is high, no discharge at all is to be seen inside the bulb; bat when the exhaustion has proceeded until the pressure of the air has fallen ton millimetre of mereury or theroubouts, a thin thread of reddish light is seen going round the bulb in the zone of the eoil. As the exhaustion proceeds still further, the brightness of this throad rapidly increases as wellas its thickness; it also changes its colour, losing the red tings and becoming white, Continuing the ex- haustion, the luminosity attains a maximum and the discharge passes as avery bright and well-defined ring. When the pressure is still further diminished, the luminosity also diminishes, until when an exceedingly good vacuum is reached no dischange at all passes, The pressure at which the luminosity is a maximom is very much less than the pressure at which the ¢lectric strength is 8 minimum in a tube provided with electrodes and comparable in size with the size of the bulb; the former pressure is in air Jesg than 1/200 of a millimotre of mercury, while the latter is about half a millimetre. 78.) We see from this result; that the difficulty which is experienced in getting the discharge to pass through an ordinary vacuum tube when the pressure is very low is not due to the difficulty of getting the electricity to pass from the electrodes into the gas, but that it also occurs in tubes without electrodes, though in this case the critical pressure is very much lower. 79.) The existence of a critical pressure can also bo easily shown by putting some mercury in the bulb, and, when the bulb haa been well exhausted, driving out the remainder of the air by heating the merenry and filling the bulb with mereary vapour. After this process has been repeated two or three times, the bulb should be fused off from the pump when fall of mercury vapour. It will only be found possible to got a discharge through this bulb within a narrow range of temperature, between about 70° and 4 - s | without electrodes shows that it does not. Numerous other ex~ periments of vory diffcront kinds point to the conclusion that a vacuum is not a conductor, Thus Worthington (Nature, 27,p. 434, 1883) showed thnt electrostatic attraction was exerted neross the best vacuum he could produce, and that a gold-leaf electroscope: would work inside it. Ayrton and Perry (Ayrton's Practical Electricity, p. 310) have determined the electrostatic a condenser in a vacuum in which they estimated the pressure to be only -001 mm. of mercury. If the air at this pressure had been a good conductor the electrostatic capacity would have been infinite, instead of being, as they found, less than at atmospheric pressure, Again, if we accept Maxwell's Theory of Light, a vacuum cannot be a conductor or it would bo opaque, and we should not receive any light from the sun or stars. 81.] The discharge has coneiderable difficulty in pasting across the junction of » metal and rarefied gas. This ean easily be shown by placing 1 metal diaphragm across the bulb in whieh the discharge takes placo, care being taken that the diaphragm ex- tends right up to the surface of the glass. In this ease the discharge does not cross the metal plate, but forms two separate closed circuits, one circuit being on one side of the diaphragm, the other on the other. The nature of tho discharge is shown in Fig. 30, in which it is seon that it travels through a comparatively long dis~ tance in the rarefied gas to avoid the necessity of cross ing a thin plate of a very good conductor. If the bulb, instead of merely Fig. 30. being bisected by one dia- phragm, is divided into #ix ‘or more regions by a suitable number of diaphragms, it will be found a matter of great difficulty to get any discharge at all through it. The metal plate in fact behaves in this case almost = — — 102 ‘THE PASSAGE OF [85. off the effects of the primary, depends upon the length of spark passing between the jars, and so upon the electromotive intensity acting on the gas: in other words, conduction through theac gases does not obey Ohm's law: the conductivity instead of being constant increases with the electromotive intensity. This is what we should expect if we regard the discharge through the gas as due to the splitting up of its molecules: the greater the electromotive intensity the greater the number of molecules which are split up and which take part in the conduction of the electricity. 85.] Another method by which we ean prove the great con- ductivity of these rarefied gases at tho pressures when thoy conduct best is by mensuring the energy absorbed by & secondary cirenit made of the rarefied gas when placed inside a primary cirenit conveying a rapidly alternating current. We shall see, Chapter IV, that when a conductor, whose conductivity is com- parable with that of electrolytes, is placed inside the primary coil, the amount of energy absorbed per unit time is proportional to the conductivity of the conductor; so that if we measure the absorption of energy by equal and similar portions of two electrolytes we can find the ratio of their conductivities. In the case of these clectrodeless discharges we can ensily com- pare the absorption of energy by two different secondary cir cuits in the following manner. In the primary circuitconneoting the outside coatings of two jars, two loops, Aand B, Fig. 33, are made, a standard bully tl) is placed in A and the substance to be oxamined in 8, When a large amount of energy is absorbed by the secondary in 8, the brightness of the discharge through a the bulb placed in A is diminished, and by observing the brightness of this dis- Pig. 88. chargo we can estimate whether the absorp- tion of energy by two different secondaries placed in @ is tho samo. If, now, an exhausted bulb be placed in B, the brightness of the discharge of the A bulb is at once cant indeed it is not difficult so to adjust the apark by which the jars are discharged, that » brilliant discharge pasos in a when the 8 bulb is out of its coil, and no visible when it isinside the coil. To compare the absorption of energy = = = = r 104 ‘TRE PASSAGE OF (ss. A and 8, the discharges «and through 8 are in rathor than. in series: in other words, that the polarized chains of molecules, which are formed before the dis- passes, consist some of A molecules and some of B mole~ cules, but that the chains conveying tho discharge do not consist partly of A and partly of 8 molecules. ‘Thus, if the discharge is passing through a mixture of hydrogen and nitrogon, the chains Bet ihiek tie wileealea split up and along which the electricity: passes may be either hydrogen chains or nitrogen chains, but not ehains containing both hydrogen and nitrogen. This seems to be indicated by the fact that when the discharge passes through a anixtuns of hydrogen and nitrogen, the spectrum of the discharge may, though a considerable quantity of nitrogen is present, show nothing but the hydrogen lines. Crookes’ observations on the stritions in a mixture of gases (Presidential Address to the Society of Telegraph Engineers, 1891) seem also to point to the conclusion that the discharges through the different gases in the mixture are separate; for he found that when several gases are present in the di: tube, different sets of striations, Art. 99, are found when the discharge passes through the tube, the spectrum of the bright portions of the strine in one set showing the lines of one, and only ‘one, of the gases in the mixture; the spectrum EY Fig. bh ture are distinet. 88.) When the dis- charge can continue in the same medium all the way it can traverse remarkably long distances, even though the greater portion of the secondary may be of such a shape as not to add anything to the electromotive force acting round it, ‘Thus, for example, the discharge will pass through a very long secondary, even though the tube of which this secondary is made — il VY | 106 THE PASSAGE OF [or this line. This discharge produces a supply of dissociated mole- cules along which subsequent discharges can pass with greater ease. The gus is thuk in an unstable state with regard to the discharge, since as soon as any amall discharge passes through it, it becomes electrically weaker and less able to resist subse~ quent discharges, ‘When, however, the gas is ina magnetic field, from the line of maximum electromotive intensity; thus subse- quent discharges will not find it any easier to pare along this line in consequence of the passage of the previous discharge. There will not therefore be the same unstability in this case as there is in the one where the gas is free from the action of the magnetic force. A confirmation of this view is afforded by the appearance presented by the discharge when the intensity of the magnetic field is reduced until the discharge just, but only just, passes when the magnetic field is on: in this case the discharge instead of passing as a steady fixed ring, flickers about the tube in a very undecided way, Unless some displacement of the line of easiest discharge is produced by the motion of the dissocinted molecules under the action of the magnetic force, it is difficult to under- stand why the magnet should displace the discharge at all, unless the Hall effect in rarefied gases is very large. 91.] In the preceding case the discharge was retarded because it had to flow across the lines of magnetic foree, when however the lines of magnetic force run along the line of discharge the action of the magnet facilitates the discharge instead of retard- ing it, This effect is easily shown by an arrangement of the following kind. A square tube asco, Fig. 35, is placed out- side the primary EFGH, the lower part of the discharge tube being situated between the poles tM of an electromagnet. By altering the longth of the spark between the jars, the electromotive intensity acting on the secondary cireuit can bo adjusted until no discharge passos round the tube ABCD when the magnet is off, whilst a bright discharge oceurs as long as the magnet ix on. The two effects of the magnet on the dis~ charge, viz. the stoppage of the discharge across the lines of force and the help given to it slong these lines, may be prettily illustrated by placing in this experiment an exhausted bulb N inside the primary, The spark length can be adjasted so that 108 Electric discharge through rarefied Gases when Electrodes ——y TRE PASSAGE OF [oa are used. 4] When the discharge passes between electrodes through a rare gas, the appearance of the discharge at the positive and negative electrodes is so strikingly different that the discharge Joses all appearance of uniformity. Fig. 36, which is taken Pig. 38. from a paper by E, Wiedemann (Phil. Mag. [5], 18, Pp. 36, 1884), represents the appearance presented by the dischargo when it passes through a gas at @ pressure comparable with that due to half a millimetre of mercury. Beginning at the negative electrode & we meet with the following phenomena. A velvety glow runs often in irregular patehes over the surface of the negative electrode; a wire placed inside this glow casts a shadow towards the nega- tive electrode (Schuster, Proc, Roy, Society, 47, p. 557, 1890), Next to this there is a comparatively dark region 1b, called sometimes ‘Crookes’ space’ and sometimes the ‘first dark space;’ the length of this region de- pends on the density of the gas, it gets longer as the donsity diminishes. Puluj’s experiments (Wien. Ber, 81 (2), p. 64, 1880) show that the length does not vary directly as the reciprocal of the density, in other words, that it is not proportional to the mean free path of the molecules, The luminous boundary / of this dark space is approximately such as could be got by tracing the locus of the extremities of normals of constant length drawn from the nega- Fig. 97. tive electrode: thus, if the electrode is x dise, the luminous boundary of the dark spuee is over a grvab part of its surface a 3 110 THE PASSAGE OF snaking them luminous. Tt ix an objection, iach poten all a fatal one, to this xe, iat tas cee 6 Oe very much greater than the mean free path of the molecules. We shall see later on that if the luminosity is due to gas shot from the negative electrode, this gas must be in the atomic: and not in the molecular condition ; in the former condition its | free path would be greater than the value caleulated from the ordinary data of the Molecular Theory of Gases, though if wo take the ordinary view of what constitutes a collision we should not expect the difference to be so great as that indicated by Puluj’s experiments. 96] The size of the dark space doer not seem to be much affected by the material of which the negative electrode is made, as long as it is metallic, It is however considerably shorter over sulphuric acid electrodes than over aluminium ones (Chree, Proc. Camb, Phil, Soc. vii, p. 222, 1891), Crookes (Phil. Trans, 1879, p- 137) found that if a metallic electrode is partly costed with Jamp black the dark space is longer over the Jamp-blacked portion than over the metallic. Lamp black however absorba gases a0 readily that this otfect may be duc to a change in the gas and not to the change in the electrode. The dark space is also, — as Crookes has shown (loc. cit.), independent of the position of the positive electrode. When the cathode is a metal wire raised to m temperature at which it is incandescent, Hittorf (Wied. Ann. 21, p.112, 1884) has shown that the changes in luminosity * which with cold electrodes are observed in the neighbourhood of the cathode disappear. here is a difference of opinion as to whether the dark space exists when the discharge passes through mereury vapour, Crookes maintaining that it doos, Schuster that it docs not. 97.] Adjoining tho ‘dark epace” isa lnminous space, bp Fig. 36, called the‘ negative column,’ or sometimes the ‘negative glow; the Iength of this is very varinble even though the pressure is constant, The speetrum of this part of the discharge exhibits peculiarities which are not in general found in that of the other luminous parts of the discharge. Goldstein (Wied. Ann. 15, p. 280, 1882) how- ever has found that when very intense discharges are used, the peculiaritios in the spectrum, which are usually confined to the negative glow, extend to the other parts of the discharge. 98.) The negative glow is independent of the position of the = — = 112 THE PASSAGE OF [9. creases a3 the diameter of the discharge tube increases, provided the striations reach to the sides of the tube. Goldstein (l.c.) eu tArN CAKE AUUECKEUA CEU ULALUE( LAMAR CHE EEE MEMO CECE CEO ECEC CLO EEL EE ECE CEEOL ECEEOOEOEMG FOCECOLC OC CO ECEEOEEO Pe eC CC eee cece cr CRLECEERCEC EECA TE CeeRCEEC CTC GCORFECEEELECeO EEE ret EE TT REEL ETERS BPUPRPRBRPPRPERRPDS ~~» OePaeries Fig. 40. found that the ratio of the values of d at any two given pres sures is the same for all tubes. If the dischange takes place 4 114 THE PASSAGE OF [103-) stondiness of the striations is a maximum (De la Rue and — Hugo Miller, Comptes Rendus, 86, p. 1072, 1878). Crookes has found (Presidential Address to the Society of Te 1891) that when the discharge passes through a mixture of dif-— ferunt gases there is a = separate set of strin~ tions for each gus: the colour of the striations in each seb being different. Crookes proved this by observing the spec- tra of the different striae. A full ac- count of the different colouredatriationsob- served in air is given by Goldstein (Wied. Aan, 12,p.274, 1881). 102.) When we consider the gction of a magnet on the striated positive oo- lumn we shall seo reasons for thinking that any portion of the positive column between the bright parts of consecutive. striations constitutes ‘4 separate discharge, and that the dis- Fig, dl. charges in the several portions do not occur simultaneously, but that the one next the anode begins the discharge, and the others follow on in order. 108.] ‘Tho positive column bears a very much more important relation to the discharge than either the negutive dark space or the negative glow. The latter effoctx are merely local, they do not depend upon the position of the positive electrode, nor do they =i = = a | 118 THE PASSAGE OF [107. and the distance between 8c and Gx, the rate of propagation of the luminosity may be calculated. The displacement of the images showed that tho luminosity always travelled from the positive to the negative electrode. When AB was the negative electrode, the luminous discharge arrived at GH, a place about 25 fect from the positive cleetrode, bofore it reached BC, which was only a few inches from the cathode, and as the interval between ita appoarance at these places was about tho samo as when the current was reversed, we may conclude that when as is tho cathode the Inminosity at a place gc, only a few inchos from it, has staried from the positive electrode and traversed a path enormonsly longer than its distance from the eathode, The velocity of the discharge through air at the pressure of about 4 a millimetre of mercury in a tube 6 millimetres in diameter was found to be rather more than half the velocity of light. 106.| The preceding experiment was repeated with a great varioty of electrodes; the result, howover, was the same whether the clectrodes were pointed platinum wires, carbon filaments, flat surfaces of sulphuric acid, or the one olectrode a flat liquid sur- face and the other a sharp-pointed wire. The positive luminosity travels from the positive electrode to the negative, even though the former is a flat liquid surface and the latter a pointed wire, The time taken by the luminosity to travel from BC to GH was not affected to an appreciable extent by inserting between BC and GH a number of pellets of mercury, so that the discharge had to pass from the gas to the mercury several times in its passage between these places: the intensity of the light was however very much diminished by the insertion of the mercury, 107,] The preceding results bear out the conclusion which Pliieker (Pogg. Ann. 107, p. 89, 1859) arrived at from the con- sideration of the action of a magnot on tho discharge, viz. that the positive column starts from the positive electrode ; they also confirm the result which Spottiswoode and Moulton (Phil Trans. 1879, p. 165) deduced from the consideration of what they have termed ‘relief’ effects, that the time taken by the negative electricity Lo leave the cathode is greater than the time taken by the positive luminosity to travel over the length of the tube. 120 THE PASSAGE OF [110 the shadow of the screen remaining dark while the glass round the shadow phosphoresces brightly. In this way many very beautiful and brilliant effects have been produced by Mr. Crookes and Dr. Goldstein, the two physicists who have devoted most attention to this subject, One of Mr. Crookes’ experiments in which the shadow of a Maltese cross is thrown on the walls of the tube is illustrated in Fig. 43. Fig. 43, 110,] As we have already mentioned, the colour of the phos- phorescence depends on the nature of the phosphorescing sub- stance; if this substance is German glass the phosphoroscence Fig. is green, if it is load glass the phosphorescence is blue. Crookes found that bodies phosphorescing under this action of the negative electrode give out characteristic band spectra, and he has developed this observation into a method of the greatest ea 122 THR PASSAGE OF tubo from any external electrostatic action. Crookes (Phil. Trans, 1879, Pt. I, p. 662) has shown, moreover, that | pencils of these rays repel each other, as they would do each pencil consisted of particles charged with same kind - of vlectricity. ‘The experiment by which this shown represented in Fig. 45; a, b ure metal dises either or both of which may be made into esthodes, a diaphragm with two {=> Fig. 45. openings d and ¢ is placed in front of the dise, and the path of the rays is traced by the phosphorescence they excite in a chalked plate inclined at « staal angle to their path. When @ is the cathode and 4 is idle, the rays travel along the path df, and when } is the cathode and « idle they travel along the path ef, but when a and 8 are eathodes simultancously the paths of the rays are dg and eh respectively, showing that the two streams have slightly repelled each other. 118,] Crookes (Phil. rane. 1879, Part IL, p. 647) found that if a dise connected with an clectroscope is placed in tho full line of fire of theae rays it receives a charge of positive elee~ tricity. This is not, howover, a proof that these mays do not consist of negatively electrified particles, for the experiments doscribed in Art, 81 show that electricity does not pass at all readily from a gas to a metal, and the positive electrifica- tion of the disc may be a secondary effect arising from the same cause as the positive electrification of a plate when exposed to the action of ultra-violet light. Vor sinoe the action of these rays is the same as that of ultra-violet light in producing phos- phorescence in the bodies upon which they fall, it seema not unlikely that the rays may resemblo ultra-violet light still further and make any metal plate on which they fall a cathode. Hertz (Wied, Ann. 19, p. 809, 1883) was unable to discover that these rays produced any magnetic effect. ‘The paths of the negative rays are governed entirely by the shape and position of the cathode, they are quite independent Z & | — ———— 124 THE PASSAGE OF ment is sufficient to show the inadequacy of a theory that sometimes been advanced to explain the phosphorescence, that the particles shot off from the electrode are nob particles, but bits of metal torn from the cathode; the phorescence being thus due to the disintegration of the negati electrode, which ia a well-known fenture of the discharge in vacuum tubes, The preceding experiment shows that this theory | ix not adequate, and Mr. Crookes has still further disproved | it by obtaining the characteristic effects in tubes when the eleotrodes were pieces of tinfoil placed ouside the glass. 115.] Goldstein (Wied. Ann. 11, p. 838, 1880) found that a sudden contraction in the cross section of the discharge tube produces on the side towards the anode the same effect as a cathode. These quasi-cathodes produced by the contraction of the tubo are accompanied by all the effects which are observed with metallic cathodes, thus we have tho dark space, the phos- phorescence, and tho characteristic behaviour of the glow in a magnetic field. 116.] Spottiswoode and Moulton (Phil. Trans. 1880, pp. 615~ 622) have observed a phosphorescence accompanying the positive column. They found that in some cases when this strikes the gas the latter phosphoresces. They ascribe this phosphorescence to a negative discharge called from the sides of the tube by the positive electricity in the positive column, Mechanical Effects produced by the Negative Rays, 117.] Mr. Crookes (Phil. Trans, 1879, Pt. I, p. 152) bas shown that when these rays impinge on vanes mounted like those in a radiometer the vanes are set in rotation. This can be shown by making the axle of the vanes run on rails as in Fig. 48, When the discharge passes through the tube, the vanes travel from the negative to the positive end of the tube. It is not clear, however, that this is a purely mechanical effect; it may, as suggested by Hittorf, be due to secondary thermal effects making the vanes act like thoge of a radiometer. In another experiment the vanes aro suspended asin Fig. 49, and can bo screoned from the negative rays by the screen ¢; by tilting the tube the vanes can bo brought wholly or partially out of the shadow of the screen, When the vanes are completely out of the shade they do nob rotate as the bombardment is symmetrical; when, how- —- - 126 THE PASSAGE OF Lr: 1879, Part I, p. 151) found that-a thin film of quartz, which it to ultra-violet light, produced the same effect. last result is of great importance in connection with a which has received powerful support, viz. that these ‘rays’ kind of ethereal vibration having their origin at the cathode. this view were correet we should not expect to find a thin q plate throwing a perfectly black shadow, as quartz is transparent to ultra-violet light, To make the theory agree with the facts we have further to assume that no substance has been dis- covered which is appreciably transparent to these vibrations * ‘The sharpness and blacknoss of these shadows are by far the strongest arguments in support of the impact theory of the mee. 119.] Though Crookes’ theory that the phosphorescence is due to the bombardment of the glass by gaseous particles projected from the negative electrode is not free from difficulties, it seems to cover the facts better than any other theory hitherto advanced. On one point, however, it would seem to require a slight modi- fication: Crookes always speaks of the molecules of the gas ro- ceiving a negative charge, We have, howovor (seo Art. 3), seen roasons for thinking that a molecule of a gas is incapable of receiving 8 charge of electricity, and that free electricity must be on the atoms as distinct from the molecules, If this view is right, we must suppose that the gaseous particles projected from the negative electrode are atoms and not molecules. This does not introduce any additional difficulty into the theory, for in the region round the cathode there is a plentiful supply of dis- sociated molecules or atoms ; of these, those having s negative charge may under the repulsion of the negative electricity on the cathode be repelled from it with considerable violence. 120.] Anexperiment which I made in the course ofan inveatiga- tion on discharge without electrodes seems to afford considerable evidence that there is such a projection of atoma from the * Since the above was written, Horts (Wve. Ann. 46, p. 28, 1892) has found that thin filma of gold lonf donot cast perfootly dark shadows but allow a certain amount of phosphorescence to take place behind them, which cannot be explained by the exist= ‘ence of holes in the film. 1¢ seems possible, however, that thie is another aspeot of the phenomenon observed by Crookes (Art, 113) thata metal plate exposed to the full force of thew rays becomes cathode ; in Hertz experiments the films may have berm 20 thin that onch side noted like n cathode, and in this case the phorphoresconce om the glass would be caused by the filma acting like @ cathode on ite owm account, 128 THE PASSAGE OF i large multiple of the mean free path of the molecules of the when in a molecular condition; it is possible, however, that: passage of a mass of air by convection currents rathor than a process of molecular diffusion. We must remember, too, gas in the atomic condition would naturally be greater than when in the molecular. 128.) Strikingly beautiful as the phenomena connected with these ‘negative rays’ aro, it seems most probable that the rays are merely a local effect, and play but a small part in carrying the current through the gas. There are several reasons which lead us to come to this conclusion: in the first place, we have seen that the great mass of luminosity in the tube starts from the anode and travels down the tube with an enormously greater volovity then we can assign to these particles again, thin di- charge seems quite independent of the anode, so that the rays may be quite out of the main line of the discharge. The exact function of these rays in the discharge is doubtful, it seems just possible that they may constitute a return current of gas by which the atoms which carry the discharge up to the negative electrode are prevented from accumulating in its neighbourhood. 124.] Theee rays have been used by Spottiewoode and Moulton (Phil, Trans. 1880, p. 627) to determine a point of fundamental importance in the theory of the discharge, viz. the relative mag- nitudes of the following times :— (1) The period oceupled by # discharge. (2) The time occupied by the discharge of the positive elec- tricity from its terminal. (3) The time occupied by the discharge of the negative elec~ tricity from its terminal. (4) The time occupied by molecular streams in lenving a negative terminal. (5) The time occupied by positive electricity in passing along the tube, (6) The time occupied by negative electricity in passing along the tube, (7) The time occupied by the particles composing molecular streams in passing along the tube. 130 THE PASSAGE OF [res ‘This image was splayed out by the finger being placed on the tube. Now a magnet displaced it as a whole without any splaying out. This thon pointed to a variation in the relative SeeDgEL of the dnterticing atroart /amd tH \iteiecy Abert with, and euch variation must have occurred during the period that they were encountering one another, and were moving in the ordinary way of such streams, for it showed itself in a varia- tion in the extent to which the streams from the negative terminal were diverted. We may hence conclude that the time requisite for the molecules to move the length of the tube was decidedly less than that occupied by the discharge, but was sufficiently comparable with ib to allow tho diminution of intensity of the streams from the sides of the tube to make itself visible before the streams from the negative terminal ex= perienced a similar diminution.” 125.] This may corvo as an example of the mothod uscd by Spottiswoode and Moulton in comparing the time quantities enumerated in Art. 124. We rogret that we have not space to deseribe the ingenious methods by which they brought other time quantities into comparison, for these we must refer to their paper; we can only quote the final result of their investigation, They arrange (1. c. pp. 641-642) the time quantities in groups which are in descending order of magnitude, the quantities in any group are exceedingly small compared with those in any group abovo thom, while the quantities in the same group are of the same order of magnitude. A. The intorval between two discharges, B, The time occupied by the disebarge of the negative elee- tricity from its terminal. ‘The time occupied by negative streams in leaving a nega- tive terminal. The time occupied by the particles composing molecular streams in passing along the tube. C. The time occupied by positive electricity in passing along the tube, The time occupied by negative electricity in passing along the tubo, D. The time oveupied by positive discharge. ‘The time required for tho formation of positive luminosity at the seat of positive discharge. SESE 132 THE PASSAGE OP [129. by which we explained the effect of a magnet on the discharge without electrodes: viz. that when an electric discharge haa passed through a gas, the supply of dissociated moleculos, or of molecules in a peculiar condition, left behind in the line of the discharge, haa made that line s0 much better a conductor than the rest of the gus, that when the particles composing it are displaced by the action of the magnetic force, the discharge continues to pass through them in their displaced position, and maintains by its passage the high canduetivity of this line of particles. On this view the case would be very similar to that of a current along a wire, the line of particles along which the discharge passes being made by the discharge so much better a conductor than the rest of the gas, that the case is analogous to a metal wire surrounded by a dielectric. 128.] This view seems to be confirmed by the behaviour of a spark between electrodes when o blast of air is blown meroas it ; the spark is deflected by the blast much as a flexible wire would be if fastened at the two electrodes. On the view the explanation of this would be, that by the passage of the spark through the gas, the eleetrie strength of the gas along the line of discharge is diminished, partly by the lingering of atoms produced by the discharge, partly perhaps by the heat produced by the spark. When a blast of air is blowing across the space between the electrodes, the electrically weak gas will be carried with it, so that the next spark, which will pass through the weak gas, will bo deflected. Feddersen’s observations (Pogg. Ann, 103, p. 69, 1858) on the appearance pre- MK eS Fig. 52. trodes, seem to prove conclusively that this explanation is the true one, for he found that the first spark was quite straight, while the successive sparks got, as shown in Fig, 62, gradually more and more bent by the blast. 129] The effects produced by a magnet show themselves in different ways, at different parte of the discharge. Beginning with the negative glow, Pliicker (Pogg. Ann. 103, p, 88, 1858), who was the first to observe the behaviour of this part of the discharge when under the action of a magnot, found that the == = ii. | 134 THE PASSAGE OF [r30. paper, show the shape taken by the glow when placed in the field due to a strong clectro-magnet, the tube being placed in Figs, 54, 55 so that the lines of magnatic foree are transverse to the line of discharge; while in Figs. 53 and 56 the lino of discharge is moro or less tangential to the direction of the magnetic force. 130.] Hittorf (Pogg. Ann. 136, p. 218 et seq., 1869) found that when the negative rays were subject to the action of magnetic Fig. 57. forco, thoy were twisted into spirals and sometimes into cireular rings. In his experiments the negative electrode was fused into 4 small glass tube fused into the discharge tube, the open end of the amall tube projecting beyond the electrode. The negative rays were by this means limited to those which were approxi- Big. 58. mately parallel to the axis of the small tube, so that it was ensy to alter the angle which these rays made with the lines of mag- netic force either by moving the discharge tube or altering the position of the electro-magnet. The discharge tube was shaped #0 that the walls of the tube were ab a considerable distance from the negative electrode, Hittorf found that when the direction of the negative rays was tangential to the line of magnetic force passing through the extremity of the cathode, the rays continued +. & 136 THE PASSAGE OF {38 components parallel to the axes of a, y, = are respectively a Pare ia ine aa? ees a? where ds is an element of the path of the particle, Thus, if m is the mass of the particle, Z the magnetic force, the equations of motion of the particle are noe = yon, (1) mod =— sera, (2) eet, (8) Since the force on the particlo is at right angles to its diree- tion of motion, the volocity v of the particle will bo constant, and since by (3) the component of the velocity parallel to the axis of 2 is constant, the direction of motion of the particle muat make a constant angle, a say, with the direction of the force. Since ds/dt is constant, equations (1)—(3) may be written met #2 = eur me TL = yen Ae, ds mut =0. If p is the radiue of curvature of the path, A, », » ita direction CS a @r_d ae =p? Squaring and eaaing we po ae) (GY + GD}: _ “a ee 138 ‘THE PASSAGE OF [34 the value of » is less than six times the velocity of sound, Henee the velocity of these particles must be i in comparison with that of the positive luminosity as we have seen, is comparable with that of light. 138.] A magnet affects the disposition of the negative glow over the surface of the electrode as well as its course through the gas. Thus Hittorf (Pogg. Ann. 136, p. 221, 1809) found that when the negative electrode is a flat vertical disc, and the dis- charge tube is placed horizontally between the poles of an electro- magnet, with the disc in an axial plane of tho clectromagnet; the dise is cleared of glow by the mag- netic foreo except upon the highest point on the side most remote from the positive slectrode, or the lowest point on the side nearest that electrode according to the direction of the magnetic force, In another experiment Hittorf, using aa a cathode a metal tube about 1 cm in diameter, found that when the discharge tube is placed so that the axis of the cathode is at right angles to the line join- ing the poles of the electromagnet the cathode is cleared of glow in the neigh- bourhood of the linea where the normals are at right angles to the magnetic foree. These experiments show that the action of n magnet on the glow is the same as its action on a system of perfectly flexible = currents whose ends can slide freely over Fig. 61, the surface of the negative electrode. 134,] The positive column is also de- flected by a magnet in the same way as a perfectly flexible wire carrying a curront in the direction of that through tho dischargo tube. This is beautifully iluatrated by an experiment dus to De In Rive in which tho discharge through a rarefied gas is set in continuous rotation by the action of a magnet. The method of making this experiment is shown in Fig. 61; the two terminals @ and d sre metal rings separated from each other by an insu- lating tube which fits over a piece of iron resting on one of the poles of an electromagnet Jf This arrangement is placed in an = & J om which the air can be exhaustod, eect it ia advisable to | ‘the vapour of alcohol or turpentine, The ter /d sre connected with un induction coil, which, i reduced, along which none of the dissociated molecules yeh ‘The now discharge will thus not be along the ‘the old one, and the laminous column will there- . We can easily see why a simple gas like hydrogen ot show this effect nearly so well as a complicated one vapour of alcohol or turpentine. For the discharges of ‘coil are intermittent, #0 that to produce this rota~ issocinted molecules produced by one discharge must il the arrival of the subsequent one. Now we should that when a molecule of a stable gas like hydrogen _ by the discharge, the recombination of ite atome in a much shorter time than similar recombinn~ sx gas like turpentine vapour ; thus we should atiaiaTobiibs idisclangs to: be snore paraiteat) andl ‘the rotation more decided in turpentine vapour than in (PRL. Trans. 1879, Part TL, p. 657) has produced us rotations of the negative rays in a very d tube. The shape of the tube he employed is re 62. When the discharge wont through this tube, the negative pole was covered with two patches which rotated when the tube was placed rn Crookes found that the direction of rota~ d when the magnetic force was reversed, but that oree were not altered the direction of rotation was by mavering tho poles of the discharge tube. This expect if we remember that the bright r eet asi bea ccd pray oxi nkatln to the negative electrode; thus the re- [>= 140 THE PASSAGE OF [r36. versal of the poles of the tube does not reverse the direction of these rays; it merely alters their distance from the pole of the electromagnet, Tho curious thing about the rotation was that it had the opposite dircction to that which would have been produced by the cl action of a magnet on a current carrying electricity in the same direction as that earried by the negative rays, showing clearly that this rotation is due to some secondary effect and not to the primary action of the magnetic force on the current. 136.] An experiment due to Goldstein, which may seem inconsistent with the view we have taken, viz. that tho deflee- tion of the discharge is dus to the deflection of tho line of least oloctric strength, should be mentioned here. Goldstein (Wied. Ann. 12, p. 261, 1881) took a large dis- charge tube, 4 em. wide by 20 long, the electrodes being at opposite ands of the tube. A piece of sodium was placed in the tube which was then quickly filled with dry nitrogen, the tube was then ex- hausted until a discharge passed freely throngh the tube, and the sodium heated until any hydrogen it might have con- tained had been driven off. When this To Battery -, had been done the tubo was refilled with Fig. 63. nitrogen and then exhausted until the positive column filled the tube with a reddish purple light. The sodium was then slowly heated until its vapour began to come off, when the discharge in the lower part of the tube over the sodium became yellow as it passed sodium vapour, while the discharge at the top of the tube re. mained red aa the sodium vapour did not extend all the way across the tube. The positive discharge was now deflected by @ magnet and driven to the top of the tube out of the region ‘occupied by the sodium vapour, the discharge was now entirely red and showed no trace of sodium light. The experiment does not seem inconsistent with the view we have advocated, as wo 142 THE PASSAGE OF [39. from these experiments that the positive column does not con- sist of a current of electricity traversing the whole of its length in the way thab such a current would traverse a metal cylinder coincident with the positive column, but that it rather consists of a number of separate currents, each striation corresponding toa current which is to a certain extent independent of thore which precede or follow. The discharge along the positive column might perhaps be roughly illustrated by placing piceos of wire equal in length to the striae and separated by very minute air spaces along the line of discharge. 138,] Goldstein found that the rolling up of the striae by the magnetic force was most marked at the end of the positive coluron nearest the negative electrode: the following is a trans- lation of Goldstein's description of this process (I.c. p. 852). The appearance is very characteristic when in the unmagnetized con dition the negative glow penetrates beyond the first striation into the positive column. The end of thenegative glow is then farther from the cathode than the first striation or, even if the rarefaction is suitablo, than the second or third, Nevertheless the end of the negative glow rolls itaolf under the magnetic action up to the cathode in the magnetic curve which passes through the cathode. Then separated from this by a dark space follows on tha side of the anode « curve in which all the rays of the first striation are rolled up, then « similar curve for the second striation, and so on. We shall huve occasion to refer to these experiments again in the discussion of the theory of the discharge. On the Distribution of Potential along an Exhausted Tube through which an Electric Discharge is passing. 139,] The changes which take place in the potential as we pass along the diacharge tube are extremely interesting, as they present a remarkable contrast to those whieh take place along @ metal wire through which a steady uniform current is passing ; in this caso the potential-gradient is uniform along the wire, but changes when the current changes, being by Ohm's law proportional to the intensity of the current; in the exhausted ube, on the other hand, the potential-gradient varies greatly in different parts of the tube, but in the positive column is almost independent of the intensity of the current passing through the gas, Tho potentials measured are those of wires immersed in 144 ‘THE PASSAGE OF [140., ‘The rapidity of the intermittence can to some extent be tested — by observing whethor or not the discharge is deflected by the approach of a conductor. When the discharge is intermittent and the interval between the discharges so long that the intar- mittence of the discharge can be detected either by the eye ar by a slowly rotating mirror, the discharge is deflected when @ conductor is brought near it; when however the intermittence ig very rapid, the discharge is not affected by the approach of the conductor. This effect has been very completely investigated by Spottiswoode and Moulton (Phil, Trans, 1879, Part I, p. 166; 1880, Part II, p. 564). 140,] We shall begin by considering Hittorf's experiments on the potential gradient (Wied. Ann. 20. p. 705, 1883). Tho dis- charge tube, Fig. 63, which was 5-5 cm. in diameter and $3-7 om. long, bad aluminium wires 2 mm. in diameter f used into the ends for clectrodes, the anode, a, was 2 cm. long, the cathode, ¢, 7om. In addition to tho cloctrodes five aluminium wires, }, <, ¢, A. % half a millimetre in diameter, were fused into the tube, The difference of potential between any two of these wires could he determined by connecting them to the plates of a condenser, and then discharging the condenser through a galvanometer. The deflection of the galvanometer was proportional to the charge in the condenser, which again was proportional to the difference of potential between the wires. The discharge was produced by means of a large number of cells of Bunsen's chromic acid battery, and the intensity of the current was yoried by inserting in the circuit a tube containing o solution of cadmium iodide, which is a very bad conductor. No inter- mittence in the discharge could be detocted either by s mirror rotating 100 times a second or a telephone. The tube was filled with nitrogen, ax this gas has the advantage of not attacking the electrodes and of not being absorbed by them so greedily as = — a for this purpose the tube in Fig. 63 was not suitablo, as the saseptiyn ptovy yas yery anak Sntestered 'with/by Aaya ee tube, he therefore used a tube shaped like that in Fig. 64, which | was purposely made wide in the region round the electrode. The diameter of the positive part of the tube was 4 cm., that of the negative 12cm. The length of the negative électrode was 15 om,, that of the positive 3.cm. In only two wires, 6 and d, were inserted in the tube. The results of experiments with this tube are given in the following 148 THE PASSAGE OF f45. without electrodes, when the interval between two discharges — was long enough to give the gas through which the discharge the electromotive intensity Acai for discharge increases. 144,] The supply of dissociated molooules furnished by previous discharges alzo explains another peculiarity of these experiments, Tt will be seen from the table that at a pressure of -0007 mm. of mercury, a potential difference which gave a galvanometer deflection of 10-6, corresponding to about 63 volts, was all that occurred in a length of 12 cm. of the positive light; it does not follow however that a potential gradient of about 5 volta per centimetre would be sufficient to initiate the discharge even if the great change in potential at the cathode were absent. Jn fact the experiments previously described on the discharge without electrodes show that it requires a very much greater electro- motive intonsity than thie even when the cathode is entirely done away with. The table shows that the potential difference between a and 6, space which includes the anode, has at the higher exhaustions passed its minimum value and commenced to inereasa, 145.] Though the potential differences between wires immensed in the positive column is independent of the strength of the current passing through the tube, yet in such a tube as Fig. 63 the potential differences between wires in the middle of the tube may be affected by variations in the current if these varia- tions are accompanied by changes in the appearance of the dis- charge. Let us suppose, for example, that tho tube is filled with nitrogen ‘at n pressure of from 2 to 3 mm. of mercury, then when the in- tensity of the current is very small the tube will appear to be dark throughout almost the whole of its length, the positive column and negative glow being reduced to mere specks in the neighbourhood of the electrodes; when however the intensity of the current increases the positive column increases in length, and if the increase is great enough to make it envelop two wires which were previously in the dark Faraday apace, the differonco 4 a ad 150 THE PASSAGE OF fand g, which were still in the dark, remained unaltered. In experiment 5 the positive column reached past the middle of bd; the potential difference in [d rose fram 25 to 43, the potential differences between the wires in the dark still being unaltered. Th experiment 6 the positive light filled the whole space ad; the potential difference between 6 and d rose to 61, and that between d and ¢ also began to rise as d was now in the positive column; this difference increased vory much in experiment 7, when the positive column reached to ¢, 148,.] We now pass on to the effect of an alteration in the strength of the current on the potential difference at the cathode. | We have already remarked that if the negative glow does not spread over the whole of the cathode, the only effect of an increase in the intensity of the current is to make the negative glow spread still further over the cathode, without altering the potential dif- ference. Until the glow has covered the electrode, there is, ac- cording to Hittorf, no considerable increase in temperature at the cathode: when however the intensity of the current is in- creased beyond the point at which the whole of the cathode is covered by the glow, the temperature of the cathode begins to increase ; when the current through the gas is very strong, the cathode, and sometimes even the anode, becomes white hot, When this is the case the character of the discharge changes in a re- markable way, sll luminosity disappears from the gas, which when examined by the spectroscope does not show any trace of the lines of ita spectrum. The tube with its white hot electrodes surrounded by the dark gas presents a remarkable appearance, and it is especially to be noted that the electrodes are raised to ineandescence by a current, which if it passed through them when thoy formed part of a motallic cirenit, would hardly make them appreciably hot. Hittorf also found (Wied. Ann. 21. p, 121, 1984) that if in @ vacuum tube conveying an ordinary luminous discharge, a platinum spiral which could be raised by a battery to a white heat was placed so as to be in the path of the discharge, the latter lost all luminosity in the neighbourhood of the spiral when this was white hot. If the spiral was allowed to cool, the luminosity appeared again before the spiral had cooled below a bright red heat, 149.] For oxperiments of this kind aluminium electrodes melt = — = 152 THE PASSAGE OF [150. have probably a discharge closely resembling that in this ex- periment, the anode is also hotter than the cathode when the — current is intense, In this case the gas was quite dark. A very remarkable feature shown by it is the smallness of the potential difference botween the electrodes, not amounting to more than 100 volts, though the gas was at the pressure of 53*1 millimetres, and the distance between the electrodes 15 mm. When the electrodes were cold, the battery power used, about 1200 volts, was not sufficient to break down the gas: the discharge had to be started by sending a spark from a Leyden jar through the tube. The conduction through the gas in this case is of the same character as that described in Art, 169, 150,] Hittorf also made experiments on hydrogen and carbonic oxide; the resulta for hydrogon are given in the following table (Wied. Ann, 21, p. 118, 1884):— Experiments with Hydrogen. Distance of the Iridium electrodes 15 mn. Number fixing the id = 2 3 i 5 6 88-8 47.05 47-05 47-05 6855 400x6 | 100x6 | 6OOxd | 8008 | 800x8 41s | 807s | 9292 | .9905 | 197 the electrodes. . 100 | 107-108 110 | 100-210] 207-120] 140 In experiment 1 the pressure and the current were almost the same as for experiment 1, Art. 149, in nitrogen; the potential difference between the electrodes was however much greater in bydrogen than in nitrogen, though the potential difference required to initiate a discharge in hydrogen is considerably logs than in nitrogen. In these experiments the potential dif- ference between the electrodes for this dark discharge seems almost independent of the current and of the density of the gas. rf a 154 THE PASSAGE OF Os3. by 6 em.; in this caso the discharge was quite dark, When ten or snore cells were used # pale bluish light spread over the anode, At should be noticed that the single cell does not start the eur rent, it only maintains it: the eure rent has previously to be started by the application of a much greater potentinl difference. Hit- torf generally started the current by discharging a Leyden jar through the tube. No current at all will pass if the poles are re- versed so that the anode is hot and the cathode cold. Tn these experiments it is necessary for the eathode to be at n white hoat for an appreciable current to pass between the electrodes; very little effect seems to he produced on the potential difference at the cathode until the latter is hotter than a bright red heat. The enrront produced by a given eleetro~ motive force is greater at highor exhaustions than at low ones, but Hittorf found he could get ap- prociable effects at pressures up to 9 or 10 mm. 153.] In considering tho results of experiments in which carbon filaments or platinum wires are raised to ineandescence, we must remember that, as Elster and Geitel have shown (Art. 43), electri- fication is produced by the incandescent body, the region around. which receives a charge of electricity ; though whether the carrier of this charge is the disintegrated particles of the incandescent wire, or the dissociated molecules of the gas itself, is nob clear, This electrification often makes the interpretation of experiments in which incandescent bodies are used ambiguous. Thus for example, Hittorf in one experiment (Wied. Ann. 21, p. 197, 1824) used a U-shaped discharge tube, in one limb of which a earbon filament was raised to incandescence; the other limb of the tube contained a small gold leaf electroscope; when the Fig. 68, pressure of the gas in the tube was very low, Hittorf found that \ it a — sy 156 THE PASSAGE OF (57. Tn Warburg's experiments, the fall in potential at the cathode, by which is meant the potential difference between the cathode and a wire at the luminous boundary of the negative glow, was measured by a quadrant clectrometer. Warburg found that, so Jong as the whole of the cathode was not covered by tho negative glow, the fall in potential at the cathode was nearly independent of the density of the gas: this is shown by the following table (Lc. p. 579), in which # represents the potential difforenes be- tween the electrodes, which were made of aluminium, ¢ the poten- tial fall at the cathode, Z and ¢ being measured in volts, p the pressure of the gas, dry hydrogen, measured in millimetres of mer- cury, é the current, through the gas in millionths of an Ampére. ‘This table shows that though the fall in potential in the pasitive light decreased as the preasuro diminished, the fall in potential at the cathode remainod almost constant. 157,] In imperfectly dried nitrogen, which contained also a trace of oxygen, the cathode potential difference depended to some extent on the metal of which the electrode was mado; platinum, xine, and iron electrodes had all practically the same potential fall; for copper electrodes the fall waa about 3 per cont. and for aluminium electrodes about 1 per cent, less than for platinum, Tn hydrogen which contained a trace of oxygen, the potential fall for platinum, silver, copper, zinc, and steel was practically the same, about 300 volts, In the case af the last three metals, however, the value of the cathode potential fall at the beginning of the experiment was much lower than 300 volts, and it was not until after long sparking that it rose to its normal value; Warburg attributed thia to the presence at the beginning of the experiment of a thin film of oxide which gradually got dissipated by the sparking; he found by direct experiment that the potential fall of a purposely oxidised steel electrode was less than the value reached by bright steel electrode after fall of potential at tho cathodo in nitrogen which e traces both of moisture and oxygen was 260 volts, eg estes bah cy ete died tore wode fall of 343 volts: thus, in this case, a mere trace of had diminished the cathode fall by 26 per cent., the of the trace of oxygen produced equally remarkable ‘Art 160. ‘This points clearly to the influence exerted 158 ‘THR PASSAG) p [162. In hydrogen, Warburg found that a trace of aqueous vapour increased the potential difforence at the cathode instead of diminishing it as in nitrogen. 160.] Warburg (Wied. Ann. 40, p. 1, 1890) also investigated the effects produced by removing from the nitrogen or hydrogen any trace of oxygen that might havo beon present. This was done by placing eodium in the discharge tube, and then after the other gas had been let into the tubo, heating up the sodium, which combined with any oxygen there might be in the tube, ‘Tho effect of removing the oxygen from the nitrogen waa very remarkable: thus, in nitrogen free from oxygen, the fall of potential at the cathode when platinum electrodes were used was only 232 volts as against $43 volts when there was a trace of oxygen present; when magnesium electrodes were used the fall in potential was 207 volte; in hydrogen free from oxygen the fall of potential was 300 volts with platinum electrodes, and 168 yolts with magnesium electrodes; thus with platinum electrodes the potential fall in hydrogon is greater than in nitrogen, while with magnesium electrodes it is lees. 161,] Warburg also investigated a enso in which the conditions for chemical change at the cathode were as simple as possible, one in which the gas was mereury vapour (with possibly m trace of air) and the cathode a mercury surface; he found that the negative dark space was present, and that the cathode fall was very considerable, amounting to about 340 volts; this, at the pressures used in these experiments between 3-5 mm. and 14-0 mm., was much greater than the potential difference in a portion of the positive light about half as long again as the piece at the cathode, for which the potential fall was measured. 162.] In air free from carbonic acid, but containing a little moisture, Warburg (Wied. Ann, 31, p.559, 1887) found that the potential fall was about 340 volts: this is very nearly the value found by Mr. Peace for the smallest potential difference whieh would send ® spark between two parallel plates. When wo consider the theory of the discharge we shall see that there are reasons for concluding that it is impossible to produce a spark by a smaller potential difference than the cathode fall of potentin} in the gas through which the spark has to pass, The rescarches made by Hittorf on the distribution of poten- tial along the tube show, as we have soen, Art, 140, that the = = = jeowull 166.) EIROTRICITY THROUGH GASES. 161 after passing between the glass and the plates reached right up to the negative glow, which was above the negative plate: the space between the plates was quite dark and free from glow. a 6 Fig. 60. Lehmann {Molehularphysik, bd. 2, p. 295) has observed with a i the appearance of tho discharge passing between electrodes of different shapes, placed very close together; they exhibit in a very beautifol way the same pecnliarities as those just described ; Lehmann’s figures are represented in Fig. 69. M ——y 164 THE PASSAGE OF [r71. and Perry (Phil. Mag. @ 18, p. 848, 1988) taing's formula | which is identical with the preceding one if the sparks are not b6=21°6 volts, if / is measured in centimetres. The value of a probably depends on the quality of the carbon of whieh the elec- trodes are made, as other observers, who have also used carbon electrodes, have found considerably smaller values for «. When more volatile substances than carbon are used the values of u are smaller, the more volatile the substance the smaller in general being the value of «. This is borne out by the following deter- minations made by Lecher (Wied. Ann. 33, p, 625, 1888); the length ¢ in these equations ia measured in centimetres, and V in volta:— Horizontal Carbon Electrodes =. =. © V = 384 454. Vertical Carbon Electrodes... V = 35:54.571, Platinum Electrodes, (-5 cm. in diameter) V = 284414 Tron Electrodes, (-55 cm. in diameter) . V = 20+504 Silver Electrodes, (49 cm. in diameter). V = 8+60/. 170,] The form of the expression for V shows that the potential required to maintain the current between two incandescent electrodes cannot fall short of a certain minimum value, however short the are may be, The preceding measurements for « show that this potential difference, though small compared with the ‘cathode fall’ when the electrodes are cold, is much greater than that which Hittorf in his experiments (see Art. 152) found necessary to maintain a constant current when the cathode was incandescent ; we must remember, however, that in Lecher's experiments the gas was at atmospheric pressure, while in Hittorf’s the pressure was very low. 171.) Lecher (1. c.) investigated tho potential gradient in the are by inserting a spare carbon electrode, and found that it was far from uniform : thus whon the difference of potential between the anode and the cathode was 46 volte, there was a fall of 36 volts close to the anode, and n smallor fall of ton volts near the cathode. The result that the great fall of potential in the are discharge occurs close to the anode is confirmed by an experi- ment made by Fleming (Proc. Roy. Soc, 47, p. 123, 1890), in which w spare carbon electrode was put into the arc; when this electrode was connected with the anode sufficient current went ELECTRICITY THROUGH GASES. 165 he new cirouit to ring an electric bell, but when it was to tho cathode the current which went round the mit was not appreciable. _ 172] The term in the expression for the potential in Art. 1s, is independent of the length of the are, and whieh |) ievollves an expenditure of energy when clectricity travels acrons ‘su infinitesimally small air space, is probably connected with the work required to disintegrate tho electrodes, since the more Yolatile are the electrodes the smaller is this term, 173. The disintegration of the clectrodes is a very marked “feature of the are discharge, and it is not, as in the case when ‘epall currents pass through a highly exhausted gas, con- fined to the negative elec- trode; in fact, when carbon electrodes are used, the loss in weight of the anode is greater than that of the cathode, the anode getting hollowed out and taking erater-like form. 174.) Perhaps the most interesting examples of the ‘are discharge are those which ocour when we are able by ‘means of transformers to pro- duce a great difference of potential, say thirty or forty thousand volts between two ‘electrodes, and also to trans- mit through the are a very considerable current. In this Fig. 7h ease the arc presents the illustrated in Fig. 71. The dischango, instend of pass- ing in # straight line between the electrodes, rises from the electrodes in two columns which unite at the top, where strin- tions aro often seen though these do not appear in the photo. | graph from which Fig. 71 was taken. The vertical columns are sometimes from eighteen inches to two feet in length, they flicker slowly about and are very easily blown out,» very slight puff of =— 166 ‘THE PASSAGE OF (176. air being sufficient to extinguish them. ‘The air blast apparently, breaks the continuity of the belt of dissociated molecules along which the current passes, and the current is stopped just aa & current through « wire would be stopped if the wire were cut, The discharge is accompanied by a crackling sound, as if & number of minute sparks were passing between portions of the — are temporarily separated by very short intervals of space. 175.] The relation between the losses of weight of the anode and the cathode in the are discharge depends however very much on the material of which the electrodes are made; thus Mat- teucci (Comptes Rendus, 30, p. 201, 1850) found thnt for copper, silver and brass electrodes the cathode lost’ more than the anode, while for iron the loss in weight of the anode was greater than that of the cathode. ‘The electrodes in the are discharge are at an exceedingly high temperature, in fact probably the highest temperatures we can produce are obtained in this way. With carbon electrodes the anode ia much hotter than the cathode (compare Art, 149). Sinco the tomporature of the electrodes is so bigh, it is probable that they are disintegrated partly by the direet action of the heat and not wholly by purely electrical processes such ms those which occur in electrolysis; for this reason, we should not expect to find any simple relation between the loss in weight of the electrode and the quantity of electricity which has lari through the aro, Grove (Phil, Mag. [8] 16, p. 478, 1810), who used a zine anode sufficiently large for the temperature not to riye about its melting point, came to the conclusion that the amounts of zine lost and oxygen absorbed hy the electrode were chemically equivalent to the oxygen liberated in a voltamoter placed in the circuit. On the other hand, Herwig, (Pogy. Anim 149, p. 521, 1873), who investigated the relation between the loss of weight of a silver electrode in the are and the amount of chemical decomposition in a voltameter pluced in the same circuit, was however unable to find any simple law connecting the two. The brightness of the light given by carbon electrodes is much increased by soaking them in a solution of sodium sulphate. 176.) The particles projected from the electrodes in the are are presumably charged with electricity, since they are deflected by a magnet; thus some of the electricity passing for mere inerease of tem| : hy chemical changes seems to have little effect in increasing the luminosity of a gas; thus in one of Hittorf's already men- would secm that the interchange of atoma between the males cules which probably goes on when the discharge passes through the gas is much more effective in making it luminous than mere increase in temperature unaccompanied by chemical changes. 178.] Many experiments have been made by G, and E. Wiede- mann, Hittorf, and others on the distribution along the line of discharge of the heat produced by the spark. Hittorf's experi- ments are the easiest to interpret, since by means of a large battery he produced through the discharge tube « carrent which, if not absolutely continuous, was so nearly so, that no want of continuity could be detected either by a revolving mirror or by a telephono; the gas had therefore a much botter chance of getting into a steady state than if intermittent discharges such ‘as those produced by an induction coil had been used. Hittorf (Wied. Ann, 21, p. 128, 1884) inserted three thermo- meters in the discharge tube, one close to the cathode, another in the bright part of the negative glow, and the third in the positive column, He found, using small eurrenta and low gaseous pressures, that the temperature of the thermometer next the cathode was the highest, that of the one in the negative glow the next, and that of the one in the positive column the lowest. The distribution of temperature depends very much upon the intensity of the current, Hittorf found that when the strength wan increased the ditference between the temperatures of his thermometers increased also. When however the increase in the current is 80 great that the discharge becomes an arc discharge, then, at any rate when carbon electrodes are used, the temperature at the anode is higher than that of the cathode; with weak currents we have seen that it is lower. E. Wiedemann (Wied. Ann. 10, p. 225 et seq.,1880) found that the distribution of temperature along the discharge depended on the pressure. In his experiments the temperature at the anode | 170 ‘THK PASSAGE OF [s8r. ‘Thus, if for example the electrodes are spheres of different sizes, Faraday (Zwperimental Researches, § 1480) found that the spark length was greater when the smaller sphere was positive than when it was negative. We may express this result by saying that when the electric field is not uniform the gas does not break down so easily when the greatest electromotive intensity is at the cathode as it does when it is at the anode, Macfarlane’s measurements (Phil. Mag. [5] 10, p. £03, 1880) of the potential difference required to start a discharge between a ball and a dise are in accordance with this result, as he found that for a given longth of spark the potential difference between the electrodes was smaller when the ball was positive than when it was negative. Fig. 74. Fig. 75. 180.] De la Rue and Hugo Miller (Phil. Trans. 1878, Part I, p. 55) observed analogous effects in the experiments they made with their large chloride of silver battery on the sparking distance between a point anda dise, They found that for potential differ- ences between 5000 and 8000 volts the sparking distance was greatest when the point was positive and the disc negative, while for smaller potential differences they found that the opposite result was true. The appearance of the discharge at the positive point they found was different from that at the negative. The discharge at the negative point is represented in Fig. 74, that at the positive in Fig, 75. 181.] Wesendonck (Wied. Ann, 38, p. 222, 1889), however, concludes from his experiments that there are no polar differ- ences of this kind when the discharge passes entirely as a spark, and that the differences which have been observed are due to the coexistence of other kinds of discharge such on o brush and glow. The existence of this kind of discharge would put the gas into a condition in which it is electrically weak and thue ill- E 172 THE PASSAGE OF ~~ [183. Lichtenberg’s Figures and Kundt'a Dust Figures. 183] Very tangible differences between the discharges from the positive and negative electrodes at ordinary pressures are obtained if we allow the discharge from one or other of the electrodes to pass on to a non-conducting plate covered with Wig. 77. some badly conducting powder. If, for example, we powder @ plate with a mixture of red minium and yellow sulphur and then cause a discharge from a positively eloctrified point to pass to the plate, the sulphur, which by friction against the minium is negatively electrified, adhores to the positively electrified parts of the plate, and will be found to be ——= arranged ina star-like form like that represented in Fig. 77. If, on the other hand, the discharge is taken from a negatively electrified body the appearance of the minium on the plate is that represented in Fig. 78. These are} known as Lighton- borg’s figures ; the positive ones are larger than the negative, If the electrodes are made of very bad conductors, such as wood, there is no difference between the positive and the negative figures. _ _- BLECTRICITY THROUGH Gasks. 173 steed Hh oa pea at wi 0 coil made to touch the powdered surface of the glass. When the discharge passes the powder arranges itself in patterns which are finoly branched and have a moss-like appear- | ance at the anode and a more feathery or lichenons appearance at the cathode. The accompanying figuro is from a paper by Joly (Proe. Roy. Soe. 47, p. 84, 1890) ; the negative electrode is on the Toft. Fig. 19. 185.] As Lehmann hos remarked (Molekularphysik, b. 11, ‘p. 303), the differences between the positive and negative figures ‘are what we should expect if the discharge passed as a brash from the positive electrode and as a glow from the negative one. He has verified by diroct obsorvation that this is frequently the case. ” Fig, 80. is also, I think, thrown on the difference poe and wou figures by Fig, 80, whieh Hugo Miller (Phil. Trans, 1878, arge produced by 11,000 of their rile | 174 THE PASSAGE OF (187. chloride of silver cells in free air. It will be noticed that there is at the negative electrode a continuous discharge superposed on the streamers which are the only form of discharge at the positive, this continuous discharge will fully account for the comparative want of detail in the negative figure, 186.] Kundt’s figures are obtained by scattering non-conduct- ing powders over a horizontal metal plate, inatead of, as in Liehtonborg’s figuros, over a non-conducting one. If tho plate be shaken after a discharge has passed from a negative point to the positive plato, it will be found that the powder will fall from every part of the plate exeept a small circle under the negative electrode, where the powder sticks to the plate and forms what is called Kundt’s ‘dust figure.’ The dimensions of this circle are very variable, ranging in Kundt’s original experiments (Pegg. Amn. 136, p. 612, 1869) from 10 to 200 mm. in diameter. If the point is positive and the plate negative Kundt’s figures are only formed with great difficulty. Mechanical Effects produced by the Discharge. 187.] We have already considered the mechanical effects pro- duced by the projection of particles from the cathode: many other such effects are however produced by the electric discharge. One of the most interesting of these is that described by De la Rue and Hugo Miiller (Phil. Trans. 1880, p. 86): they found that when the discharge from their lange chloride of silver battery passed through air at the pressure of 63 mm. of mercury, the pressure of the air was increased by about $0 per cent., and they proved, hy meastring the temperature, that the increase in pressure could not be accounted for by the heat produced by the spark. ‘This effect con easily be observed if a pressure gauge is attached to any ordinary discharge tube, the gas inside being moat conveniently at a pressare of from 2 to 10 mm. of mercury. At the passage of cach spark there is a quick movement of the liquid in the gauge as if it had been struck by a blow coming from the tube; immediately after the passage of the spark the liquid in the gauge springs back to within a short distance of its position of equilibrinm, and then slowly creeps back the rest of the way. his creeping effect is probably due to the slow escape of the heat produced by the passage of the spark. \ 4 = 194) ELECTRICITY THROUGH GASES. iit intervals; 4 magnified representation of these is shown in Fig. 82, taken from Joly’s paper, When the air betwee the Plates was replaced by hydrogen these furrows hadm tendency to be more widely separated. 193.) The explosive effects produced by the spark are well luatrated by an experiment due to Hertz (Wied. Ann. 19, P- 87, 1985), in which tho anodo was placod at the bottom of # glass tube with a narrow mouth, while the cathode was placed Fig. £2 ‘utaide the tubo and close to the open ond. The tube and the electrodes were in a bell jar filled with dry air at a pressure of 40-50 mm. of mercury. When the discharge from a Leyden jarcharged by an induetion-coil passed, the glow aceompanying it was blown out of the tube and extended several centimetres from the open end. In this experiment, as in the well-known ‘electric wind," the explosive effects seem to be more vigorous at the anode than they are at the cathode. Chemical Action of the Electric Discharge. 194.] When the electric discharge passes through a gas, it produces in the majority of cases perceptible chemical changes, though whether these changes are due to the electrical action of the spark, or whether they are secondary effects due to a great increase of temperature occurring cither at the electrodes or along the path of the discharge, is very difficult to determine when the discharge takes the form of a bright spark. = Sapient ty ctipeching tied ana a Seat wants cbt wines aaa through liquid electrolyte the substances Hhenied at the electrodes are im consequence of secondary chemical acticns frequently different from the ions which carry the current. 196.] A very convenient method of producing discharges as free 8 possible from great heat is by using a Siemens’ azonizer, repro sented in Fig. 83, Two glass tubes are fused together, and the gas through which the discharge takes place circulates between them, entering by one of the side tubes and leaving by the othor; the inside of the inner tube and the outside of the outer are coated with tin-foil, and are connected with the poles of an "rr | so ‘THE PASSAGE OF [199. fluid; nitrogen partly combines with ammonia: carbonic oxide and hydrogen give a solid product: carbonic oxide and marsh gas n resinous substance: nitrogen and hydrogen ammonia. _ Dextrine, benzine, and sodium absorb nitrogen under the influence of the dischargo, and enter into chemical combination with it. Hydrogen forms with benzine and turpentine resinous 198,] Perthelot (Annales de Chimie et de Physique, [5], 10, ). 65, 1877) has shown that the absorption of nitrogen by flechsine takes plang adder vecy anal slacecen Mpeg? he showed this by connecting the inside and the outside of the ozonizer to points at different heights above the surface of the ground, and found that this difference of potential, which varied in the course of the experiments from +60 to —180 volta, was sufficient to produce in the course of a few weeks an appre- ciable absorption of nitrogen by a solution of doxtrine in contact with it, The potential differences in these experiments were 80 amall, and their rate of variation so slow, that it seems im- probable that any discharge could have passed through the nitrogen, and the experiments suggest that chemical action between a gas and a substance with which it is in contact can he produced by the action of a variable electric field without the passage of electricity through the bulk of the gas. Berthelot suggests that plants may, under the influence of electricity, absorb nitrogen by an action of this kind, This suggestion also raises the very important question as to whether the chemical changes which accompany the growth of plants can have any influence on the development of atmospheric electricity. 199.] We must now consider the relation between the quantity of electricity which passes through a gas and the amount of chemical action which takes place in consequences, It is neces- sary here to make a distinetion, which has been too much neglected, between the part of this action which occur at the electrodes and the part which occurs along the length of the spark. When a current of electricity passes through a liquid electrolyte the only evidence of chemical decomposition is to be found at tho electrodes. When, however, the electric dis- charge passes through a gas tho chemical changes are not con- finod to the electrodes but occur along the line of the discharge as well, This is proved by the fact that when the electrodeless - ELECTRICITY THROUGH GASES, 18t Aischarge passes through oxygen ozone is produced, as is testified ‘by the existence for several seconds after the discharge has passed of beautiful phosphorescent glow: the came thing is also proved by the behaviour of the discharge whon it passes through acetylene ; the first two or three sparks are of a beautiful Tight green colour, while all subsequent discharges are a kind of whitish pink, showing that the first two or three sparks have the gas. 200,] Since chemical decomposition is not confined to the elec- trodes its amount must depend upon the length of the spark ; ‘this has been proved by Perrot (Annales de Chimie et de Phy- sigue [3], 61, p. 161, 1861), who compared the amounts of water ‘yapour decomposed in the same time in a number of discharge tabes placed in series, the spark lengths in the tubes ranging from two millimetres to four centimetres; he found that the yolumes of gas decomposed varied from 2 ce. to 52 ce., and that neither the longest nor the shortest spark produced the maximum effect. By placing a voltameter in the cireuit Perrot found that ‘in one of his tubes the amount of water vapour decomposed by the sparks was about 20 times the amount of water decomposed ‘im the voltometer. It is evident from this that if we wish to ‘arrive at any simple relation betweon the quantity of electricity ‘passing through the gas and the amount of chemical decomposi- tion produced we must separate the part of the latter which ‘occurs along the length of the spark from that which takes place electrodes, 201.) This seems to have been done in a remarkable investi- made more than thirty years ago by Perrot (Le.), which ‘does nob seem to have attracted the attention it merits, and whieh would well repay repetition. The apparatus used by Perrot in his exporiments is represented in Fig. 84 from his paper. The spark passed between two platinum wires sealed into gloss tubes, ef 7, df g, which they did not touch excopt at the ‘places where they wore sealed: the open ends, ¢, «, of theao tubes “were about 2 mm. apart, and the wires terminated inside the tubes at a distance of about 2 mm, from the ends. The other ends of these tubes wore inserted under test tubes e ¢, in which ‘the gases which passed up the tubes were collected, Tho air was aa eat etistscn Un gales races Unougn viaak the discharge passed was obtained by heating the water in the ¥ nEiaiii | 182 | THE PAssacy oF " bor ‘vessel to about 90°C. : special precautions were taken to free this water from any dissolved gas. The stream of vapour arising from this water drove up the tubes the gases produced by the passage of the spark; part of these gases was produced along the length of the spark, but in thie case the hydrogen and Fig. 84. oxygen would be in chemically equivalent proportions ; part of the gases driven up the tubes would however be liberated at the olectrodes, and it is this part only that we could expect to bear any simple relation to the quantity of electricity which had passed through the gan. When the sparking had ceased, the gases which had collected in the test tubes ¢ and ¢ were analysed; in the tirst place they were exploded by sending a strong spark through them, this at onee got rid of the hydrogen and oxygen which existed in chemically equivalent proportions and thus got rid of the gas produced along the length of the spark. After the explosion the gases loft in the tubes were the hydrogen or oxygen in excess, together with a small quantity of nitrogen, due to a little air which had leaked into the vessel in the course of the experimonta, or which had been absorbed by the water. The results of these analyses showed that there was always an excoss of oxygen in the test tube in connection with the positive olectrode, and an excess of hydrogen in the test tube connected with the negative electrode, and also that the amounts of oxygen and hydrogen in the respective tubes were very nearly chemically equivalent to the amount of copper deposited from « solution of copper sulphate in a voltameter placed in series with the discharge tube. — | 1st THE PASSAGE OF [202 Gas in the test tube over the negative electrode 27-70 c.c. ; excess of hydrogen 18 ¢.c.; nitrogen +21 ec. 6th experiment. Duration of experiment 34 hours Copper ‘ited in voltameter 6 milligrammes, chemically equivalent to "242 co. of hydrogen and to 1-06 ec. of oxygen. Gas in the test tube over the positive electrode 30-20 cc. ; excess of oxygen -90 c.c.; nitrogen -2 c.c, Gas in the test tube over the negative pole 2-50 ¢.c,; excess of hydrogen 2-05 ¢.c.; nitrogen +2 ¢.c. These results seem to prove conclusively (assuming that the discharge passed straight between the platinum wires and did not pass through a layer of moisture on the sides of the tubes) that the conduction through water vapour is produced by ehemi- ca} decomposition, and also that in a molecule of water vapour the atoms of hydrogen and oxygen are associated with the same clectrica} charges as they are in liquid electrolytes, 202.] Anotherway in which the chemical changes which accom- Pany the passage of the spark through a gas manifest themselves is by the production of a phosphorescent glow, which often laste for several seconds after the discharge has ceased. Tnagreat many gases this glow does not oceur, it is however extremely bright in oxygen. A convenient way of producing the glow is to take a tube about a metre long filled with oxygen at a low pressure, and produce an electrodeless discharge at the middle of the tube. From the bright ring produced by the discharge a phospho- rescent haze will spread through the tube moving sufficiently slowly for its motion to be followed by the eye. The haze seems to come from the ozone, and the phosphorescence to be due to the gradual roconversion of the ozone into oxygen. This view is borne out by tho fact that if the tube is heated the glow is not formed by the discharge, but as soon as the tube is allowed to cool down the glow is again produced: thus the glow, like ozone, cannot exist at a high temperature. The spectrum of this glow in oxygen is « continuous one, in which, however, few bright lines can be observed if very high dispersive power is used. The glow is also formed in air, though not so brightly as in pure oxygen. When electrodes are used it seems to form moat readily over the negative electrode, especially if thie is formed of « flat surface of sulphuric acid. Ihave experimented with a large number of gases in order to | 186 ‘ THE PAassaoh OF machine led toa thin wire placed about6 mm. above « plate e which ‘was connected to the earth. A glow discharge passed between the- wire and the plate, and the difference of potential between the in- side and outside coatings of the jar 8 was constant and equal to about 12 electrostatic units, When the knobs of the air-break F ‘were pushed suddenly together a spark about 5 mm. in length was produced at F, and in addition a bright spark 5 mm. long jumped across the sir spaco at ¢ whore there was previously only a glow, The passage of the spark at F put the two condensers 8 and c into olectrical communication, and this was equivalent to increasing the capacity of g by about one part in a thousand ; this alteration in the eapacity produced a correspand- ing diminution in the potentinl difference between its coatings. This disturbance of the electrical equilibrium would give rise to amall but very rapid oscillations in the potential difference be- tween the wire and the plate ¢, and this variable field seemed able to send a spark across e, where when the potential was steady but a glow was to be seen, 204.) It thue appears that gus ia electrically weaker under oscillating electric fields than under stondy ones, for it is not apparent why the addition of the capacity of the small condenser to that of 8 should produce any considerable difference in the electromotive intensity ate. Tt is trae that while the discharge is oscillating the tubes of electrostatic induction are not distri- buted in the same way as they are when the field is steady, and sme concentration of these tubes may very likely take place, but it does not seem probable that the disturbance produced by so small a condenser would be sufficient to account for the lange effects observed by Jaumann, unless, as he supposes, the gas is electrically weaker in variable electric fields. , Another point which might affect the electromotive intensity ate is the following: the comparatively small difference of potential between the wire and the plate is partly due to the glowing air-space at ¢ acting as a conductor, this eonductivity is due to dissociated molecules produced by the discharge, and it is likely that this would exhibit what are called ‘unipolar’ properties, that is, that its conductivity for a current im one direction would not be the same as for one in the opposite. Even when the change produced in the distribution of elec- tricity is not so great as that duo to an actual reversal of the of the constituents of the electrolyte and in the closo connection, oxpreseod by Faraday’s Laws, between the quantity of electricity transferred through the electrolyte and the amount of chomical change produced, that né one ean doubt the importance of the part played in this case by chemical decomposition in the trans- mission of the electric current. 209.] When electricity passes through gases, though there is (with the possible exception of Perrot’s experiment, see Art, 200) no one phenomenon whose interpretation is so unequivocal as some in electrolysis, yet the consensus of evidence given by the very varied phenomena shown by the gascous discharge seems to point strongly to the conclusion that here, as in electrolysis, the discharge is accomplished by chemical agency. Perrot, in 1861, secms to have been the first to suggest that the discharge through gases was of an cloctrolytic nature. In 1882 Giese (Wied. Ann. 17, pp. 1, 236, 519) arrived at the same conclusion from the study of the conductivity of flames. Before applying this view to explain in detail the laws govern- ing tho clectrie discharge through ynses, it seems desirable to mention one or two of the phenomena in which it is most plainly suggestod, The experiments bearing most directly on this subject are those made by Perrot on the decomposition of steam by the dis- charge from a Rubmkorff’s coil (see Art, 200). Perrot found that when the discharge passed through steam there was an excess of oxygen given off at the positive pole and an excess of hydrogen at the nogative, and that these excesses were chemically oquiva~ lent to each other and to the amount of copper deposited from a voltameter containing copper sulphate placed in series with the discharge tube. If this result should be confirmed by subsequent: researches, it would be a direct and unmistakeable proof that the passage of clectricity through gases, just as much as through electrolytes, is effected by chemical means. It would also show that the charge of electricity associated with an atom of an Z — “192 THE PASSAGE OF 3. charge is accomplished beh an hat mention Art. 38, that the halogens chlorine, bromine, and iodine, which ‘are dissociated at high tomperatures, and which at ench tem- peestanie hayes seedy andetgns Oech ee to conduction, have then Jost all power of ioeelation and allow electricity to pass through them with ense. Then, again, we have the very interesting result discovered by R. y. Helmholtz (Wied. Ann, 32, p. 1, 1887), that a gus through which electricity is passing and one in which chemical changes are known to be going on both affect a steam jet in the same way. 212.) Again, one of the most striking features of the discharge through gases is the way in which one discharge facilitates the passage of a second; the result is trae whether tho discharge passes between electrodes or as an endless ring, as in the experi~ monts deseribed in Art. 77, Closely connected ‘with this effect ig Hittorf's discovery (Wied. Ann. 7, p. 614, 1879) that a few galvanic cells are able to send a current throngh gas which ix conveying the electric discharge. Schuster (Proceedings Royal Soc, 42, p. 371, 1887) describes a somewhat similar effect. A large discharge tube containing air at a low pressure was divided into two partitions by a metal plate with openings round the perimeter, which served to screon off from one com~ partment any electrical nection occurring in the other, if a vigorous discharge passed in one of these compartments, the electromotive force of about one quarter of a volt was sufficient to send a current through the air in the other, Since such electromotive forees would not produce any dis- charge through air in its normal state, these experiments suggest that the chemical state of the gas has been altered by the dis- change. 213.] We shall now go on to discuss more in detail the conse- quences of the view that dissociation of the molecules of a gas always accompanies eloctric dischargo through gases, We notice, in the first place, that the separation of one atom from another in the molecule of a gas is very unlikely to be produced by the an- aided agency of the external electric field. Let us take the case ofa molecule of hydrogen as an example; we suppose that the molecule consists of two atoms, one with a positive charge, the other with an equal negative one. The most obvious assumption, \ — 194 THE PASSAGE OP [214 that the resultant effect is very small; when, however. the medium is polarized, order is introduced into tho arrangement of the molecules, and the inter-molecular forees by all tending in the anme dircetion may produce very large effects. 214.) The arrangement of the molecules of a gas in the electric field and the tendency of the inter-molecular forces may be illus- trated to some extent by the aid of a model consisting of a large number of similar small magnets suspended by long strings attached to their centres. The positive and negative atoms in the molecules of the gas are represented by the poles of the magnets, and the forces between the molecules by those between the magnets. The way the molecules tend to arrange them- selves in tho eloctric field is represented by the arrangement of the magnets in a magnetic field. The analogy between the model and the gas, though it may serve to illustrate the forces between the molecules, is very im- perfect, a8 the magnetsare almost stationary, while the molecules are moving with great rapidity, and the collisions which occur in consequence introduce effects which are not represented in the model. The magnets, for example, would form long chains similar to those formed by iron filings when placed in the magnetic field ; in the gas, however, though some of the molecules would form chains, they would be broken up into short lengths by the bombardment of other molecules. The length of these chains would depend upon the intensity of the bombardment to which they were subjected, that is upon the pressure of the gas ; the greater the pressure the more intense the bombardment, aud therefare the shorter the chain. We shall call these chains of molecules Grotthus’ chains, because we suppose that when the discharge posses through the gas it passes by the agency of these chains, and that the same kind of interchange of atoms goes on amongst the molecules of these chains as on Grotthus' theory of electrolysis goes on between the molecules on a Grotthus’ chain in an electrolyte. ‘The molecules in such a chain tend to pull each other to pieces, and the foree with which the last atom in the chain is attracted to the next atom will bo much smaller than the force betwoon two atoms in an isolated molecule ; this atom will therefore be much more easily detachod from tho chain than it would from a single molecule, and thus chemical change, and therefore electric — 4 — 196 THE PASSAGE OF [au many experiments have shown that there is no appreciable dis- eee ta the lines of the spectrum of the gas in the discharge tube when the discharge is observed end on, while if the mole- cules were moving with even a very small fraction of the volosity of light, Dippler's principle shows that there would be a measur- able displacement of the lines. It does not indeed require spee~ analysis to prove that the molecules cannot be moving with half the velocity of light; if they did it ean easily be shown that the kinetic energy of the particles carrying the discharge of a condenser would have to be greater than the potential energy in the condenser before discharge, When, however, we consider the discharge as passing along these Grotthus’ chains, since the recombinations of the ditieront molecules in the chain go on simultaneously, the electricity will pass from one ond of the chain to tho other in the time roquired for an atom in one molecule to travel to the oppositely charged atom in the next molecule in the chain, Thus the velocity of the discharge will exesed that of the individual atoms in the pro~ portion of the length of the chain to the distance between two adjacent atoms in neighbouring molecules. This ratio may be very large, and we can understand therefore why the velocity of the electric discharge transcends so enormously that of the atoms. 217.] We thus see that the consideration of the smallness of the electromotive intensity required to produce chemical change or discharge, as well as of the enormous velocity with which the discharge travels through tho gas, has led us to the conclusion that a small fraction of the molecules of the gas are held together in Grotthus’ chains, while the consideration of the method by which the discharge passes slong these chains indicates that the spark through the gas consists of a series of non-contemporaneous discharges, the discharge travelling along one chain, then waite ing for a moment before it passcs through the next, and so on. It is remarkable that many of the physicists, who have paid the greatest attention to the passage of through gases, Sagas feat ici git hry, hads’ chenrvasinsstojthin ceca the electric diechange is made up of a lango number of separate The behaviour of atris under the action of magnetic foreo is one of the chief reasons for coming to this eonelusion. On this point Spottiswoode and Moulton (Phil. Trans. 1879, part 1, 219,] As far as phenomena connected with the electric discharge are concerned, the Grotthus’ chain is the unit rather than the swhere the/fandamontal length ia that of the mean free path of the molecules. 220.) Peace's discovery that the density—which we shall eall the critical density—at which the ' electric strength” of the gas is a minimum depends upon the distance between the electrodes, proves that the gas, when in an electric field sufficiently intense to produce discharge, possesses a structure whose length scale is comparable with the distance between the electrodes when these aro near enough together to influence the critical density. As this distance is very much greater than any of the lengths recognized in the ordinary Kinetic Theory of Gases, the gas when under the influence of tho electric ficld must have a structure very much coarser than that recognized by that theory. eect eames consists in the formation of Grotthus’ —" 222] ‘ELECTRICITY THROUOM GASES, 199 221] The striations are only clearly marked within somewhat narrow limits of pressure. But it is in accordance with the con clusion which all who have stadied the spark have arrived at— ‘that there is complete continuity between the bright well-defined spark which occurs at high pressures and the diffused glow which represents the dischargo at high exhaustions—to suppose that they always exist in the apark discharge, but that at high pressures they are so closo together that tho bright and dark ‘parts cease to be separable by the eye. ‘The view wo have taken of tho action of the Grotthus’ chains ‘in propagating the electric discharge, and the connection between these chains and the striations, does not require that every dis- only be under somewhat exceptional circumstances that the con- ditions would be regular enough to give rise to visible striations, 222,] We ehall now proceed: to consider more in detail the wchegens of the preceding ideas to the phenomena of the The first case we shall consider is the caleu- ‘the path rapa eich im ‘The striations on the proceding view of the discharge may, sinco they are equivalent to a bundle of Grotthuy' chains, be regarded as forming a series of little electrolytic cells, the beginning and the end of a strin corresponding to the electrodes of the cell. Let #” be the electromotive intensity of the field, A the length of a strin, then when unit of electricity passes through the stria the work dono on it by tho olectric field is FA, Tho passage of tho olec- tricity through the etria is accompanied just as in the case of the electrolytic cell, by definite chemical changes, such as the ition of a certain number of molecules of the gaa; thus if w is the increase in the potential energy of the gas due to the changes which occur when unit of electricity passes through the stria, then neglecting the heat produced by the current we have by the Conservation of Energy Pr=w, or the difference in potential between the beginning and end of a strin is equal tow. If the chemical and other changes which take place in the consecutive strim are the same, the potential difference due to each will be the same also. There is however one stria which is under different conditions from the others, viz. that next the negative electrode, i.o. the negative dark space, For in the body of the gas,the ions set free at an extremity of the stria, are set free in close proximity to the ions of opposite sign at the extremity of an adjacent stria, In tho stria next the electrode the ions at one end are set free against a metallic surface. The experiments described in the account we have already given of the discharge show that the chemical changes which take place ot the cathode are abnormal; ono reason for this no doubt is the presence of the metal, which makea many chemical changes possible whieh could not take place if there were nothing but gas present. This stria is thus under excep. tional cireumstances and may differ in size and fall of from the other stria:, Hittorf's experiments, Art. 140, show that the fall of potential at the cathode iaabnormally great, If we call this potential fall and consider the case of discharge between two parallel metal plates ; the discharge on this view, starting from the positive cloctrodo, goes consecutively across a number n of a | 202 THE PASSAGE OP [223 molecules of the gas ; hence we see why the change in the ‘eleetric strength’ of a gas takes place when the spark length is very large in comparison with lengths usually recognized in the Kinetic Theory of Gases. According to formula (3), the curve representing the relation between electromotive intensity and spark length isa rectangular hyperbola ; this is confirmed by the curves given by Dr, for air, carbonic acid, oxygen and coal gas (see Fig. 19), and by those given by Mr. Peace for air. 228.] The preceding formule are not applicable when the dis- tance between the electrodes is less than A, the length of the atria noxt the cathode. But if the discharge passes through the gas and is not carried by metal dust torn from the electrodes we ~ ean onsily sco that tho electric strength must increase as the distance between the electrodes diminishes. For as wehave seen, the molecules which are active in carrying the discharge are not torn in pieces by the direct action of the electria field but by the attraction of the neighbouring molecules in the Grotthus’ chain. Now when we push the electrodes so near together that the distance between them is less than the normal length of the chnin, we take away some of the molecules from tho chain and 80 make it more difficult for the molecules which remain to split up any particular molecule into atoms, so that in order to effect this splitting up we must increase the number of chains in the field, in other words, we must inerease the electromotive intensity. Peace's curves, Fig. 27, showing the relation between the potential difference and spark length are exceedingly flat in the neighbourhood of the eritical spark length. This shows that the potential difference required to produce discharge increases very slowly at first as the spark length is shortened to less than the length of a Grotthus’ chain. We now procoed to consider the relation between the spark potential and tho pressure. As wo havo already remarkod, tho length of a Grotthue’ chain depends upon the density of the gas; the donsor the gas the shorter the chain: thia is illustrated by the way in which the strim lengthen out when the pressure is reduced. The experiments which have been made on the connection between the length of a stria and the density of the gas are not sufficiently decisive to enable us to formulate tho pressure diminishes. There will thus be a density at which tho electric strength of the gas is a minimum, and that density will be the one at which the length of the stria next the eathode is equal or nearly equal to the distance between the electrodes. ‘Thus the length of a stria at the minimum strength will have to be very much less when the electrodes are very near than when they ore far apart, and since the stria-length is less tho density at which the ‘electric strength’ is a minimum will be very much greater when the electrodes are near together than when they are far apart. This is most strikingly exemplified in Mr. Peace’s experiments, for when the distance between the electrodes was reduced from 1/5 to 1/100 of a millimetre the critical pressure was raised from 30 to 250 mm. of mercury. ‘The mean free path of a molecule‘of air at ® pressure of 40 mm. is about 1/400 of n millimetre. 225.] The existence ofa critical pressure, or pressure at which the electric strength is a minimum, when the discharge passes between electrodes can thus be explained if we recognize the formation of Grotthus’ chains in the gas, and the theory leade to the conclusion which, a8 we have seen, is in secordance with the facts, that the critical pressure deponds on the epark Jength. 226.] Wo have seen that when the distance between the elec- trodes is lees than the length of the stria next the negative electrode, the intonsity of the field required to produce discharge will increase as the distance between the electrodes diminishes. Peace’s observations show that this increase is so rapid that the potential difference between the electrode when the spark passes increases when tho spark length is diminished, or in other words, that the electromotive intensity increases more rapidly than the reciprocal of the length of a Grotthus' chain. This will explain the remarkable results observed by Hittorf (Art. 170) and Lohmann (Art, 170) whon the electrodes wore placed very near together in a gas at a somewhat low preseure. In such cases it was found that tho dischange instead of passing in the straight line between the electrodes took a very roundabout course. To oxplain this, suppose that in the experiment shown in Fig. 68 = 206 THE PASSAGE OF [228. those noar tho anode, Those results might at first sight seem in- consistent with the experiments we have described (Art. 40) om the cloctrical effect on metal surfaces of ultra-violet light and incan- descence. In these experiments we saw that under such influences nogative electricity eseaped with great ease from a metallic electrode, while, on the other hand, positive electricity had great difficulty in doing so In the ordinary discharge through gases it seems, on the contrary, to be the positive electricity which escapes with case, while the negative only escapes with great difficulty. We must remember, however, that the vehicle conveying the olec- tricity may not be the same in the two eases. When ultra-violet light is incident on a metal plate, there seems to be nothing in the phenomena inconsistent with the hypothesis that the negative electrification is carried away by the vapour or dust of the metal. In the caso of vacuum tubes, however, the electricity is doubtless conveyed for the most part by the gas and not by the metal, In order to get the electricity fram the gas into the metal, or from the metal into the gas, something equivalent to chemical combination must take place between the metal and the gas. Some experiments have been made on this point by Stanton (Proc. Roy. Soc. 47, p. 659, 1890), who found that a hot copper or iron rod connected to earth only discharged the electricity from a positively electrified conductor in its neigh- bourhood when chemical action waa visibly going on over the surface of the rod, e.g. when it was being oxidised in an atmo- sphere of oxygen. When it was covered with a film of oxide it did not discharge the adjacent conductor; if when coated with oxide it was placed in an atmosphere of hydrogen it discharged the cloctricity as long as it was being deoxidised, but as soon as the deoxidation was complete the leakage of the cloctricity stopped. On the othor hand, when the conductor was nogatively electrified, it leaked even when no apparent chemical action was place. Ihave myself observed (Proc. Roy. Soc, 49, p. 27, 1891) that the facility with which electricity passed from a gus to 2 metal was much increased when chemical action took place. If this is the case, the question as to the relative ease with which the electricity escapes from the two electrodes through « vacuum tube, depends upon whether # positively or negatively electrified surface more readily enters into chemical combination with the adjacent gas, while the sign of the electrification of a metal ad 212 CONJUGATE FUNCTIONS. [235- whore fand F are known functions; eliminating ¢ between these equations we get é ot =x(otey), which gives ue the solution of our problem. 235,] We shall now proceed to consider the application of this method to some special problems. The first ease we shall consider is the one discussed by Maxwell in Art, 202 of the Electricity and Magnetism, in which a plate bounded by a straight edge and at potential Vis placed above and parallel to an infinite plate at zero potential. The diagrams in the = and w planes are given in Figs. 90 and 91 respectively. — Di=to i= T=-1 Ls fal Fig. 90. Ls H F an Fig @. ‘The boundary of the < diagram consists of the infinite straight line AB, the two sides of the line cp, and an are of a circle stretching from «= — on the line AB to @ = + on the line cp. We may assume arbitrarily the values of ¢ corresponding to three corners of the diagram, we shall thus aseume tf = — © at the point #= — on the lino AB, f= —1 at the paint 2=+ on the same line,and ¢=0atc. The internal angles of the polygon are zero at a and 27 at c; hence by equation (1), Art. 231, the Schwarzian transformation of the diagram in the ¢ plane to the real axis of the ¢ plane is dz t at ‘The diagram in the w plane consists of two parallel straight lines ; the internal angle at G, the point corresponding to t = —1, is zero; hence the Schwarzian transformation to the real axis of t is dw 1 Feed Es (4) = E [SSE 216 CONJUGATE FUNCTIONS. [236 value, On the upper side of the plate, however, when « is a large multiple of 4, ¢ is approximately equal to ne od #0 that the density varies inversely as the distance from the edge of the plate. The capacity of a breadth a of the upper plate, i.o. the ratio of the charge on both surfaces to V, is parkities x wee fteF tlog(1+ Th) We see by the principle of images that the distribution of electricity on the upper plate ia the same as would ensue if, instead of the infinite plate at zero potential, we had another semi-infinite parallel plate at potential —V, at a distance 2h below the upper plate, and therefore that in this case the capacity of a breadth o, when a/A is large, of either plate is app oximately cd Aik ne past St og {1+ 52 410g (14 5 *)t]- 286,] The next case we shall consider is the one discussed by Maxwell in Art. 195, in which a semi-infinite conducting plane is placed midway between two parallel infinite conducting planes, maintained at zero potential; we shall suppose that the potential of the semi-infinite plane is V. The diagrams in the ¢ and w planes are given in Figs, 92 and 93 rospectively. tn tcoF- E test —_—___—_———— # ‘tao 12 tence A Beat Fig. 02, tay roe | fte+1 fate =o 7a =i Fig. 99. The boundary of the ¢ diagram consists of the infinite line AB, the two sides of the semi-infinite line cD, and the infinite 220 CONJUGATE FUNCTIONS. [237. the value unity y increases by H —h. When ¢ is nearly unity we may put t=14Re% where Fis small, and @ changes from 7 to zero as ¢ passes through unity. When ¢ is approximately 1, equation (13) becomes dz. 1 Fried cia hence the increase in z as t passes through 1 is IC —a!} [log R+.0}, =— ‘7% 0(1—a%)t, =-9 c(1—a*)t, but since the increase in z when ¢ passes through this value is «(H—h), we have H-h=—C5(1—a"}t, When ¢ changes from + © to —, z diminishes by 12H; but when ¢ is very large, equation (13) becomes de _C at’ z=Clogt. Now t=Re'%, where R is infinite, and 6 changes from 0 to 7 as ¢ changes from + to —e; but as ¢ changes from plus to minus infinity, 2 in- creases by C log R+ 0}, =iCn, and since the diminution in z is 1.2 H, we have Gi H=-C;- Thus h= H{1— Vi—a}, or a =)/ ee 244 CONJUGATE FUNCTIONS. [249. side of aB is (without any limitation as to the value of k) equal to 1 do 4m da’ and since z= beng, da ag om odng = Foe ie oath =- hop VEP.EP; hence the surface density is equal to 6 1 ~4ak CRAP. BE 249.] We pass on now to consider the transformation ety © ® =en(o+y), where ¢ is taken as the potential and y as the stream function. Over the equipotential surface for which $ = 0, we have ety e & =en(y) Pana ~ en(y, Fk) Hence y=0, +2b, +20b,...5 while 2 ranges from 0 to infinity. For the equipotential surface for which ¢ = K, we have ees e & =en(K+y) yp mh®) =o ange) Hence ystitnb, t3nb, +$nb..., while ranges from minus infinity to a value 2 given by the equation En ak, Thus this transformation givos the distribution of electricity | 248 CONJUGATE FUNCTIONS. [es2. ‘The surface density at a point q on EF may be shown in a similar way, using (36), to be equal to b 2 “aie ey Riko me i ~ 47 {AQ.BQ.cQ.zQ]! 252.) If we put wih ©? =dn(o+uy), and take as before g for the potential and y for the stream function, then since, when = 0, +4 e © =da(y) _ an (y,®) en (y, BY wwe havo y = 0, y=47b, y =+27b..., while x ranges from 0 to +. Thus the equipotential surfaces for which ¢ vanishes are 4 pile of parallel semi-infinite plates stretching from the axis of y to infinity along the positive direction of x, the distance between two adjacent plates being xb. When } = K, we have atiy « © =dn(K+w) ae), thus y= 0,y=i7b,y=+2zb..,, while « ranges from —= to —, where 2, {a given by the equation * eo F=Kk (38) ‘Thus the oquipotential surfaces for which ¢ = X are a pile of parallel semi-infinite plates stretching from — to a distance o, from the previous set of plates. The distance between adjacent plates in this set is again 7b, and the planes of the plates in this set are the continuations of those of the plates in the set at polential zero. This system of conductors is represented in Fig. 106. a | 254 ELECTRICKL WAVES AND OSOILLATIONS. [257 The case we shall first consider is that of an infinite con- ducting plate bounded by the planes x = 4, x =—h, immersed in 4 dielectric. We shall suppose that plane waves of clectromotive intensity aro advancing through the dielectric, and that these waves impinge on the plate, We shall suppose also that the waves fall on both sides of the plate and are symmetrical with to it. These waves when they strike against the plate will be reflected from it, so that there will on either side of the be systems of direct and reflected waves. Tet P and R denote the components of the electromotive intensity parullel to the axes of z and = respectively, the eom- ponent parallel to the axis of y vanishing since the case is one in two dimensions. Then in the dielectric the part of R due to the direct wave will be of the form Be (mettstp6), whilo the part due to the reflected wave will be of the form Cet lne=le+ pe) Thus in the dielectric on one side of the plate R= Betis tte +70 5 (yet (mente + pO, Q) If V is the velocity with which olectromagnetic disturbances are propagated through the dielectric, we have by equation (5). Art, 256, since yp’ K’ = 1/V3, GR CR_1dR de * at Vide’ iia heneo B4m? = iy: If A is the wave length of the incident wave, 0 the angle between the normal to the wave front and tho axis of a, we have, since Qn p=" 2a an. b= 0088, m= sind. Since Q vanishes, we have ap dR, dx " dz 256 ELECTRIOAL WAVES AND OSCILLATIONS. pe Wo can get the expression for the magnetic force in the dielectric very simply by the method given in Art. 9. In the incident wave the resultant electromotive intensity is B_ (netics yl) cos 6 hence the polarization is Kk’ B elles les ph) Tr cos 0° z where K’ is the specific inductive capacity of the diclectric. The Faraday tubes are moving with the velocity V, hence by equa- tions (4), Art. (9), the magnetic forces due to their motion is vn BL simestet et), 008 0 The magnetic induction corresponding to this magnetic force is equal, since y’K’ equals 1/V?, to B ¢ met le pt) Vcosd 2 which is the first term on the right in equation (6). We may show in a similar way that the magnetic force due to the motion of the Faraday tubes in the reflected wave is equal to the seeond term on the right in equation (6). We must now consider the conditions which hold at the junction of the plate and the dielectric, These may be expressed in many different ways: they are, however, whon the conductors are at rest, equivalent to the conditions that the tangential electromotive intensity, and the tangential magnetie foree, are continuous. Thus when « = /, we must have both R and yy continuous. The first of these conditions gives A(e*¢e-™) = Be pe, (7) the second mmm yak why _ (+m) py cho ith RPE A (eM) = TN wee), (8) Since atm = cenlp, , o and Bam Jase, tC Xco8 8 : a | 258 ELECTRICAL WAVES AND OSCILLATIONS. [257- ‘Thus corresponding to the current J, cos (pt-+me) in the plate, =p B= Ls c05 (pt +2), alyman P=—f, sin (pt +ma), in the plate. b= 4xply 5,008 (pt +-m2) ‘Thus, since ma is exceedingly small, we see that the maximum electromotive intensity parallel to the boundary of the plate ix exceedingly large compared with the maximum at right angles to it, Tn the dielectric we have R= The cos (pt +s) cos (21) —2nT, Vcos 0 sin (pt + mz) sind (a—A), P= a , ban Osin ( ph-+maz) sind (@—A) —27I, Vsind cos (pt+ms) cost (x—A), b == 5772 sin (pt +e) sind (2—h) +27J,cos (pt + ms) cost (e—A). Thus at the surface of the plate where a =h R= To con(pt-+m2}, P=—2x1,Vsiné cos(pt+me), b= 27J, cos (pt+me). Thus at tho surface of the plate P/R = —42Vhotsing. If the plato is metallic this quantity is exceedingly large unless the plate is oxcessively thin or 0 very small, so that in the dielectric the resultant electromotive intensity at the surface of the plate is along the normal, this is in striking contrast to the effect inside the plate where P/R is very small. The Faraday tubes in the dielec- tric close to the plate are thus at right angles to the plate, while in the plate they are parallel to it; hence by Art. 10 the electric momentum in the diclectric close to the plate is parallel to the axis of z, or parallel to the plate, while in the plate itslf it is parallol to the axis of «, or in the dircction of motion from the ontside of the plate to the inside, If inVhcos@=c, then 260 ELECTRICAL WAVES AND OSCILLATIONS. = vanish as soon aa 7, (4—2) is asmall multiple of unity. We thus get the very interesting result that an alternating current does not distribute itself uniformly over the cross-section of the conductor through which it is flowing. but concentrates iteelf towards the outside of the conductor. When the vibrations are very rapid the currents are practically confined to a thin skin on the out- side of the conductor, The thickness of this skin will diminish as 7 meronses; we shall tako 1/n, as the measure of its thickness. This inequality in the distribution of alternating currenta ix explicitly stated in Art. 690 of Maxwell's Electricity and Magnetism, but its importance was not recognised until it was brought into prominence and ils consequences developed by the investigations of Mr, Heaviside and Lord Rayleigh, and the ex- periments of Professor Hughes. The amount of this concentration is very remarkable in the magnetic metals, even for comparatively slow rates of alternation of the current. Let us for example take the case of a current making 100 vibrations per second, and suppose that the plate is made of soft iron for which we may put » = 10%,¢—= 10%, In this case p= 2x10", and n, or {27~p/c}* is approximately equal to 20; thus at adepth of half a millimetre from the surface of such a plate, the maximum intensity of the current will only be 1/e or +368 of its value at the surface. At the depth of 1 milli- metre it will only be -135, at 2 millimetres -018, and at 3 milli- metres -0025, or the 1/400 part of its value at the exterior. Thus in such a plate, with the assigned rate of alternations, the currents will practically cease at the depth of about 2mm. and will be reduced to about 1/7 of their value at the depth of one millimetre. Thus in this case the currents, and therefore the magnetic force, are confined to a layer not more than 3 millimetres thick. ‘The thickness of the ‘skin’ for copper is about 13 times that for soft iron. The preceding results apply to currenta making 100 vibra~ tions per second ; when we are dealing with such alternating currents as are produced by the discharge of a Leyden Jar, where there may be millions of alternations per second, the thickness of the ‘skin’ in soft iron is often lees than the hundredth part of a millimetre, i = 262 BLECTRICAL WAVES AND OSCILLATIONS, ~ : Peviodie Currents in Cylindrical Conductors, and the Rate of Propagation of Electric Disturbances along them. 259.] We shall now proceed to consider the caso which is moat easily realized in practice, that in which electrical disturbances are propagated along long straight cylindrical wires, such for example as telograph wires or sub-marine cables. A peculiar feature of electrical problems in which infinitely Jong straight cylinders play a part, is the effect produced by the presence of other conductors, even though these are such a long way off that it might have appeared at first sight that their influence could have been neglected. ‘This is exemplified by the well-known formula for the capacity of two coaxial cylinders. Tf a and b are the radii of the two cylinders the capacity per unit length is proportional to 1/log (b/s). Thus, oven though the cylinders were so far apart that the radius of the outer cylinder was 100 times that of the inner, yet if the distance wore further increased until the outer radius was 10,000 times the inner, tho capacity of the condenser would be halved, though similar changes in the distances between concentric § would hardly bave affected their capacity to an appreciable extent. For this reason we shall, though it involves rather more complex analysis, suppose that our cylinder is surrounded by conductors, and the results we shall obtain will enable us to determine when the effects due to the conductors can legitimately be neglected. 260,] The case we shall investigate is that of a cylindrical metallic wire surrounded by a dielectric, while beyond the dielectric we have another conductor; the dielectric is hounded by concentrie cylinders whose inner and outer radii are a and b respectively. If b/a is a very large quantity, we have a ease approximating to an aerial telegraph wire, while when b/a ix not large the case becomes that of a sub-marine cable. In the dielectric between the conductors there are convergent and divergent waves of Faraday tubes, the incidence of which on the conductors produces the currents through them. 261,] We shall take the axis of the eylindors ns tho axis of =, and suppose that the electric field ia symmetrical round this axis; then if the components of tho electric intensity and magnetic rr — | 264 ELECTRICAL WAVES AND OSCILLATIONS. L wire, ho asia of the-wirs i tau a ths oF 9 PONE the components of the electromotive intensity parallel to the axes of a, y, = respoctively 5 a, b, ¢ are the components of the respectively the magnetic permeability and specific resistance of the wire, p’, 0” the values of the same quantities for the external conductor, K is the specific inductive capacity of the diclectrie between the wire and the outer conductor. We shall suppose that the mag- netic permeability of the dielectric is unity, and that V is the velocity of propagation of electromagnetic action through this dielectric, We shall begin by considering the equations which hold in the dielectric: it is from this region that the Faraday tubes come which produce the currents in the conductor. We shall assume, as before, that the components of tho electromotive vary as ¢! "+28, The differential equation satisfied by R, the s component of the electromotive intensity in the dislectrio, is (Art. 266) GR @R @R_ 1 dk ae * ay * ae = ae” or, since R varies as (+P), mee aR +g —BR=0, (3) pre as where ke =m Fy If we introduce cylindrical coordinates 7, 0,5, this equation may be written GR idk, 10R ar * > dr * 8 do but since the electric field is symmetrical about the axis of 2, R is independent of ¢, henco this equation becomes oR, 1G pro, —-K?R=0; ae the solution of which by Art, 261 is, C and D being constanta, R= {Ody (shr)+ DK, (ker) et m2, ) Both the J and K functions have to be included, as 7 can neither vanish nor become infinite in the dielectric. This equation == = SS 262] BLEOTRICAL WAVES AND OSCILLATIONS. 269 Making these substitutions, equation (15) reduces to =~ Fon (G+ aloe 36) 3718} oh 2 Ky ‘n’b) moat (5+ te blog Fe) (ene) p one'n’., a» di(na)K,(n'b)] 1 ~ aera BOT Ym) Ry Gn) gto)” Now since both ka and kb are very small, Hatlog + keto 5 will be exceedingly small quantities unless a is so znuch smaller than b that log (2y/cka) is comparable with 1/M*b*. This would require such a disproportion between b and a as to be scarcely realizable in practice on a planot of the sizo of the earth; we taay therefore write the proceding equation in the form ptr a Jolene) yp K,(on’b) bah lieittm) ~ ab Ks (en) P ope gs) HB So(vna) K,(en'd} 1 bi pal 9) 25 ale J, (ens) K,Gn'e)| Tog(eay "? 8) where we have put n?= 4 zpep/o,n?= 4zy'ip/o’. Woshowedin Art. 258 that we were justified in doing this when the electrical vibrations are not so rapid as to be comparable in frequency with those of light. We seo from (16) that i* is given by an equation of the form =~ F(e—y—1F (ot—0') fr): (164) We remark that for all electrical oscillations whose wave lengths are large compared with the radii of the cable, p*(b*—a*)/V* is an exceedingly small quantity, since it is of the order (b*—s*//A%, where A is the length of the electrical wave. Th equation (16*) we see that we can neglect the third term inaide the bracket as long as both £ and y are small compared with 27°%/p7(b*—a). Now STACI ~ na Jy(ima) a0 that the large values of € occur when na is small; and im ‘his A a Wm 270 ELECTRICAL WAVES AND OSCILLATIONS. = ae a gee re viet §=- a= ioe ro ~ an ime iS Now for cables of practicable dimensions and materials con- veying oscillations slower than those of light 2p (b*§—a*)/V#a is an excoodingly small quantity, so that for such cuses £ is very small compared with eerie oun, 1 a the large values of y occur when n'bis small. Substituting the approximate values for K,, KX,’ we find i 2 =—ip' log (=): This is very small compared with »’/n’b, and may, as in the preceding case, be shown for all practicable cases to be very small compared with 2V*%/p*(b'—a*). Hence, as both € and 7 are small compared with this quantity, we may neglect the third term inside tho bracket in equation (16), which thus reduces to ep? fm To (ine) wo Ky (ind) LS (7) =~ ye ora Tei(ma) na) 0 (end) We shall now proceed to deduce from this equation the velocity of propagation of electrical oxcillations of different frequencies. Slowly Alternating Currents. 263.] The first case we shall consider is the one whore the frequency is so small that na is a small quantity, In this case, since we have approximately To (ena)/To (in equation (17) becomes is paket) Li ae Vilna? nd Ry (ur'd)) log (b/a) ) The first term inside the bracket is very largo, for it is equal to 2:p/n®a* and na is emall; the second term in the bracket vanishes if is infinite, and even if b is so small that n'd is a small quantity, we see, by substituting the values for Ky and K,’ when the variable is small, that the ratio of the =—%ne, rr = = length. The distance to which a disturbance travels before 1/c of ite original value is, on substituting the value of 8, seem to be yk tarp thus the distanco to which a disturbance travels is inversely proportional to the square root of the product of the frequency, the resistance, and capacity per unit length. If we take the case of a cable transmitting telephone messages of such a kind that 22/p, the period of the electrical vibrations, is 1/100 of a second, then if the copper core is 4 millimetres in diameter and the external radius of the guttapercha covering about 2-5 times that of the core, R is about 13 x 10-* Ohms, or in absolute measure 1-3 x 104, Cis about 15x 10-. Substitu these values for B and I, we find that the vibrations will travel over about 128 kilometres before falling to 1/e of their initial value, The velocity of propagation of the phases is about 80,000 kilometres per second, If we take an iron telegraph wire 4 mm. in diameter, B is about 9.4% 104; the capacity of wuch a wire placed 4 metres above the ground is stated by Hagenbach (Wied. Ann. 29. p. 377, 1886) to be about 10-* per centimetre, hence the distance to which electrical vibrations making 100 vibrations per second would travel before falling to 1/e of their original value would be {1-3 x 15/9-4}4, or 1-43 times the distance in the preceding case: thus the messages along the aerial wire would travel about half as far again as those along the cable, the increased resistance of the iron telegraph wire being more than counterbalanced by the smaller electrostatic capacity. Since vibrations of difforont frequencies die away at different rates, a message such as a telephone message whieh is made up of vibrations whose frequencies extend over a somewhat wide range will lose its character as soon as there is any appreciable decay in the vibrations. We see from this investi< gation that the lower the pitch the further will the vibrations travel, so that when a piece of music is transmitted along a telophone wire the high notes suffer the most, 274 ELECTRICAL WAVES AND OSCILLATIONS. In the dielectric, wo have by equations (7), (18), amd (24), assuming that kr is small, R= 3 Tf —2 Ver log =} «—* con(—az+pl), since from (13) and (14) D=2rVvA, The electromotive intensity along the radius, {P?+Q?}4, is equal to 2 V8 {ratt/po}t 2, © Tye-* 008 (—as+ pt— D In this ense the radial electromotive intensity is very lange com= pared with the tangential intensity, so that in the dielectric tho Faraday tubes are approximately at right angles to the wire. ‘Tho resultant magnetic induction is equal to 20 «-** o90(—a8-+-pt) © 265.] The interpretation of (17) is easy when mais very small, since in this case the first term inside the bracket is very lange compared with the second; as ma increases the diseussion of the equation becomes more difficult, since the second term in the bracket is becoming comparable with the first. It will facilitate the discussion of the equation if we consider the march of the function ina J, (ina)/J,(ina@), Perhaps the simplest way to do this is to expand the function xJy(a)/Jo'(w) in powers of «. Since J, () is a Bessel’s function of zero order, we have Je" (a) + 2d (0) +Jo(2) = 0, ==1-04 ig Ji(2) since J,'(z)=—J,(x), J, (a) being Bessel's function of the first order, Let 0, &), tg, @y... be the positive roots of the equation J, (2) =0, then ()= 2(1- 5) (0-5) a—4).. — 276 ELECTRICAL WAVES AND OSCILLATIONS. - a, and since n* = 4+y:p/r approximately, we have wna geiteh 2 HE (eaapatle) + sayag (4D ae) =f A rupatoe)— sygq (trmrate? + aaasean({mmPatoynd. (19) The values of ina J, (cna)/J, (na) fora few values of & ru pat/r are given in the following table:— atzppart/e || iad, (ena) (ina) 3 —2 {1-001 +-0621} 1 =2 {10054-1251} 15 =2 {10124-1864} =2 {1021-4254} 25 —2{1-082+-S2s} 3 =2 41-0454 97¢} From this table we see that even when 47ppa°/r is as lange as unity, we may still as an approximation pub enad, (cnay/Jy’ (ena) equal to —2, and /* will continue to be given by (18). 266.] We must consider now the relative values of the terms inside the bracket in (18) when 7 is comparable with unity, In the case of acrial telegraph wires it is conceivable that there may bo cases in which though na is not large n’b may be #80; but when this is the ease we have by Art. 261 K, (en’b) == Ker’), so that since n’b is very large the second term inside the bracket in equation (18) will be small compared with the first, hence we have 1 ¥=-2,*, V? Bra? log (b/a) which is tho samo value as in Art. 263. In all telegraph cables where the external conductor is water, and in all but very elevated telegraph wires whore the external conductor is wet earth, the value of o” will so greatly exceed that of w that unless b is more than a thousand times 278 BLECTRICAL WAVES AND OSCILLATIONS. (67. | place we sea that to our order of approximation both the of propagation ot the phase al these ot aR vibrations are independent of the resistance of the wire. These quantities depend somewhat on the resistance of the external conducter, but only to a comparatively small extent even on that, as o” only enters their expressions as a logarithm, Tho velocity of propagation of the phases only varies elowly with the frequency, ak p only occurs in its expression as a logarithm. The rate of decay, i.e. the real part of «m, is proportional to the frequency and thus varies more rapidly with this quantity than when na is small, as in that case the rate of decay is proportional to the square root of the frequency (Art. 263). We see from the preceding investigation that for sending periodic disturbances along a cable, the frequency being such as to make n’b a very small quantity, we do not gain any ap- preciable advantage by making the core of a good conductor like copper rather than of an inferior one like iron witless the conditions are such as to make a small compared with unity, We see too that the distance to which the disturbance travels before it falls to 1/e of its original value increases with the resistance of the external conduetor, We shall show in a subsequent article that the heat produced per second in the external conductor is very large compared with that produced in the same time in the wire, thus the dissipation of energy is controlled by the external conductor and not by the wire. The preceding results will continue true as long as nb is small, even though the frequency of the olectrical vibrations gets £0 large that na/u is a very large quantity; for when 7a is large we have by Art, 261, J%q (ena) = —edy (ena), 80 that equation (16) becomes = Kae 1 . ie Pigs Hig ave gto) Since na/. is large and n’b small the second term inside the bracket is large compared with the first, so that we get the same value of * ox that given by equation (20), 267,] The next case we have to consider is that in which both 267. ELECTRICAL WAVES AND OSCILLATIONS. 279 wea and n’b are very Jange; when this is the case wo know by Art. 261 that J’ (ena) =—1Jy (ena), K', (en'b) = cK, on’). Making these substitutions, equation (17) becomes eoBlketlacbay mma h his (Stee) cgiat =F ee 5+ n/a eel approximately. Since the second term inside the bracket is small compared with unity, extracting the square root we have, mE gl VS icemh ‘This represents a vibration travelling approximately with the velocity V and dying away to 1/< of its initial value after tra- versing a distance n/n + n/ MEY oat): Since the imaginary part of m is small compared with the real part, the vibration will travel over many wave lengths bofore its amplitude is appreciably reduced. From the expression for the rate of decay in this case we sec that when the wire is surrounded by a very much worse conductor than itself, as is practically always the case with cables, the distance to which those very rapid oscillations will travel will be governed mainly by tho outside conductor, and will be almost independent of the resistance and parmeability of the wire ; no appreciable advantage therefore would in this case be derived by using a well-con- dueting but expensive material like copper for the wire. In serial wires the decay will be governed by tho conductivity of the earth rather than by that of the wire, unless the height of the wire above the ground, which we may take to be comparable with b, is so great that y’c’/b* is not large compared with po/a*, Experiments which confinn the very important conclusion ‘that these rapid oscillations travel with the velocity V, that is 4 Ee 280 ELECTRICAL WAVES AND with the volocity of through the dieleotri, will be deseribed 5 pa Ce 8 268.) As rapidly alternating currents are now 1 employed, it will be useful to determine the the electromotive intensity both in the wire and in the dieleetrie in terms of the total current passing through the wire. Let this current at the point > and time ¢ be represented by the real part of J,¢"*", ‘The line integral of the magnetic foree taken round any circuit is equal to 47 times the current | that circuit. Now by equation (10) the magnetic force at the surface of the wire is ST AJ’ (ena) emt, Since the line schaceal oh Oita noe of the wire is equal to 4 rJye™ 72", we have =n _Iy ~ Bra J (ine) Substituting this value for A in equation (9), wo find that in the wire wn Ty ~ 2a Fytena) where the real part of the expression on the right-hand side is to be taken. When 7a and nr are very large, we have by Art. 261 J’ (ena) =—t Jo (vnr) = Req Toon) eine +P; (24) « vianr a Vizna substituting these values in (24), wo find Rm PREY Fae Beale D cos (4), (28) where yams + pt—(2rup/o)) (o—r) + 5° Similarly, we find by equation (9) that the radial eleetro- motivo intensity eds is given by tho equation 1=-2 eo aeenie} tenn —_ {4 Qy gon 7 xin(y—])- (28) The resultant magnetic force is by equation (10) equal to 2 -Jon 4 aar) Joke {2nup/o}4 a— eos (y—F)+ === 268,] «ELECTRICAL WAVES AND OSCILLATIONS. 281 Sinee all these expressions contain the factor «~(2*#7/*)4(a—r), we see that the magnitudes of the electromotive intensity and of the magnetic foree must, since na—and therefore (2 xup/o)'a— is by hypothesis very large, diminish very rapidly as the distance from the surface of the wire increases. The maximum valuce of those quantitice at the distance (¢/2z,4p)! from the surface are only i/c of their values at the boundary, and they diminish in geometrical progression as the distance from the surface inereases in arithmetical progression. Thus the currents and magnetic forces are, ax in Art, 258, practically confined to a skin on the outside of the wire. We have taken (0/27up)? as the measure of the thickness of this ‘skin.’ For currents making 100 vibrations per second, the skin for soft iron having @ magnetic permeability of 1000 is about half a millimetre thick, for copper it is about thirteen times as great. For currents making a million vibrations per second, such as can be produced by discharging Leyden jars, the thickness of the skin for soft iron —since wo know that this substance retains ite magnetic proper- ties even in these very rapidly alternating magnetic fields (J. J. ‘Thomson, Phil. Mag. Nov. 1891, p. 460)—is about 1/200 of a ‘millimetre, for copper it is about 1/15 of a millimetre. Tn these eases there is enormous concentration of the current, and since the currents produced by the discharge of a Leyden jar, though they only last for a short time, are very intense whilst they last, the condition of the outer layers of the wires whilst the discharge is passing through them is very interesting, as they are convey- ing currents of enormously greater density than would be sufficient to melt thom if tho currents were permanont instead of transient, ‘This concentration of the current, or‘ throttling’ as it is some- times called, produces a grest increase in the apparent resistance of the wire, since it reduces so largely the area which is available for the passage of the current. If in equation (25) we put r =a, ‘we get maximum value of R =(zpo/za*)' x (maximum value of the current through the wire), thus we may look upon (upo/sa*)* as the apparent resistance per unit length of the wire to these currents. This resistance increases indefinitely with tho rate of alternation of the curront; we seo too that it is inversely proportional to the circumference of the wire instead of to the ares as for steady currents. This is what we shold 4 282 ELECTRICAL WAVES AND OSCILLATIONS, — be | expect, since the currents are concentrated in the region of the cireumference. The resistance of the solid wire to these alter- nating currents ia the samo as that to steady currente of a tube of the game material, the outside of the tube coinciding with tho outside of the wire, and the thickness of the tube being 1/2 times the thickness of the skin. We see by comparing equations (26) and (26) that the eloctro- motive intensity parallel to the axis of the wire is very large compared with the radia) electromotive intensity in the wire, 80 that in the wire the Faraday tubes are approximately parallel to its axis, 269.) Let us now consider the expressions for the electromotive intensities and magnetic force in the dielectric; wo find by equations (8) and (24), assuming ‘a and kb small, na, n’b large, D=2V2k Typ. Hence, using (22), we have in the dielectrie when kr is small, Bem fee) — [CE + ERE) Sey taco where b=ms+pt+ ts while the radial electromotive intensity is 27 con (me +H), and the resultant magnetic force 20608 ms-+p!) We see that the maximum value of the radial electromotive intensity is very great compared with that of the tangential, so that in the dielectric the Faraday tubes are approximately radial. The momentum dune to these tubes is, by Art, 12, at right angles both to the tubes and the magnetic foree, so that in the dielectric it ia parallel to the axis of the wire, while in the wire itself it is radial. Thus for these rapidly alternating currents the momentum in the dielectric follows the wire, The radial polarization in the dicleetric is K/4z times the radial electro- motive intensity, and since =Yv4, 270.) ELECTRICAL WAVES AND OSCILLATIONS, 283 it is equal to if oa Te Ve (met Pe). Af the Faraday tubes in the dicleetrie are moving with velocity V at right angles to their length, i.e. parallel to the wire, the magnetic force due to these moving tubes is, by Art. 9, ab right angles both to the direction of motion, i.e, to the axis of the wire, and to the direction of the tubes, i.e. to the radius, and the magnitude of the magnetic force being, by (4), Art 9, #= 0 times the polarization, is 210 goa (me+p!), which is the expression we havo already found. Hence we may regard the magnetic force in the field as due to the motion through it of the radial Faraday tubes, these moving parallel to the wire with the velocity with which electromagnetic disturb- ances are propagated through the dielectric. In the outer conductor when x'r is largo pea aa [Be —aete/eh -¥) oon a, Re feect 7 2#H 8/24 (F-®) 606 4, where o= matpt—(22p'p/o)+ T+ ‘The radial electromotive intensity is 1 oT, vo] ’ 2) ae cos (+ 7). ‘The resultant magnetic force is perpendicular to rand equal to BT, —(senpid i (r—) 7 ethan eas We see from these cquations that unless po’ is comparable with V? the tangential electromotive intensity will be large compared with the radial. Transmission of Arbitrary Disturbances along Wires. 270,] Since vibrations with different periods travel at different rates, we cannot without further investigation determine the rate at which an arbitrary disturbance communicated \o @ 4. _ 284 ELECTRICAL WAVES ae | q limited portion of the wire will travel along it, deduce an expression which would represent way in which an arbitrary disturbance is propagated, we should have to make use of the general relation between m and p given by equation (18). This relation is however too complicated to allow of the necessary integrations being effected. The com~ plication arises from the vibrations whose frequencies are 50 great that 2mypai/r is no longer a small quantity; such vibra- tions however die away more rapidly than the slower ones, so that when the distance from the origin of disturbance is con- siderable the latter are the only vibrations whose effects are felt. For such vibrations, we have by Art. 263 m? =- EP Hence a term in the expression for R of the fori F(a)e7 ar" cos m (e—a), where a is any constant and J’ (a) denotes an arbitrary function of o, will satisfy the electrical conditions, By Fourier's theorem, however, = eae ** Pla)" Ht conm (ee) dnd, (27) is equal to F(z) when t=0. Hence this intogral, since it satistics the equations of the electric ficld, will be the expreesion for the disturbance on the wire at 2 at the time ¢ of the disturb- ance, which is equal to (2) when t= 0, When the disturbance is originally confined to a space close to the origin, F(a) vanishes unless a is very small ; the expression (27) becomes in this case +o my F <7 BF' cosmedm, (28) where P=/[Flo)ds. e Since fe“ con Blade = Feat, we sec by (28) that the disturbance at timo ¢ and place = will ‘be equal to F{sErjt}beP2Ae, (29) 270.) ELECTRICAL WAVES AND OSCILLATIONS. 285 Thus at & given point on the wire the disturbance will vary as where c is a constant. The rise and fall of the disturbance with the time is represented in Fig. 108, where the ordinates represent Fig. 108. the intensity of the disturbance and the abscissae the time. It will be noticed that the disturbance remains very small until ¢ approaches c/4, when it begins to increase with great rapidity, reaching its maximum value when ¢ = 2c; when ¢ is greater than this the disturbance diminishes, but fades away from its maximum value much more slowly than it approached it. Since the disturbance rises suddenly to its maximum value we may with propriety call 7, the time which elapses before this value is attained at a given point, the time taken by the disturb- ance to travel to that point. We see from (29) that T= 42 Rr. (30) Thus the time taken by the disturbance to travel a distance 2 is proportional to 2’, it is also proportional to the product of the resistance and capacity per unit length. By dividing « by 7 we get the so- current along the wire;’ this by (30) is 2 zRr” ‘Tho velocity thus varies inversely ns pee ‘pproximate equation vibrations of infinite freq with infinite velocity, in reality we have seen (Arte they travel with the velocity V. These very however die away very quickly, and whec sro got tle eae equal to a small multiple of 2/VRT they will practically haye disappeared, and at such distances we may trust the ex- prossions (31). A considerable number of experiments have been made on the time required to transmit messages on both aerial and submarine cables; the results of some of these, made on aerial telograph iron wires 4 mm. in diameter, are given in the accompanying table taken from a paper by Hagenbach (Wied. Ann. 29. p. 877):— ot ant of nn io. | Te han fee mae ‘ag i HEgesgege Hagenbach proved by making experiments with lines of different lengths that the time taken by a message to travel along a line was proportional to the square of the length of the Tine, If we apply the formula T= }2kr aml ——————— 270) ELECTRICAL WAVES AND OSCILLATIONS, 237 to Hagenbach’s experiment in the above table, where 5 = 2848 x 108, R= %4x10', and (by estimation) T= 10-*, we find T = -0033, whereas Hagenbach found -0017. The agree~ ment is not good, but we must remember that with delicate receiving instruments it will be possible to detect the disturb- anc before it reaches its maximum value, so that we should expect the observed time to be less than that at which the effect is a maximum, In Hagenbach’s experiment the line was about 4 times the length which, according to the formula, would have made the disturbance travel with the velocity of light, so that it would seem to have been long enough to warrant the ap- plication of a formula which assumes that the shorter waves would have become so reduced in amplitude that their effects might be neglected. When the wire is of length 1, we know by Fourier’s Theorem that any initial disturbance R may be represented by the equation R=(A,sin™ +B, 0057) + (A,sin “7 + B,c0n =F + (Aysin 3 ie © ment! tone Since ip =—m/RT, the value of 2 after a time ¢ has elapsed will be represented by the oquation R=(Aysin™= +2,cos%2) oP RF + (Ay sin 77 + Bycos 2) Fae tiie 1 (A, win! 4 Hea Het For a full discussion of the transmission of : ; signals along cables the reader is referred to a series of papers by Lord Kelvin at the beginning of Vol. IT of his Collected Papers. a 271,] Wo havo hitherto only considored the total electromotive intensity and have not regarded it as made up of two 3, one due to external causes and the other due to the of the alternating currents in the conductors and dielectric, For some purposes, however, it is convenient. to separate the electromotive intensity into these two parts, and to find the relation between the currents and the external electromotive intensity acting on the system. We may conveniently regard the external electromotive intensity as arising from an electrostatic potential satisfying the equation V7 = 0. We suppose that, as in the preceding investigation, all the variables contain the factor \"**?". Since varies as ¢”*, the equation V7 = 0 is equivalent to &o ide c Pr + er —m'p = 0. The solution of this is, in the wire = Ly (ur) 429, in the dielectric = (MJ, (er) 4K, (vmn)} Me *P9, in the outer conductor = SK, (umn) elm +29. Tf, as before,-a and b are the radii of the internal and external boundaries of the dielectric, we have, since is continuous, LJ, (uma) = MJ (cma) +NK, (una), SK, (mv) = MJ, (omb)+NK, (mb). The excess: of the normal electromotive intensity due to the electrostatic potential in the dielectric over that in the wire is equal to om (LJ (uma) — (My (sma) + NK (vma)} eet; substituting the value for L—M in terms of WV from the preceding equation, this becomes nay (ema) (ema) —Jo (sma) Ky’ (uma). b By equations (10) and (12) Qna om) a, a (uma) fe, in ane (ar) BK, (enters are respectively the line integrals of the magnetic force round the circumference of the wire and the inner cireumference of the outer conductor, henes they are respectively 4 times the current through the wire, and 47 times the eurrent the wire plus that through the dielectric. Unless however t radius of the outer conductor is enormously greater than that of the wire, the current through the wire is infinite in eomparigon with that through the dielectric; for the electromotive intensity R ig of the same order in the wire and in the dielectric ; the current density in the wire is R/c, that in tho dielectric (K/d =) dR/dt, or KipR/4m, or epR/4x V*. Now for metals o is of the order 104; and since V* is 9% 10%, we see thet even if there are a million alternations per second the intensity of the eurrent in the wire to that in the dielectric is roughly ns 2x10" is to unity; thus, unless the area through which the polarization currents flow exceeds that through which the conduction currents flow in a ratio which is impracticable in actual ex- periments, we may neglect the polarization currents in comparison with the conduction ones, so that so) 4 Igina) = teem PKj(on’b). (34) Returning to equations (32) and (33), we notice that (mn? Y/u (ki? —m?), which is equal to 4mVYop, is very large when op is small compared with F*. Now ¢ for 272) ELECTRICAL WAVES AND OSCILLATIONS. 298 A dt or since pla as z =e tsar. If L is the coefficient of self-induction and R the resistance of a circuit through which a current J is fowing, we have external eloctromotive foros = L974 RI. By the analogy of this equation with (39) we may call P the solf- induction and Q the resistance of the cable per unit length for these alternating currents. @ has been called the ‘impedanco’ of nnit length of the cireuit by Mr. Heaviside, and this term is preferable to resistance as it enables the latter to be used ex- clusively for steady currents. By comparing (39) with (38), we sce that Pa 2og? + Sum Lwrpatalfet)+ De (up ntat/o%) — oy e= Sh + hie state!) — Fag opt xtatfet) +} These results are the same aa those given in equation (18), Art. 690, of Maxwell's Electricity and Magnetiem, with the exception that » is put equal to unity in that equation and in it A is written instead of 2 log (b/a). We see from these equations that as the rate of alternation ineresses, the impedance increases while the self-induction diminishes ; both these effects are due to the influence of the rate of alternation on the distribution of the current. As the rate of alternation incroases the current gets more and more concen- trated towards the surface of the wiro; the offective arca of the wire is thus diminished and the resistance therefore inereased. ‘On the other hand, the concentration of the current on the surface of the wire increases the average distance between the portions of the currents in the wire, and diminishes that between the currents in the wire and those flowing in the opposite direction in the onter conductor; both these effects diminish the self- induction of the system of currents. ‘The expression for Q does not to our degree of approxi- (40) = | 204 ELECTRICAL WAVES AND | mation involve b at all, while b only enters into of the expression for P, which is independent of the fi i; thus, as long ax aa is very small, the presenee of the outer conductor does not affect the impedance, nor the way in whieh the self-induetion varies with the frequency. When p= 0 the self-induction per unit length is 2log(b/a)+}4. Since u for soft iron may be as great as 2000, the self-induction per unit length of straight iron wires will be enormously greater than that of wires made of the non-metallic metals, 278.] We shall now pass on to the cnse when aa is large and nb small, so that ned (-na)/pa Jy (ina) is small compared with 2 oK,(:n'b)/pbX,’ (cn'b). These conditions are com- patible if the specific resistance of the outer eondustor is very smuch greater than that of the wire. In this ease equation (37) becomes 1_ ne’ K,(en’d) aan flog + Tep izp b KJ (mb I Since n’b is small, we have approximately K,(en’b) = log (2y/n’d), Ky’ (en'b) =—1n'b; hence B= 2p flog h/a+ blog (y/W mb" pie). 1d Fh ‘Thus the coefficient of self-induction in this case is 2 log (b/a) + 2p" log (>/ my" pb*/o'), and the impedance fap. It is worthy of remark that to our order of approximation neither the impedance nor the self-induction depends upon the resistance of the wire. This is only what wo should expeet for the self-induction, for since 7a is large the currents will all be on the surface of the wire; the configuration of the currents has thus reached a limit beyond which it is not affected by the resistance of the wire. It should be noticed that the conditions ne large and nb small make the impedance } py’ large com- pared with the resistance «/ra* for stoady currents. 274.) ELECTRICAL WAVES AND OSCILLATIONS. 295 Very Rapid Currents. 274.] Wo must now consider the case whore the frequency is 20 great that na and ’b are very large; in this ease, by Art. 261, Sf (ena) =F, (ona), Ky (en’) = LK, (en'd), #0 that equation (37) becomes smrofor [spa + hI ~ 2} we seo from this equation that the self-induction P is given by the equation P= 2log(b/a)+(ow/2xpa")h+(o'p'/2mpb*), (42) and the impedance Q by = (oup/2aa*)' + (o'u'p/2xb')t. (43) In a cable the conductivity of the outer conductor is very much less than that of the coro, so that «’/b* will bo large com- pared with o/a*; thus the self-induction and impedance of a cable are both practically indopondent of the resistance of the wire increased is 2 log (b/a); as this does not involve p it is the same for iron ws for copper wires, The difference between the self- induction per unit length of the cable for infinitely slow and infinitely rapid vibrations is by equations (10) and (42) equal to » The impedance of the circuit increases indefinitely with the frequency of the alternations, If we trace the changes in the values of the self-induction and impedance as the frequency p increases, we soc from Arts. 272, 273, 274 that when this is so small that na is a amall quantity the self-induction decreases and the impedance in- creases by an amount proportional to the square of the frequency. When the frequency increases so that ma is con- siderable while n’b is small, the self-induction varies very slowly with the frequency while the impedance is directly proportional to it. When the frequency is so great that both me and n’b are large the self-induction approaches the limit 2 log (b/a), while the impedance is proportional to the square root of the frequency. $e Sinco b = a+d, where d is very small compared with a, bid log 5 = 5 spproximately. Making these substitutions, equation (41) becomes E= ap{at JS = 2al’. ‘Thus, in this case the self-induction per unit length is anfds JB 1n,/Mb. 276.) Though, as we have just seen, it is possible to regard the case of two paralle! metal slabs as a particular ease of the cable, yet inasmuch as the geometry of the particular case is much ‘simpler than that of the cable, the ease is one where points of theory are most conveniently discussed ; it is therefore advisable and the impedance If the total current through « lab per: sented by the real part of 7, <9, then, when | is so great that op/87/*p is a small quantity and real part of m large compared with the im 5 have since 10179 =f” Ba Co ah a(ne + pl), = aa +. Osene hy; ——_ hhenoe by (14) As—Batale. = We have therefore in the dielectric R= ol, V3n' (a/h) cos (ms+pt+ 2) P=47mI,(V*%/p)cos(me+pt), b =—42I, cos (me +p), where nl = {27 p/s)*. In the metal slab we have on the side where a is positive, R= ol, S20" con (ms +pt—mn' (wh) + 5) P=— lyme” “sin (me +-pt—n' (w—h)), & == Aap” © 008 (ms + pt—w' (w—h))- We see from theso equations that //R is very largo im the diolectric and very small in the metal slab, thus the Faraday tubes are at right angles to the conductor in the dielectric and parallel to it in the metal slab. Mechanical Force between the Slabs, 277.) This may be regarded as consisting of two parts, (1) an attractive foree, due to the attraction of the positive of one slab on the nogative of the other, (2) a repulsive force, — due to the repuleion between the positive currents in one slab and the negative in the other. To calculate tho first foree we notice that since /po is very lange, the value of P in the con- SSS 277) BLECTRIOAL WAVES AND OSCILLATIONS. 301 duetor is very small compared with the valne in the dielectric, and may without appreciable error be neglected; hence if « is the surface density of the electricity on the slab and X the specific | inductive capacity of the dielectric, 4ze =—K4Anm(V*/p) I, cos (ms +pt). ‘The force on the slab per unit area is equal to Pe/2 ; substituting tho values of P and ¢ this becomes 22K m* (Vp) 1,2 cos? (me + pt). The force due to the repulsion between the currents in the slabs per unit volume is equal to the product of the magnetic induction } into w, the intensity of the current parallel to >. Since db srw =~ > the foree per unit volume is equal to 1 ait ~ Bap da’ hence the repulsive force per unit area of the surface of the slab 1 dite EN rate =" ), = a = Amul ,%cos" (mz-+pt). ‘When tho alternations are so rapid that the vibrations travel with the velocity of light Vint =p, and since X = 1/V%, the attraction between the slabs is equal to 2a," cos" (ms-+, while the repulsion is’ seeahii 2QzuT,* cos* (ms + pt), hence the resultant repulsion is equal to 2 (u—1)I,* cos" (ms +pt). ‘If the slabs are non-magnetic » = 1, 90 that for these very rapid vibrations the eloctrostatic attraction just counterbalances the electromagnetic repulsion. Mr. Boys (Phil. Mag. [5], 31,p. 44, 1891) found that the mechanical forees between two conductors carrying very rapidly alternating currents wax too xmall to be = detected, even by the n Sp etal ts ronc hls al p enabled him to detect forees com: those due to the electrostatic charges or to: the currents. Propagation of Longitudinal Waves of Mi Induction along Wires. 278.] In the preceding investigations along the wire and the lines of magnetic foree have series of co-axial circles, the axis of these circles the wire. Another case, however, of considerable pr portance is when these relations of the magnetic force and © are interchanged, the current flowing in circles round the the wire while the magnetic force is mainly along it. condition might be realized by surrounding a portion: 4 by a short co-axial solenoid, then if alternating currenta az throngh this solenoid periodic magnetic forces “ wire will be started. ‘We shall in this artiele investigate Jaws which govern the transmission of such forces along The problem has important applications to the construction transformers ; in some of these the primary coil is wound round one part of a closed magnetic circuit, the another. This arrangement will not be efficient if there considerable leakage of the lines of magnetic force . primary and the secondary. We should infer from general siderations that the magnetic leakage would increase rorieie rato of alternation of the current through the primary. suppoxe that an alternating current passes through an [ ring imbedded in a cylinder of soft iron surrounded by air, the straight axis of the ring coinciding with the axis of the cylinder, _ ‘The variations in the intensity of the current through this ring will induce other currents in the iron in its neighbourhood; the magnetic action of these currents will, on the whole, cause the com= ponent of the magnetic force along the axis of the eylinder to be | Jess and the radial component greater than if the current through _ the ring were steady; in which case there are no currents in the iron, Thus the effect of the changos in the intensity of the current through the primary will be to squeeze as it were the lines of | TE 278.) ELECTRICAL WAVES AND OSCILLATIONS. 303 magnetic force out of the iron and make them complete their cireuit the air. Thus when the field is changing quickly, the lines of magnetic force, instead of taking a long path throngh the medium of high permeability, will take a short path,even thongh the greater part of it is through a medium of low permeability such asair. The case is quite analogous to the difference between the path of a steady current and that of a rapidly alternating one. A steady current flows along the path of least resistance, @ rapidly alternating ono along the path with loast self-induction. ‘Thus, for example, if we have two wires in parallel, one very Jong bot made of euch highly conducting material that the total resistance is amall, the other wire short but of such a nature that the resistance is large, then when the current is steady by far the greater part of it will travel along the long wire; if however the current is a rapidly alternating one, the greater part of it will travel along the short wire because the self induction is smaller than for the long wire, and for these rapidly alternating currents the resistance is a secondary con- sideration. In the magnetic problem the iron corresponds to the good conductor, the air to tho bad one. When tho fiold is steady tho lines of force prefer to take a long path through the iron rather than a short one through the air; they will thus tend to koop within the iron; when however the magnetic field is a very rapidly alternating ono, the paths of the lines of force will tend to be as short ax possible, whatever the material through which they pass. The lines of foree will thus in this case leave the iron and complete their circuit through the air. We shall consider the case of n right circular soft iron cylinder where the lines of magnetic force are in planes through the axis taken as that of 2, the corresponding system of currents flowing round circles whose axis is that of the cylinder. The cylinder ig eurrounded by a dicloctric which extends to infinity. Let a, b,c be the components of the magnetic induction parallel to the axos of @, y, ¢ respectively; thon, since the component of the magnetic induction in the xy plane is at right angles to the axis of the cylinder, we may put ‘Let us suppose that a, 0, ¢ all vary as ¢**™), 278.) ELECTRICAL WAVES AND OSCILLATIONS. 305 since the radial magnetic induction is continuous, we have ™ Ady (ona) = "2 OK, (ska). Eliminating A and C from these equations, we get Jy (ena Ky (ck an equation which will enable us to find m when p is known. Let us begin with the case when the frequency of the alter- nations is small enough to allow of the currents being nearly uniformly distributed over the cross-section of the cylinder. In this case we have approximately Ty(ina)=1, Iy(ena) = dine, 80 that equation (49) becomes "Ke (ko) (50) Since for soft iron 2/u is a small quantity, the right-hand side of this equation and therefore ka must be small ; but in this case we have approximately K,(vka) = log (2y/ika), Ky (ku) =~ 3, 80 that (60) becomes = 2 = Ko! log (2y/:ka). (51) ‘To solve this equation consider the solution of awlogz =—y, when y is mall. If = —y/logy, thon Jog log(1/y) loge =—y {r+ i va, but when y is small log log (1/y) is small compared with log (1/y), so that an approximate solution of the equation is 2 =—y/ logy. x ST SY 3068 ELECTRICAL WAVES AND 0 If we apply this result to equation approximate solution of that equation is oe Bam—%, and since the value we have just found for k is in any practicable case very large compared with p'/V3, we seo that k* =m ap- proximately, so that aitft_4 y. 8 ti log (uy*)! ‘Thus sinee in the expression for c there is the factor 2 Aid eime or e-a* lceuant A wo see that the magnetic force will die away to 1/e of ite valuo aus ae ta {wlog (uy’)}* from its origin. 279]. In the last case the current was uniformly dis tributed over the cross-section. We can investigate the effect of the concentration of the current at the boundary of the eylinder by supposing that ma is large compared with unity though small compared with z. In this case, since approximately Jo’ (sma) = —0J, (ena), equation (49) becomes Since the left-hand side of this equation is small, :a is algo emall, so that by Art, 261 we may write this equation aa 2 a = Ma? log (2y/ka). (52) This equation gives n value for k* which is very Jarge compared with p’/V*, so that approximately m =k. We alsosee that & or mv is small compared with , we may therefore put n= (4mmep/o}h by the currents in the wire. Dissipation of Knergy by the Heat produced by Alternating Currents. 280.) A great deal of light is thrown on the laws which govern the decay of currents in conductors by the consideration of the circumstances which affect the amount of heat produced in unit timo by these currents. As we haye obtained the expressions for these currents wo could determine their heating effect by direct integration; wo shall however proceed by a different method for the sake of introducing a very important theorem due to Professor Poynting, and given by him in his paper ‘On the Transfer of Energy in the Electromagnetic Field, Phil. Trans, 1884, Part II, p. 343, The theorem is that Effet 1088+ 2!B ai = Eft ae +05 +7G)dedyds + fh if if (Xé+ ¥o+2Z2)dadyds+ /) if, / (Pp+Qq+Rr)dzdyds = af [URAC +m (P'y—Wo)+n(Ya-PA)}as, ae! ——— = ‘ELECTRICAL WAVES AND OSCILLATIONS. 309 where the volume integrals on the left-hand side are taken ) seetesipte volume contained by the closed surface S, of which dS is an clement and J, m,n the direction cosines of the normal drawn outwards. P, Q, FB aro the components of the olectromotive intensity. a, 8, y those of the magnetic force. X, ¥, Z those of the mechanical force acting on the body in consequence of the passage of currents through it. %, % @ the components of the velocity of point in the body. P, q, 7 the components of the conduction currents. P,Q, BR’ tho parts of the components of the olectromo- tive intensity which do not depend upon the motion of the body. K the specific inductive capacity and » the magnetic per- meability. The following proof of this theorem is taken almost verbatim from Professor Poynting’s paper. Let u, v, w be the components of the total current, which is the sum of the polarization and conduction currents ; we baye, since the components of the former are respectively KdP KdQ KdR dxdt’ dxdt’ 45 dt’ Henee Efff pi? 49% 18 BoP) dedyde = J i if (P(u—p)+Q(v—q)+ BR (w—r)} dedyds = fi if fi (Pu+Qu+ Ru) dzdyds —[f[f@vrearRndedyis (63) — — ey 0+ Rew stp eles i ‘vo — wh) b+ (wa — ue) + (ud — i 4 Y 9428) +P ut vt. where X, Y, Z are the n mite er Ye A Substituting this value for Pu+Qu+ posing, we obtain £f if, ie Pee +Qu 10 aE dedyds +f [[(xes v9+28)dndyde+ | [n+ =f [feurevrne) re ar 4nv= % - 2 tow a BS da. Substituting these values for u,v, w in the right Mi ‘equation (54), that sido of the equation becomes. alle G- H+eg- =B G | = ffl e082 ete a Now =hffuere—enamerr—iey which is the theorem we set out to prove. Now the electrostatic energy inside t 5 f[[ereatRmaeay as, preines r=Hp, o=KQ, a= Kn, eS [fer aR ded af i (aa+bp-+oy)dedyde, [flere aadyas. Thus the first two integrals on the left-hand side of e 280.) --ELROTRICAL WAVES AND OSCILLATIONS. 313 ‘Thus we may regard the change in the energy inside the closed surface as due to the transference of energy across that surface; the energy moving at right angles both to H, the resultant magnetic force, and to E, the resultant of J”,Q', R. The amount of energy which in unit time crosses unit area at right ‘angles to the direction of the energy flow is HE siné/4x, where 0 is the angle between H and B. Tho direction of the energy flow is related to those of H and E in such a way that the rotation of a positive screw from EB to H would be accompanied by a translation in the direction of the flow of energy. Equation (55) justifies us in asserting that we shall arrive at correct results as to the changes in the distribution of energy in the field if we regard the energy as flowing in accordance with the Jaws just enunciated; it does not however justify us in asserting that the flow of energy at any point must be that given by these laws, for we can find an indefinite number of quantities U,, V,, W, of the dimensions of flow of onergy which satisfy the condition J[feenctme+auyas =0, where the integration is extended over any closed surface. Hence, we ee that if the components of the flow of energy were Rp-Qy+Su, instead of R'B—Qy, P’y—Ra+ Sv, instead of P'y—Ka, Ga—P’g+ Ew, instoad of Q’a—P’g, the changes in the distribution of energy would still be those whieh actually take place. Though Professor Poynting’s investigation does not give unique solution of the problem of finding the flow of energy at any point in tho clectromagnetic field, it is yet of great value, as the solution whieh it does give is simple and one that readily enables us to form » consistent and vivid representation of the changes in the distribution of energy which are going on in any actual caso that we may have under consideration. Several appli- cations of this theorem are given by Professor Poynting in the paper already quoted, to which we refer the reader, Woe shall now proceed to apply it to the determination of the rate of heat production in wires at rest traversed by alternating currents, 4 B14 ELECTRICAL WAVES AND OSCIELATIONS. = | 281.] Since the currents are periodic, P*, Q%, R%, a, 6%, 7* will be of the form A . +B cos (2 pt + 8), where A and B do not involve the time; hence the first two integrals on the left-hand side of equation (55) will be multiplied by factors which, as far as they involve ¢, will be of the form sin (2pt+0); bence, if we consider the mean valuc of these terms over a time involving a great many oscillations of the currents, they may be neglected: tho gain or loss of energy ropresonted by these terms is periodic, and at the end of a period the energy {s the same as at the beginning. Tho third term on the left hand side vanishes in our ease because the wires are at rest, and since 2, g, + vanish P’, Q’, R’ become identical with P,Q, R ‘Thus when the effects are periodic we see that equation (55) leads to the result that the mean value with respect to the time of [f[fevrearen dadyds is equal to that of Ff [ UBA—Oy) +m Py—Ra) +n (Qa—PA)} aS, ‘The first of these expressions is, however, the mean rate of heat production, and in the ease of a wire whose electrical state is symmetrical with respect to its axis, the value of the quantity under the sign of integration is the same at each point of the circumference of a circle whose plano is at right angles to the axis of the wire; hence in this case we have the result: ‘The mean rate of heat production per unit length of the wire is equal to the mean valuo of 4a (tangential clectromotive intensity) x (tangential magnetic force), (56) a, as before, being the radius of the wire, 282.] Let us apply this rosult to find the rate of heat pro- duction in the wire and in the outer conductor of a cable when the current is parallel to the axis of the wire. By the methods of Art. 268, we see that if the total current through the wire at the point ¢ is equal to the real part of Te met29, EE eet een 282.) “ELECTRICAL WAVES AND OSCILLATIONS, 315 or if m=—a-+ef, to Tye~** cos (—az+pt), then, Art. 268, equation (24), the electromotive intensity J in the wire parallel to the axis of 2 is equal to the real part of tom Jy (snr) - gs —as4pt) = ara dg(ns) Coors (7) If we neglect the polarization currents in the dielectric in comparison with the conduction currents through the wire, then the line integral of the magnetic force round the inner surface of the outer conductor must equal 49 J,¢(***); using this le we see that H in equation (11), Art. 262, equals =«n'o' I,j270K,' (en’b), and hence the electromotive intensity parallel to = in the outer conductor is equal to the real part of ton’ Ky(on'r) > 02 o(—a2+ph, ~ o5b % i) al 2 (58) the notation being the same as in Art. 262. The tangential magnetic force at the surface of the wire is (Art. 262) or. te 00s (— as + pt), (59) while that at the surface of the outer conductor is, if we neglect the polarization currents in the dielectric in comparison with the conduction currents through the wire, FF 5 (— 08+ (60) Let us now consider the case when the rate of alternation of the current is so slow that both ns and n’b are small quantities. When na is small J,/(¢na)=—ina/2, while J,(¢na)=1 ap- proximately; hence, putting r= a in (57), wo find that the tangential electromotive intensity is ele ™ 00s (—a2+pt). Hence by (56) and (59) the rate of heat production in the wire is equal to the mean value of sat ot 7** cost (—as + pt), 4 — | 816 BLECTRICAL WAVES AND OSCILLATIONS. = that is to salt, Let us now consider the rate of heat production in the outer conductor; since x'b is very small, we have approximately K,(in’b) = log (2y/en’b), Ki,’ (vn'b) =—1/in'b, Making these substitutions in (68), we see that the tangential electromotive intensity at the surface of the outer conductor ix equal to the real part of 2 pk log (2y/in') Tye-Pt t+, and since rn’ = 4,’ cp/o’, tho real part of this expression ia 2p’ p log (y-V 77a pb) [ye ** sin (—a2 +-pt) —ru'plye—** con (—a2+pt). Hence by (56) and (60) the rate of heat production in the outer conductor is equal to tou'p]ye™*, since the mean value with respect to the time of sin(—a=+pt) cos(—az+>pt) is zero, Thus, when n’b is small, the rate of heat production in the outer conductor is independent both of the radius and specifie resiatance of that conductor. Tho ratio of the heat produced in unit time in the wire to that produced in the outer conductor is thus 20/377a*y’p, which is very large since we have assumed that n'a, ie. 4zypa%/c, is a small quantity; in this caso, therefore, by far the larger proportion of the heat is produced in the wire. This explains the result found in Art, 263 that the rte of decay of the vibrations is nearly independent of the resistance of the outer conductor and depends almost wholly upon that of the wire, 288.] When the frequency is so grent that mo is large n’b is still small, then J, (cna) = iJ,’ (na), 80 that by (57) the tangential electromotive intensity at the surface of the wire ia equal to the real part of Fi ttzmp/eyie Beg (arp) — = — 284.) ELECTRICAL WAVES AND OSCILLATIONS. B17 which is equal to Sara |27HP/o}¥ Tye~* (005 (—a2+ pt)—sin (—a2 +p}. Hence by (56) and (59) the mean rate of heat production in the wire is equal to = Tra Since n’b is supposed to bo small the rate of heat production in the outer conductor is as before (2zpp/o)!T,2 «244, Fephte™, hence the ratio of the amount of heat produced in unit time in the wiro to that produced in the outer conductor is he 2 lie if ‘Thus, since n*at and so 4xppa*/ris very largo by hypothesis, we ~~ see that unless y/y’ is very large this ratio will be very amall; in other words the greater part of the heat ix produced in the outer conductor; this is in accordance with the result obtained in Art. 266, which showed that the rate of decay of the vibrations was independent of the resistance of the wire. 284.] When the frequency is so high that both na and »’b are Jarge, then the expression for the heat produced in the wire is that just found. To find the heat produced in the outer con- ductor we have, when n’b is very large, K, (on'b) = 1K (nb); heneo by (68) the tangential electromotive intensity in the outer conductor is equal to the real part of pe ~ 33b which is equal to = (Au ple) Tye-P* {000 (a5 + pt)—sin (—02+ pl). Hence by (55) and (60) the mean rate of heat production in the outer conductor is (Aap apy’ ge Pt @ oer), Sep Ot De Le. 318 ELECTRIOAL WAVES AND OSCILLATIONS, ‘Thus the ratio of the heat produced in unit time in the wir to that produced in the same time in the outer conductor is {ay/fet: so that if, ax is generally the case in cables, o” is very much greater than o, by far the larger part of the heat will be pro- duced in the outer conductor, Heat produced by Foucault Currents in a Transformer, 285.] We shall now proceed to consider the case discussed in Art. 278, where the lines of magnetic force are in planes through the axis of the wire, the currents flowing in circles in planes at right angles to this axis. This case is onc which is of great practical importance, as the conditions approximate to those which obtain in the soft iron cylindrical core of an in- duetion coil or a transformer; in this ease the windings of the primary coil are in plancs at right anglos to the axis of the iron cylinder, while the lines of magnetic foree due to the coil are in planes passing through this axis, Whon » variable eurrent is passing through the primary coil, currents are induced which heat the core and the heat thus produced is wasted as far as the production of useful work is concerned; it is thus a matter of importance to investigate the laws which govern its development, so that the apparatus may be designed in such a way as to roduce this waste tom minimum. We shall that the magnetio force parallel to the axis at the surface of the wire is represented by the real part of Heist, or if m= a+cf, by He-** con (az + pt), ‘The magnetic force at the surface of the cylinder is the most convenient quantity in which to express the rate of hoat pro- duction, for it is due entirely to the external field and is not, when the field is uniform, affected by the currents in the wire itself. Using the notation of Art. 278 we see by the results of that article that in the wire = AS, (une) dP, honee @ =P ATi (nr) emer, but sinoe at the surface of the wire, c is equal to the real part of He Pt tated), wo s00 that at the surface © = real part of pFd (oma) Bs .<(aze pl) STitona) Oeste, (61) Let us first take tho case when the radius of the wire is so small that na is small; in this case we have, since approxi- , oot J()=1- at ae Jy (2) =—tef1 == and n® = dinpip/o, © =real part of hap fr —FAPE Le Beeierrro = }uapHe-** sin (az+ pt)— pre cos (ae+pt). But by equation (56) the rate of heat production in the wire per unit length is equal to the mean value —}aO@He—** cos (az+pt); where the minus sign has been taken because (Art. 280) @ H is proportional to the rate of flow of energy in the direction of translation of a right-handed screw twisting from © to H; in this case this direction is radially outwards. ‘Thus the rate of heat production in the wire is te -ape Fagtetptat te, and is thus proportional to the conductivity, 20 that good conductors will in this case absorb more energy than bad ones. = ee Let us now apply this result to find the in tho core of a transformer or induction coil. wuppose that the eore consists of iron wiro of circular the wires being insulated from each other by the when the magnetic force due to the primary coil is u along the axis of the coil and over its cross-section. external magnetic foree is uniform alongs, the axis of a currents induced in the wire by the variation of the mag force flow in circles whose planes are at right angles to the intensities of the currents are independent of the Under these circumstances the currents in the wire do not rise to any magnetic force outside it. The magnotic outside the wires will thus be due entirely to the primary coil, and as this magnetic force is uniform over the cross-section it will be the same for each of the wires, 80 that we can apply the preceding investigation to the wires separately. In order to use the whole of the iron, the magnetic fores must be approximately uniformly distributed over the cross-section of the wires; for this to be the case ma must be small, as we have seen that when wa is large the magnetic foree is eon- fined to a thin skin round each wire. For soft iron, for which we may put » = 10%, « = 104, the condition that wa is small implies that when the primary current makes one hundred alternations per second, the radius of tho wire should not be moro than half a millimetre. If now the total cross-section of the iron is kept constant so as to keep the magnotic induction through the core constant, we have, if N is the number of wires, A the total cross-section of the iron, Naa’ =A. The heat produced in all the wires per unit length of core in one second is, if 7 is the maximum magnetic force due to the coil, N ere, x Be Teron PM, and is thus inversely proportional to the number of wires We may therefore diminish the waste of energy due to the heat L — 286] ETROTRIOAL WAVES AND oscrtnations, B21 produced by the induced currents in the wires by increasing the number of wires in the core. We thus arrive at the practical rule that to diminish the waste of work by eddy currents the core should be made up of as fine wire as possible. In many trans- formers the iron core is built up of thin plates instead of wires ; when this is the case the advantage of a fine sub-division of the core is even more striking than for wires, for we can easily prove that the work wasted by eddy currents is inversely pro- portional to the square of the number of plates (see J. J. Thom- | son, Electrician, 28, p, 599, 1892). | If y is the current flowing through the primary coil and V | the number of turns of this coil per centimetre, then Hoospt = 4Ny, and 32? = 167*V* (mean value of y*), thus in the case of a cylindrical coro of radius a the heat pro- OSL ale lie cea ad 7 u*p*at N22 (mean value of y/o. re Paicinnine ofa cirenit (Art. 272) the heat produced in unit time is equal to Q (mean value of 7); thus the core will increase the impedance of the primary coil by Qn pt pat NAY /, 286.] Let us now consider the case when na islarge ; here we havo Sy (ena) =—«J, (ina), and since nt = Azpip/a, we see by (61), putting a and 8 equal to zoro, that © = real part of — [Pee Tet dz Bae = =— “ga HE lcoapt sin pt}. But by equation (56) the rate of heat production per unit Jength is equal to the mean value of — $80H cos pt, and is thus equal to i [P= om, eee | ‘We can show, as before, that this corresponds to an increase in the impedance of the primary circuit equal to 40° N® {puo/2n}ta. In this case the heat produced is proportional to the square root of the specific resistance of the core, so the worse the eon- ductivity of the core the greater the amount of heat produced by eddy currents, whereas in the case when na was small, the greater the conductivity of the core the greater was the loss due to heating. When na is large, the heat produced varies as the eireumfer- ence of the core instead of, as in the previous case, as the square of the area; it also varies much more slowly with the and magnetic permeability, This is due to the fact that when na is large the currents are not uniformly distributed over the core but confined to a thin layer on the outside, the thickness of this layer diminishing aa the magnetic permeability or the fre- quency increases; thus, though an incroase in » or p may be accompanied by an increase in the intensity of the currents, it will also be attended by « diminution in the area over which the currents are spread, and thus the effect an the heat produced of the increase in p or » will not be so great as in the previous ease when 7a was small, and when no limitation in the area over which the current was spread accompanied an increase in the frequency or magnetic permoability. If we compare the absorption of energy when ”a is large by cores of iron and copper of the same size subject to alternating currents of the same frequency, we find—since for iron w may be taken as 10° and o as 10‘, while for copper p= 1, ¢ = 1600,— that the absorption of energy by the iron core is between 76 and 80 times that by the copper. The greater absorption by the iron can be very easily shown by an experiment of the kind Ggured in Art. 85, in which two coils are placed in the circuit connecting the outer contings of two Leyden Jars; in one of these coils an exhausted bulb is placed, while the core in which the produced is to be measured is placed in the other. When oscillating current produced by the discharge of the jars through the coils a brilliant discharge passes through the hausted bulb in A, if the coil B is empty or if it tt copper cylinder; if however an iron eylinder of the same Ke ay 287.] ELECTRICAL WAves AND OSCILLATIONS, 323 feplaces the copper onc, the discharge in the bulb is at once extinguished, showing thet the iron eylinder has absorbed a great deal more cnergy than the copper one. This experiment also shows that iron retaine its magnetic properties even when the forces to which it is exposed are reversed, as in this ex- periment, millions of times in a second. 287.] Another remarkable result is that though a cylinder or tube of a non-magnetic metal does not stop the discharge in the bulb in A, yet if a piece of glass tubing of the same size is coated with thin tinfoil or Dutch metal, or if it has « film of silver deposited upon it, it will check the discharge very decidedly. We are thus led to the somewhat unexpected result that a thin layer of metal when exposed to very rapidly alternating currents may absorb more energy than a thick layer. The following investigation affords the explanation of this, and shows that there is a certain thickness for which the heat produced is a maximum. This result can easily be verified by the ment just described, for if an excessively thin film of silver is deposited on a beaker very little offect is produced on the dis- charge in the bulb placed in A, but if successive layera of very thin tinfoil are wrapped round the beaker over the silver film the brightness of the discharge in A at first rapidly diminishes, it however soon increases again, and when a fow layers of tinfoil have been wrapped round the beaker the discharge becomes almost as bright as if the beaker wore away. To investigate the theory of this effect we shall calculate the energy absorbed by a metal tube of cireular cross-section, when placed inside a primary coil whose windings are in planes at right angles to the axis of the tube; this coil is supposed to be long, and uniformly wound, so that the distribution of magnetic force and current is the same in all planes at right angles to its axis. We shall use the same notation as before; the only bols which it is necessary to define again are a and b, are respectively the internal and external radius of the and V the velocity with which eloctromagnetic action propagated through the diclectrie inside the tube, The fic force outside the tubo is represented by the real part He, and this foree is due entirely to the currents in the coil, ¥2 = = 287.) ELECTRICAL WAVES AND OSOILLATIONS. ‘To determine A we have the condition that when rab, y= er, hence He = {BJ, (mb) + 0K, (enb)} Tn order to find the heat produced in the tube we require the value of @ when r= b; but here e=- = {BJ,i (snd) + CK, (smb) } cP. Eliminating B and C from these equations, we find -9O= real part of Het?! “<2 x i (on) Ky 09) = ob) EY (09) + 2° unm) Ke Tho offect we are considering is one which is observed when ‘the rate of alternation of the current is very high, so that both na and nb aro very large; but whon this is the case na Ji (una) = oe Je (una) =~ E,(\na)=<"* = Ey (ina) = e-** " — un Vaunb Vinnd ao [= Ky ai fa. Ko(enb) =« nl gest o (snb) = te tsi making these substitutions and writing k for b—a, we find — Q= real part of wh nh, Mh nh, nh Sits ers Jy (enb) = J, (enb) =— He? (62) Now since ma is very large, na/u is also very large for the non-magnetic metals, and even for the magnetic metals if the frequency of the currents in the primary is exceedingly large, - 326 ELECIRIOAL WAVES AND OSCILLATIONS, ‘but when this is the case, then, unless h is so small that no longer large, we may write equation (62) as = 0 = real part of why mh + ats 3) Since n = {4mpip/r}4, we may write n=, (141), where n, = {2a yp/c}4, and equation (63) becomes on, 2h —-2mb ies +2 cin Bh F308 pt de .2m¥ IF 2 con 2m,h 4am etm 2m asin amy h py . , Aa md, 3h 9 con anh ae. In calculating the part of the energy flowing into the tube which is converted into heat, we need only consider the part which flows across tho outer surface of the tubo, because the energy flowing across the inner surface is equal to that which flows into the dielectric inside the tube, and sinco there is no dissipation of energy in this region the average of the flow of energy across the inner surface of the tube must vanish. Hence the amount of heat produced in unit time in the tube is by equa- tion (30) equal to the mean value of — 4bOT cos pt, where the — sign has been taken because the translatory motion of a right-handed screw twisting from © to H is radially outwards ; this by (64) is oqual to ansb (29% 4 2nin 2h) pry. 16m mks 24 _Doosomh | when 2h is vory large this is equal to omb Ter’ which is (Art, 286), as it ought to be, the same as for a solid cylinder of radius b. When fis small and n*ah/y not large we must take into account terms which we bave neglected in arriving at the preceding expression. L 287.) ELECTRICAL WAVES AND OSCILLATIONS, 827 In this case, we find from (62) that tepPet W/r) He Hein, on Era tinepeye! — () so that the rate of heat production is 4 (xpta'bh/o) H? 14 dnt pathy ‘Thus it vanishes when A = 0, and is a maximum when =ara3 the rate of heat production is then vepbaH?, and bears to the rate when the tube is golid the ratio which is equal to n,a/2y. Since n,a/« is very large the heat produced in a tube of this thickness is very much groater than that produced in a eolid eylinder, Lot us take the case of a tin tubo whose internal radius is 3 cm. surrounded by a primary coil conveying a current making a hundred thousand vibrations per second, then since in this ease gg = 13x10, a=3, p=2ax 10, p=1, the thickness which gives the maximum heat production is about 1/90 of a millimetre, and the heat produced is about 26 times as much as would be produced in a solid tin cylinder of the same radius as the tube. We see from equation (65) that the amplitude of © diminishes ‘as tho thickness of the plate increases, but that when the plate is indefinitely thin the phasea of the tangential electromotive intensity and of the tangential magnetic force differ by a quarter- period ; the product of these quantities will thus be proportional t sin 2pt, and ax the mean value of this vanishes there is no converted into heat in the tube. As the thickness of the tube increases the amplitude of © diminishes, but the phase of @ gets more nearly into unison with that of 1. We may regard © as made up of two oscillations, one being in the same phase as Hf while the phase of the other diffors from that of Hl vy =] 328 ELECTRICAL WAVES AND OSCELLATIONS. ‘a quarter-period. The amplitude of the second HEstslahed oa he \laknses of il cals alors the first reaches a maximum when h = o/27ap. “= Ia the nvtgatin of tn a red whe Xt el nays has doen assumed large. We can however easily show that unless this is the case the heat produced in w thin tube will not exceed that produced in a solid eylinder, Vibrations of Electrical Systems. 288.] If tho distribution of electricity on a system in electrical equilibrium is suddenly disturbed, the clectricity will redis- tribute itself so as to tend to go back to the distribution it had when in cloctrical equilibrium ; to effect this redistribution electric currents will be started. The currents possess kineti¢ «nergy which is obtained at the expense of the potential energy of the original distribution of electricity; this kinetic energy will go on increasing until the distribution of electricity is the same as it was in the state from which it was displaced. As this state is ono of equilibrium its potential energy-is a minimum. The kinetic energy which the system has acquired will carry it through this state, and the system will go on losing kinetic and reacquiring potential energy until the kinetic energy has all disappeared, ‘The systern will then rotrace ite steps, and if there is no dissipation of energy will again regain the distribution of electricity from which it started. The distribution of elee- tricity on the system will thus oscillate backwards and for- wards; we shall in the following articles endeavour to caleulate the time taken by such oscillations for some of the simpler electrical systems, Electrical Oscillations when Two Equal Spheres are connected by a Wire®, 289.] The first case we shall consider is that of two equal spheres, or any two bodies possessing equal electric capacities, connected by a straight wire. This case can be solved at once by moans of the analysis given at the beginning of this chapter. Let us take the point on the wire midway between the spheres as the origin of coordinates, and the axis of the wire ag * Seo J.J. Thomson, Proc. Lond. Mark. Soe. 19, p. 542, 1888, | | 290.) -BEROTRICAL WAVES AND OSCILLATIONS. 829 the axis of 2. We shall suppose that the electrostatic potential has equal and opposite values at points on the wire equidistant from the origin and on opposite sides of it. Then using the same notation-ax in Art. 271, we may put o= Le" ™) J, (cm) e*, in the wire, = 1 (em etm) et approximately, since mr will be very small. Thus B, the ox- ternal clectromotive intensity parallel to the wire, is equal to em L (eo) If 21 is the length of the wire, then the potential of the sphere at the end = J, will be 2uLsin mle”, If C is the capacity of the sphere at one end of the wire, the quantity of electricity on the ephere is 2.CLeinmle'?, and this increases at the rate —2pLsinml Now the increase in the charge of the sphere must equal the current flowing through the wire at the point z=, hence if I denotes this current, we have IT=—2CpLsinml<, but by equation (39) of Art. 272 wo have E=(:pP+Q)/, whence substituting the values for B and J when s = f, we got =2imE cosml = —(spP+@)2CpLsin mle”, or moot ml = —cp (ipP+Q)C. (66) 290.) Let us first consider the case when the wave length af electrical vibrations is very much longer than the wire; here mm is very small, so that equation (66) becomes }=—(\P+@)C. (67) ‘The values of P and Q, the self-induction and impedance of the wire, are given in equation (40) of Art. 272; they depend upon the frequency of the electrical vibrations. When this is eo t = Ee = = 291.) BLECTRICAL WAVES AND OSCILLATIONS. 331 negative electricity as there was initially on B, while the eharge on B is the same as that originally on A, When the negative charge on A has reached this value it begins to decrease, and after a time both spheres are again free from electrification. After this positive electricity begins to reappear on A, and increases until the charge on A is the same as it was to begin with; this positive charge thon decreases, vanishes, and is replaced by a negative one as before. The system thus behaves aa if tho charges vibrated backwards and forwards betweon tho spheres. The changes which take place in tho electrical charges on the spheres are of course accompanied by currents in the wire, these eurrents flowing sometimes in one direction, sometimes in the opposite, When the circuit has » finite resistance the amplitude of the oscillations gradually diminishes, while if the resistance is greater than (8 £/C) there will not be any vibrations at all, but the charges will subside to zero without ever changing sign ; in this ease the current in the connecting wire is always in one direction. 291.) If we assume that the wave length of the electrical vibrations is so great that tho current may be regarded as uniform all along the wire, and that the vibrations are so slow that the current is uniformly distributed across the wire, the discharge of a condenser can easily be investigated by the following method, which is duo to Lord Kelvin (Phil. Mag. [4], 5, p. 393, 1853), Let Q be the quantity of electricity on one of the plates of a condenser whose capacity is C’ and whose plates, like those of x Leyden Jur, are supposed to be close together ; also let R be the resistance and J the coefficient of self-induction for steady currents of the wire connecting the plates. The electromotive force tending to increase Q is —Q/0”; of this RdQ/dt is required to overcome the resistance and Ld*Q/dt* to overcome the inertia of the cireuit; honce we have Dees Boe a =0. (69) The solution of this equation is, if 1 R ” CL” 4p" =z! re Q= Ae cos Sor — He) era}, where A and 8 are arbitrary constants. Aas — = 882 BLEOTRICAL WAVES AND OSCI In this case we have an oscillatory d As equal.to 2 we (Gr-an)* 1 Re _ When er the capacity and self-induction of the circuit must be very largo when tho discharging circuit consists of metal wire of customary dimensions. Thus, to take an example, suppose two spheres each one metre in radius are connected by a copper wire 1 millimetre in diamoter, In this case = 1/9x 1078, « = 1600, a= -05, p= 1, substituting these values we find that to ensure na being amall, the self-induction of the cireuit must be comparable with the enormously large value 10", which is comparable with the self- induction of a coil with 10,000 turns of wire, the coil being about half a metre in diameter. Tho result of this example is sufficient to show that it is only Lees eapesiree of the cireuit or the capacity of the is exceptionally large that a theory based on the See teas ania ool cows ta sapien fore important to consider tho caso whore na is largo and the currents in the discharging circuit are on the eurface of the wire, 204.] The theory of this case is given in Art. 274, and we see from equations (42) and (43) of that Article that when the frequency of the vibrations is so great that na and n’b (using the notation of Art. 274, and supposing that the wire connecting the spheres is a cable whose external mdius ix b) are large quantities, equation (64) of Art. 289 becomes meat mi = —2«p {ip logb/a+(ip)!(uo/tz0")t +(pit(we'Aab HC. Retaining the condition that m/ is small, which will be the case when the wave length of the electrical vibrations is very much greater than the length of the discharging circuit, this equation becomes G1 =~! Lap Blog (ofa) + (ip)! (ne/teat)! + (w/oa sb)! 4 — | 293.) ELECTRICAL WAVES AND OSCILLATIONS. 337 which is the condition that the syatem should execute electrical vibrations. When the spheres are connected by a free wire and not by a cable o'/b* vanishes, and the condition that the system should Oeeillate reduces 9 p(y /-g%)0< $21/3/27 BC. ‘Tho results given by Ferrari's mothod for solving biquadratic equations are too complicated to be of much practical value in determining the roots of equation (71), neither, since the roots are imaginary, can we apply the very convenient method known as Hornor's method to determine the numerical value of these roots to any required accuracy. 295.] For the purpose of analysing the nature of the electrical oscillations it is convenient to consider separately the real and imaginary parts of ip, the » of equation (71). The real part, supposed negative, determines the rate at which the electrical vibrations die away, whilo tho imaginary part gives the period ‘of those vibrations, We shall now proceod to show how equa-~ tion (71) can be treated so as to admit of the real and imaginary parts of « being separately determined by Horner's method. Tf we put. s f=2-T» equation (71) becomes &407E+40¢+I—sH* =0, (72) where H, G, I are the quantities whose values we have just written down. Since the coefficient of & in this equation yanishes and since its roots are by hypothesis complex, we see that tho real part of one pair of roots will be positive, that of the other pair negative: the pair of roots whose real parts aro negative are those which correspond to the solution of the electrical problem. For if the real part of ¢ wero positive the real part of cp would also be positive, so that such a root would to an olectrical vibration whose amplitudo increased indefinitely with the time. The roots of equation (72) will be of the form Ato AMY “At, —A— Yr We shall now proceed to show how «, may be uniquely deter- mined, Since 6H, —$@,1—3H? are respectively the wums ot z = and three, and all together, we . yitye—2a? = 6H, (7) 2, (y'—y2) = 26, 4) (+4) (ei! +y¢) = I-38, or atta (y tty) +t Qty —W uh = 1-3 es yetys and yJ—y,* by a (73) amd (74), sioittaaaaibhoi Sao, or putting 2? =», tp419Hy 4 (02° —D)n_-@ = 0. 5) Since the last term of this expression is negative there is ab least one positive real root of this equation, and since the values given for H and J show that when A is positive 12 H* —7 is easen- tially negative, we see by Fourier's rule that there is only one such root. But since a, is real the value of » will be positive, so that the root we are seeking will be the unique positive real root of equation (75), which can easily be determined by Horner's method, The value of «, is equal to minus the square root of this root, and knowing 2, we can find y,*/47%, the square of the corresponding frequency uniquely from equations (73) and (74). We can in this way in any special case determine with ease the logarithmic decrement and the frequency of the vibrations. 296.] If in equation (75) wo substitute the values of G, #, and dente oe One that equation becomes @4+207(1—8q)—£(3q"—49)—g (1g)? = 0 We can by successive approximations expand (in terms of g, and thus when @S‘/Z” is small approximate to the value of ¢ ‘The first term in this expansion is C= (9/2)', or since ez =% ee SEs 5 =~ FigirA | —— 296.) BLROTRICAL WAVES AND OSCILLATIONS. 339 ‘The corresponding value of y,? determined by equations (73) ee en sree t= polar} Now S= {po/4 sa*)+(u'o//a rb*)h) 20, and, approximately, we Te and 2=— 2, hence we see, , == Fp luny,/2 m0!) + ilo"y,/2ad!)}. But by Art. 274, the quantity enclosed in brackets is equal to Q, the impedance of unit length of the circuit when the frequency of vibration is y,/2x; thus we have “= 9p whore Q is the impedance of the whole circuit. 4 s Since P= tnt Ta =a, (1—(2q)*} +1, the real part of «p differs from «, by a quantity involving q. Noglecting this term, we seo that the expression for the amplitude Ss: of the vibrations contains the factor « 2/’'. Comparing this R with the factor ¢~22", whieh oceurs when the oscillations aro #0 slow that the current is uniformly distributed over the cross- rection of the discharging wire, we find that to our order of ap- proximation we may for quick vibrations use a similar formula for the decay of the amplitude to that which holds for slow vibrations, provided we use the impedance instead of the resist- ance, and the coefficient of self-induction for infinitely rapid vibrations instead of that for infinitely slow ones. This result is, however, only true when C'S'/L is a small quantity. Now if ‘the external conductor is so far away that p’o’/b* is small com- pared with uo/n?, thon St = (21 (pe/sna®)h}t = 2228 RA, za 340 ELECTRICAL WAVES AND OSCILLATIONS. where Ris the resistance of the whole circuit to stendy current, Substituting this valuo for S* we see that the condition that CS4/L* is a small quantity is that Cl)? R?/4 L® should be small. When this is the case wo see that, neglecting the effect of the external conductor, : Su seern (noy,/2 na*)h. Since @, is proportional to p#, tho rate of decay of the yibra- tions will be greater when the discharging wire is made of iron than when it is made of a non-magnetic metal of the same resistance. This has been observed by Trowbridge (Phil. Mag, [5], 32, p. 504, 1891). 297.] We have assumed in the preceding work that the length of the electrical wave is great compared with that of the wire; we have by equation (66) moot nl =—sp {ipP +Q) 0 When the frequency is very high, «pP will be very large com- pared with Q, hence this equation may bo written as mootml = pPc, Nowif Vis the velocity of light in the dielectric, p = Vm, hence irahare cotmd _ V22PlC m oF Now 2P/ is equal to I’, the self-induction of the discharging cirouit for infinitely rapid vibrations, and V"C is equal to the electrostatic measure of the capacity of the sphere which we shall denote by [C], hence tho preceding equation may be written as cotml — L'[C] —= ml ae” Thus, if L’(C]/20 is very largo, m/ will be very small; if, on tho other hand, L’[0]/2?" is very small, cot m2 will be very small, or nl = (2) + nF approximately, where 7 is an integer. Sinco 2r/m ia the length of the electrical wave the latter will equal 4l, 41/3, 4//5..., or the half-wave length will be an odd sub- multiple of the length of the discharging wire. We are limited by our investigation to tho odd submultiplo becauso we have assumed that the current in the discharging wire is sym- SE . 298.] BLECTRICAL WAVES AND OSCILLATIONS. 341 metrical about the middle point of that wire. If we abandon this assamption we find that the half-wave length may be any submultiple of the longth of the wire. The frequencies of the vibrations are thus independent of the capacity at the end of the wire provided this is small enough to make L[C]/2/ small. In this cave the vibrations are determined merely by the condition that the current in the discharging wire should vanish at ite pact Vibrations along Wires in Multiple Arc. 298.] When the capacities of the conductors at the ends of a single wire are very small, we have seen that the gravest electrical vibration has for its wave length twice the length of the wire and that the other vibrations are harmonies of this, We shall now investigate the periods of vibration of the system when the two conductors of small capacity are connected by two or more wires in parallel. The first caso we shall consider is the ono represented by Fig. 109, where in the connection be- tween the points A and F we have the loop BCED, 4 ee oe ¥ DB Fig. 109. We proved in Art, 272 that the relation between the current F and tho externul electromotive intensity E is expressed by the syoation. B= {pP+Qil ‘Where, when as in this case the vibrations are rapid enough to make ma large, the term ipP is much larger than @, we may therefore for cur purpose write this equation as B= .pPi, (76) where P is the coefficient of self-induction of unit length of the wire for infinitely rapid vibrations. Let the position of a point on AB be fixed by the length s, measured along AB from A, that of one on BOE by the length #, measured from BZ, that of one on BDE by s, measured also from B, and of one on EF by a, measured from Z, Lat Ula potential, external electromotive intensity along a wire is —d¢/ds, and as this is proportional to the current it must vanish at the ends A, ¥ of the wire if the capacity there is, as we suppose, vory small, Hence along AB we may write, if p/27 is the frequency, $ = acosms, cospt, along BCE 4 _ (acos ma, cos ml, +b sin ms,) cos pt, along BDE 4 — (a cos ms, 00s ml, +¢sin may) 008 pt, andalong EF — = dos m(s,—1,) 008 pt. Equating the expreasions for the potential at B, we have peorlmi pe ar” 7 acos mi, cos ml, +csin ml, = deos mi, . (77) The current flowing along AB at B must equal the sum of the currents flowing along BCE, BDB, hence by (76) we have a =-3-F (78) Again, the current along EF at # must equal the sum of the currents flowing along BCL, BDE, hence we have dsinml, _beosml, asinmt, cos ml, re coos ml, RO & ay 4 £ asin mi, cos me, on We get from equations (77) and (78) é a fea cot ml, cos my — cobmisoe ml} =—de0s ml} te +m From equations (77) and (79) we get a{anml _ cot miyeos mt, cohen conned x & : =—ecnem [Same 4 Sm ill (79) SSS] 298.) ELBOTRIGAL WAVES AND OSCILLATIONS. 343 Eliminating « and d from these equations, we get pps sy ese sy a ee (80) It AB and EF are equal lengths of the same kind of wire, t, =, and Hf = P,, and (80) reduces to the simple form tan mé, cot ml, se = 4 {Seemh , comeomby, 77 B B Mee acre dg, He lave tanml, _ cot Imi, are =e + tinh, (81) if we take the lower sign, we have tanml, ___ {tantml, | tan}ml, {= > }- (82) Since m = 2x/A, where A is the wave length, these equations determine the wave lengths of the clectrical vibrations. If all the wires have the samo radius, 2 = 7, = 2, and equa- tions (81) and (82) become respectively tan 278 = cot (x) +00t(x'), (81*) and tan2x4 + tans”? + tan? =o, (e2*) From these equations we can determine the effect on the period of an alteration in the length of one of the wires. Suppose that the length of BDZ is increased by 2,, and let 8a be the correspanding increase in A, then from (81*) PS 1 soot 274 + 41, conee? ™ + 41, c08002™} = 42t,coe00!™. ‘Wo see from this equation that 4 and J, are of the same sign, so that an increase in 2, increases the wave length. If we take equation (82*), we have Minden atime minha, hhenoo, in this case also, an increase in /, increases A. If LU, ia goes on until /, vanishes, when the wave length of the gravest vibration is 41,. 299.] The currents through the wires BUZ and BDE are at B in the proportion of oo tml, , cot dm So el sty ; if we take the vibrations corresponding to equation (81), and in Mio proportion of maps Hs tan} ml, Beas for the vibration she. We can prove by the method of Art. 298 that if we have n wires between B and F, and if AB = EF, tanml cobmls cotmly _ a E E “a cosecml, , cosecmil, coace me, = + = 2S Tt follows from this equation that if any of the wires are shortened the wave lengths of the vibrations are also shortened. Exgorrican Oscriuations ox CrtinpEns. Periods of Vibration of Electricity on the Cylindrical Cavity inwide a Conductor. + 300.] If on the surface of a cylindrical cavity inside a conductor an irregular distribution of electricity is produced, then on the removal of the cause producing this currents of electricity will flow from one part of the ojlindas te another to restore the electrical equilibrium, electrical vibra~ tions will thus be started whose periods we now proceed to investigate. ‘Take the axis of the cylinder as the axis of ¢, and suppose that initially the distribution of electricity is tho same on all sections at right angles to the axie of the cylinder; it will evidently remain so, and the currents which restore the electrical —4 300] RLRCTRICAL WAVES AND OSCILLATIONS. 345 pga Waitin gt angen So sia . ‘Ife is the magnetic induction parallel to s, then in the cavity filled with the dielectric ¢ satisfies the differential equation Be Me _ 1 de dat" dyt~ V? de" where V is the velocity of propagation of electrodynamic action through the dielectric. vse ips lpertnals ages Po txude Beta “o a’ whore ¢ is the specific resistance and » the magnetic permea- bility of the substance. ‘Transform these equations to polar coordinates r and @, and suppose that ¢ varies as cos sd?" making these assumptians, the differential equation satisfied by c in the dielectric is @e, ide gf # dA * sar tap) =% the solution of which is ¢= Acs 20,( 4 rer, where J, denotes the internal Bessel’s function of the s'> order. ‘The differential equation satisfied bye in the conductor is @e ide 4zucp 8) det tart — Se? _ Sho= ‘e Let n* = 4zusp/o, then the solution of this equation is ¢ = Boos 26K, (cnr), whore X, denotes the external Bessel’s function of the e* order. Sines the magnetic force parallel to the surface of the cylinder is continuous, we have if a denotes the radius of the cylindrical ? AV,(Ha) = 2 E, (ine). (83) ‘The electromotive intensity at right angles to r is also con- tinuous. Now the current at right angles to r and ¢ is — dij mpd, = torand zis —cde/ixpdr. In the dielectric the current is ‘to the rate of increase of the electric displacement, ie. to sp times the electric displacement or to ip K/tm times the electro motive intensity; we see that in the dicleetric the electromatire intensity perpendicular to r is ~ pap Ip bane wo hava Aa Pra) = 3 enok’ (ona) (et) Eliminating A and B from (83) and (84), we got re a) a Gr Sas (8s) Gp) Now K = py and o=*"*'P oo that (85) may be written » K(na), (24) a E,(ma)” Now the wave length of the electrical vibrations will be com- | parable with the diameter of the cylindor, and the value of p corresponding to this will bo sufficiont to make na oxceedingly large, but when na is very large we have (Heine, Kugelfunc- tionen, vol. i, p. 248) EK, (ena) = (-'em, / approximately, hence K’,(:ma)=+X, (ina); thus the right-hand side of (86) will be execcdingly small, and on approximate solution of this equation will be Vv J, G =0. ‘This signifies that the tangential electromotive intensity vanishes at the surface of the cylinder, or that the tubes of electrostatic induction cut its surface at right angles. The roots of the equation Iiley= 301] ELECTRICAL WAVES AND OSCILLATIONS. 347 | fors= 1, 2, 3, are given in the following table taken from Lord | Rayleigh’s Theory of Sound, Vol. IL, p. 266:— Thus, when s = 1, the gravest period of the electrical vibra- tions is given by the equation ce ia = 1.841, or the wave length of the vibration 2% V/p =-543 x 27a, and is thus more than half the circumference of the cylinder. In this case, as far as our approximations go, there is no decay of the vibrations, though if we took into account the right-hand side of (86) we should find there was a small imaginary term in the expression for p, which would indicate a gradual fading away of the vibrations, If it were not for the resistance of the conductor the oscillations would last for ever, as there is no radiation of energy away from the cylinder, The magnetic force vanishes in the conductor except just in the neighbourhood of the cavity, and the magnetic waves emitted by one portion of the walls of the cavity will be reflected from another portion, so thab no energy escapes. Metal Cylinder ewrrounded by a Diclectric. 01.) In this case the waves starting from one portion of tho cylinder travel away through the dielectric and carry energy with thom, so that the vibrations will dio away independently of the resistance of the conductor. ‘Using the same notation as before, we have in the conducting be c= A cosséJ, (cnr) &?, and in the surrounding dielectric ¢= Booss0K,(Fr) e™, _— 301.) BEEOTHIOAL WAVES AND OSoILEATIONS, 349 When «= 1, i it 3 15 105 Kia=o sf +a” Baap * 2(eeP ~ nf: 5 erie 7 87 195 Kia) =r + sie * 128 (a) — Tosca a To approximate to the roots of the equation K,’(«)= 0, put i# = y, and equate the first four terms inside the bracket to zero ; we get 7 5T 195 ¥+ 59+ i999 tora = @ cubic equation to determine y. One root of this equation is real and positive, the other two are imaginary; if a is the positive root, B4vy the two imaginary roote, then we have 7 o+2B=—5r 2Bathtry= sa. 196 a(F +7) = tona” We find by the rules for the solution of numerical equations that « = +26 approximately, hence A=- 66, y= HOt These roota are however not large enough for the approxima- tion to be close to the accurate values. Hence from equation (88), we see that when s = 1, Pea —-55 40-64, v or tp = (= 56 + 0-64) > ‘This representa a vibration whose period is 3.1 7a/V, and whose amplitude fades away to 1/c of its original value after a time 1.8a/V. ‘The radiation of energy away from the sphere in this case is 80 rapid that the vibrations aro practically dead beat; thus after ‘one complete vibration the amplitude is only «~*7**, or about one two hundred and fiftieth part of its value at the beginning of the oscillation. — {ss cos “64 (Zi=") 64 in 64 (E*}. ‘Thus R vanishes at all points on a series of cylinders eon- centrie with the original one whose radii satisfy the equation OR cob -64 (Se )= 113, the distance between the consecutive cylinders in this series is 1.57 7a. The Famday tubes between two such cylinders form elosed curves, all cutting at right angles the cylinder for whieh @=0, or cos 6a (VERY 2) =I: ‘The closed Faraday tubes moye away from the cylinder and are the vehicles by which the energy of the cylinder radiates into space. The axes of the Faraday tubes, io, the lines of electromotive intensity between two cylinders at which R=O, are represented in Fig. 110. a ‘The genesis of these closed! endless tubes from the unclosed ones, which originally stretched from one point to another of the — =) 352 ELECTRICAL WAVES AND OSCILLATIONS. now overpower the inside pressure and will p -) tion shown in the socond position 8 of tho tubo; this tion inereases until the two sides of the tube mect as in the third position ¢ of the tube; when this takes place the tube breaks up, the outer part D travelling out into space and forming one of the closed tubes shown in Fig, 111, while the inner part & rans into the cylinder. Devay of Magnetic Force in a Metal Cylinder. 808.] In addition to the very rapid oscillations we have just investigated there are other and slower changes which may occur in the electrical state of the cylinder. Thus, for example, a uniform magnetic field parallel to the axis of the eylinder might suddenly be removed; the alteration in the magnetic foren would then induce currents in the cylinder whose magnetic action would tend to maintain the original state of the magnetic field, so that the ficld instead of sinking abruptly to zero would dio away gradually. The rate at which the state of the system changes with the time in cases like this is exceedingly slow compared with the rate of change. we have just investigated. Using the same notation as in the preceding investigation, it will be slow enough to make pa/V an exceedingly small quantity; when however pa/V is very small, K’,(pa/V) is execedingly large compared with K,(pa/V), since (Heino, Kugelfwnctionen, vol. i. p. 237) K, (0) is equal to (nop yas thus since whon @ is small ,/0) is proportional to log 6,, K,(po/V) is proportional to (V/pa), and K,'(pa/V) to (Vjpa)'*; heneo the right-hand side of equation (87) is oxceed- ingly large, so that an approximate solution of that equation pailhe J (oma) = 0, We notice that this condition makes the normal electromotive intensity at the surface of the cylinder vanish, while it will be remembered that for the vory rapid oscillations tho tangential electromotive intensity vanished. As the normal vanishes there is no electrification on the surface of the cylinder in this case, The equation /,(z)=0 has an infinite number of roots all 303.) RUROTRIGAL WAVES AND OSCILLATIONS, 353, real, the smaller values of which from «= 0 to ¢ = 5 are given in the following table, taken from Lord Rayleigh’s Theory of ‘This table may be supplemented by the aid of the theorem that the large roots of the equation got by equating a Bessel's function to zero form approximately an arithmetical progression whose common difference is 7. If x, denotes a root of the equation J, (e) =0, then since p is given by the equation J, (ima) = 0, where aa HP, we see that p,, the corresponding value of p, is given by the equation See as Pa = nen ‘Thus, since :p, is real and negative, the system simply fades away to its position of equilibrium and does not oscillate about it. The term in ¢ which was initially expressed by ‘ cos 80J,(2,") | will after the lapse of a time # have diminished to Acoss0J, (a1, oye Saat If we call 7 the time which must elapse before the term sinks to 1/¢ of its original value, the ‘time modulus’ of the term, then, since rates, 354 BLECTRICAL WAVES AND OSCILLATIONS. we see that the time modulus is inversely resistance of unit length of the cylinder and tional to the magnetic permeability. Since n/a for iron i% larger than it is for copper, the magnetic force will fade away mare slowly in an iron cylinder than in a copper one. | 804,] A case of great interest, which can be solved without difficulty by the preceding equations, is the one where a cylinder is placed in a uniform magnetic field which is suddenly annihilated, the lines of magnetic force being originally parallel to tho axis of tho cylinder, We may imagine, for example, that | the cylinder is placed inside a long straight solenoid, the current — through which is suddenly broken, Since in this case everything is symmetrical about the axis of the cylinder, s = 0, and the values of :p are therefore niece < a (2-404) a (6-620)! -T,1 fe Now we know from the theory of Bessel’s functions that any function of r can for values of x between 0 and o be expanded in the form Ayo (2) + Asda (#2) +A (G2) ee where @, ty, cy... are the roots of the equation J, (a) = 0. ‘Thus, initially ole) = Ale (eZ) + dale (42) + AsTo (2) + one heneo the value of o after a time ¢ will be given by the equa- tion « 7 6 Ade (m2) awa + Ay (at) mT, 80 that all we have to do is to find the coefficients A,, A,, ete ‘We shall suppose that initially o was uniform over the section of the cylinder and equal to c,. ‘Then, sinee Sf eles 2) Fo (a Zar = 0 356 ELECTRICAL WAVES AND OSCILLATIONS. = be very approximately represented by the first term in the preceding expression; hence we have, since J, (2404) = 519, * o= 1.6 ey Jy (2-404) Peer This expression is a maximum when r = 0 and gradually dies away to zero when r= a, thus the lines of magnetic force fade away most quickly at the surface of the cylinder and linger longest at the centre, The time modulus for the first term is 47a*)/5-780, For copper rod 1 cm. in radius for which * = 1600, this is about 1/736 of a second; for an iron rod of the same radius for which # = 1000, « = 104, it is about 2/9 of a second, 305.] ‘he intensity ofthe eurrent is — 7 x, honee at # dis- ansa'r from the axis of the eylinder the {aber pee (a) Paya Jy(m) Since at the instant tho magnetic force is destroyed, ¢ is constant over the cross-section of the cylinder, the intensity of the current when t= 0 will vanish except at the aurface of tho cylinder, where, as the above equation shows, it is infinite. After some time has elapsed the intensity of the current will be adequately represented by the first term of the series, i.e. by « Taaatees ie, J, (2-404) Ime 52 This vanishes at the axis of the cylinder and, as we seo from tables for J;(a) (Lord Rayloigh, Theory of Sound, vol. I, p. 265), attains a maximum when 24045 = 1-841, or at a distance from the axis about 3/4 the radius of the cylinder. The following table, taken from the paper by Prof. Lamb on this subject (Proc. Lond. Math. Soc. XV, p. 143), gives the value of the total induction through the eylinder, and the electro- motive force round « circuit embracing the cylinder for a series of values of t/r, where + = 4p al/o— erat Oe 306.) ELEOTRICAL WAVES AND OSCILLATIONS. 357 Rate of Decay of Currents and Magnetic Force tn infinite Cylinders when the Currents are Longitudinal and the Mag- netic Force Transversal. 306.) We have already considered this problem in the special ease when the currents aro symmetrically distributed through the eylinder in Art. 262; we shall now consider the case when the currents are not the same in all planes through the axis. Let w be the intensity of the current parallel to the axis of the cylinder, then (Art. 256) in the cylinder w satisfies the differential equation aw , dw _ dxudw de * dpe dt" If w denotes the rate of increase in the electric displacement parallel to = in the dielectric surrounding the cylinder, then, singe wis equal to X %, qrhore 2 is the electromotive intensity parallel to 2 the axis of the cylinder, w’ satisfies the equation ad oe 1ew +a hae PO es akc vate os eos 7 then transforming to cylindrical coordinates r, @, tho oquation satisfied by w in the cylinder becomes Pw, idw Sree rss de tar tt at the solution of which is w= Acos sd Ten), w= 0, 307] PLECTRICAL WAVES AND OSCILLATIONS. 359 | Ja this cane pa/V is very smal o that (Art. 308) X, (Fa) is approximately proportional to (2a) " » and thus hence equation (91) becomes enad, (ona) +spJ, (ina) = 0. (92) Bessel's functions, however, satisfy the relation Fi (enn) + J, (una) = Ja (ena), so that (92) may be written 8(#—1) J, (ena) +enad,_, (ina) = 0, For non-magnetic substances «= 1, so that this equation reduces to J,_-, (ena) = 0. ‘The magnetic induction along the radius is equal to o1dw. ~ pr do’ at right angles to the radius it is equal to 807.] Let us consider the case when s= 1. Fora non-magnetic eylinder » will be given by the equation J, (ena) = 0; thus the values of «p will be the same as those in Art. 304, and we may put w= oon fa, T(z ye aa Ay (2) 4 ao bos where #,, 7, are the values 2-404, 5-520..., which are the roots of the equation Gian 308. ELECTRICAL WAVES AND OSCILLATIONS. 361 Hence, when 2, is a root of To(z) =9, [rae Dar = ie File) (08) fEn@2) ry S4,(«2) + aps 80 that oe Hf 2d; (x 2)ar=— A,[ rdi8(@,2) ar. ar Henoo by (95) and (96) - 4y== 7)" Thus by (93), the currents produced by the annihilation of a magnetic field H parallel to y are given by the equation r Hoos, 71(>5) ~ee =e ey ‘Thus the currents vanish at the axis of the cylinder; whon t = 0 they are infinite at the surfaco and zero elsowhere. When, as in the case of iron, p is very large, the equation (92) becomes approximately 7, (¢na) = 0. ‘The solution in this case can be worked out on tho seme lines as the preceeding one; for the results of this investigation we refer the reader to a paper by Prof. H. Lamb (Proc, Lond. Math. Sov. XV, p. 270). Blectrical Oscillations on a Spherical Conductor. 308] The equations satisfied in the cloctromagnetic field by the components of the magnetic induction, or of the électro- motive intensity, when these quantities vary as <'?*, are, denoting any one of them by F, of the form oat rt a aeR, (97) where in an insulator A‘ = p*/¥*, V being the velocity of pro- pagation of cloctrodynamic action through the dielectric, and in a conductor, whore specific resistance is ¢ and magnetic permeability 4, = —Axpiple, a #7428 5 Or aint oO. We can easily verify by substitution that the solution of this equation is, writing p for Ar, q a prfl d]* Ad eB omeftgr ceases, where A and B are arbitrary constants; particular solutions of this equation are thus so=r bin“ see, @ fin=oft dpen, @) ines ayy} ) sea fey iy. ® ‘The first of these solutions is the only one which doos not become infinite when p vanishes, so that it is the solution we must choose in any region whoro p can vanish; in the case of the sphere it ia the function which must be used inside the sphere; we shall denote it by S, (p). Ontside the sphere, where p cannot vanish, the choice of the function must be governed by other considerations, If we are considering wave motions, then, since the solution (y) will contain the factor e'\2"-?), it will correspond to a wave diverging from the sphere; the solation (0), which contains the factor '@+?), corresponds to waves converging on the sphere; the solutions (a) and (2) correspond to a combination of convergent and diver- gent waves; thus, where there is no reflection we must take (y) if the waves arc divergent, (8) if they are convergent, In other b el = 09.) ELECTRICAL WAVES AND OSCILLATIONS. 363 enses wo find that A is complos and of tho form p4..q3 in this case (a) and (3) will be infinite at an infinite distance from the origin, while of the two solutions f and 3 one will be infinite, the other zero, we must take the solution which vanishes when p is infinite, We shall denote (y) by Ez (p), (8) by B* (p), and when, as we shall sometimes do, we we leave the question as to which of the two we shall take unsottled until we hve deter- mined A, we shall use the expression Z,(p), which thus denotes ‘ono or other of (y) and (8). When there is no reflection, the solution of (97) is thus ex- Premed by (9) Y,«'?* inside the sphere, E, (e) Y,<° outside the sphere. In particular when Y, is the zonal harmonic Q,, the solutions are Su(0) Que? Buln) Quel”. When Y, is tho first tesseral harmonic, the solutions are Es, geer, — Zaye em Yatton, Lani teen, where « = cos, 6 being the colatitude of the intersection of the radius with the surface of the sphere. 809] We shall now proceed ta prove those properties of the functions S, and #, which we shall require for the subsequent imyestigations, The reader who desires further information about these interesting functions can derive it from the following ‘sources :— Stokes,‘ On the Communication of Vibration from a Vibrating Body to the Surrounding Gas,’ Phil. Trans. 1868, p, 447. Rayloigh, ‘Theory of Sound,’ Vol. If, Chap. XVIL G. Niven, ‘On the Conduction of Heat in Ellipsolds of Revo~ lation,’ Phil. Trans. Part I, 1880, p. 117. C. Niven, ‘On the Induction of Electric Currents in Infinite Plates and Spherical Shells” Phil. Trans. Part I, 1881, p. 307. H. Lamb, “On the Vibrations of an Elastic Sphere,’ and “On the Oscillations of « Viscous Spheroid,’ Proc. Lond. Math. Soc, 13, pp. 61, 189. H. Lamb, ‘On Electrical Motions in a Spherical Conductor, Phil. Trans. Part U, 1883, p. $19. 4 810.) We shall now proceed to the of a distribution of electricity over the us suppose that a distribution of density is proportional to a sonal harmonic produced over the surface of the sphere, and that the ducing this distribution is suddenly removed; th and electrical vibrations will be started whose period it is object of the following investigation to determine. Since the currents obviously flow in planss through of the zonal harmonic, which wo shall take for the axis Fee oo ee ee ee right angles to this axis; and since tho round a circuit is equal to the rate of diminution in the of lines of magnotic foreo passing through it, we ace that : case, since the motion is periodic, there ean be no magnetic force at right angles to such a cireuit; in other we the magnetic force parallel to the axis of > vanishes. taking a small closed circuit at right angles to a radius sphere, we soe that the eleetromotive fores round this eireu and therefore the magnetic foree at right angles to it, vanish; hence the magnetic force has no component along the radius, daalls toes ab ight covke ta bow tie acts oe so that the lines of magnetic force are a series of small circles with the axis of the harmonic for axis. Hence, if a, b, ¢ denote the components of magnetic induc- tion parallel to the axes of x, y, 2 respectively, we may put a=yx(r 4) b=—ax(r, 4H) c=0, where x (7, 4) denotes some function of 7 and p. this with the resulta of Art. 308, we see that inside the sphere a=A Lor) et = Zsa, Gos c=0, | where A? =—4npip/s, and A is a constant. ——E ; ut.) ELECTRICAL WAVES AND osomLaTIONs. «867 Outaide the’ sphere, net BYE een, (ime BEE, (ur) oot, (108) e=0, whore \ = p/V, and B is a constant. Since the tangential magnetic force is continuous, we have if a is the radius of the sphere, As, (Na) = BE, (Aa). (108) To get another surface condition we notice that the electro- motive intensity parallel to the surface of the sphere is con- tinuous. Now the total current through any area is equal to 1/4n times the line integral of the magnetic force round that area, hence, taking as the area under consideration an elemen- tary one drr sin dd, whose sides are respectively parallel to an element of radius and to an element of a parallel of latitude, wo find, if q is the current in a meridian plane at right angles to the radiug, 1d trg= 7 7) whore y is the resultant magnetic force which acts tangentially to a parallel of latitude. ‘The electromotive intensity parallel to g is, in the conduetor q and in the dielectric oes pk Henee, since this is continuous, we have Acd Bard TF das) = Te ag hE ah (106) Eliminating A and B from equations (105) and (106), we get ot tesa) 4% 1aB,(an)) =—° (107) Bia) pK EB, (aa) 811,] The oscillations of tho surface electrification woul Yee = e— 368 ELECTRICAL WAVES AND OSCILLATIO state of uniform distribution aro pi Jongth must be comparable with the radius of such rapid vibrations as those howevor A’a, or {[—4m, is very lurge, but when this is the case, we ace from the equation ain=- bal a that 8,’(x"a) is approximately equal to ‘saa, so that tho left-hand side of equation (107) is of the order oan / ~ THe EE and thus, since 1/K = V%, 4 ak,(s0)} is of the order oe 1 aeeep ie qareee af SEP, a ov sinee p is comparable with V/a. ‘This, when the sphere conducts as well as iron or copper, is extremely small unless a is Jess than the wave length of sodium light, while for a perfect conductor it absolutely vanishes, hence equation (107) is very approximately equivalent to J fan, (na)) =0. (108) This, by the relation (101), may be written Eq (0a)—™** B, (ra) = 0, which is the form given in my paper on ‘ Electrical Oscillations” Proc, Lond. Math, Soc. XV, p. 197. This condition makes the tangential electromotive intensity vanish, so that the lines of electrostatic induction are always at right angles to the surface of the sphere. $12.] In order to show that the equations (103) and (10%) in the radiue vector varies as Q,, for the difference between the 312) ELECTRICAL WAVES AND OSCILLATIONS. 369 radial currents in the sphere and in the dielectric is proportional to the rate of variation of the surface density of the electricity ‘on the sphere, and therefore, since the surface density varies ax ¢'**, it will be proportional to the radial current. Consider a small area at right angles to the radius, and apply the principle that 4 times the current through this area is equal to the line integral of the magnetic force round it, we get, if p is tho current along the radius and « = cos 0, srp=1 7 (sind), (109) whero y, a8 before, is the resultant magnetic force which acts along a tangent to parallel of Jatitude. By equation (103), y is proportional to ing ten sind", dp so that p is proportional to @ § ing On apnea ; bat. faint 5224 + n(n +1) Q,= 0, hence p, and therefore the surface density, is proportional to Q,. We shall now consider in more detail the case x = 1. We have y y= ’ We shall take as the solution of the equation p'/V? =a? $=™ and we shall take 2; (A7) as our solution, as this corresponds to & wave diverging from the sphere, Thus, equation (108) bo- comes ad Gq (Er a) = 9, or substituting for Hy (Aa) tho value given in Art, 309, oo 1 ean? ai id (Aa)@—1Aa = 1, if Nes tse Bb —_— ‘870 ELECTRICAL WAVES AND Hence p=Zft+ GI, taking tho positive sign since the wave is di Hence, the time of vibration is 4za/+/3¥, and length 40/3. ‘The amplitude of the vibration falls to its original value after a time 2a/V, that is after the by light to pass across a diameter of the sphere. In occupied by one complete vibration the or about 1/35 of its original value, thus tho vibrations will tadependent of the resiatanns of the conductor and is due to the emission of radiant energy by the sphere. Whenever these sleotrics]) vibrations: oan ‘radiate: froaly 2 they scs>] yan immense rapidity and are practically dead beat, If we substitute this value of A in the expressions for the magnetic force and electromotive intensity in the dicleetric, we shall find that the following values satisfy the conditions of the problem. If y is the rosultant magnetic force, acting at right ae ee -1) (ve ome Ce where t= 43 (yn), If © is the electromotive intensity at right angles to 7 in the meridional plane, K the specifi induetive capacity of the dix electric surrounding the sphere, then hae Konstan s) fists tte oF ” con (0-42, where tan = ate ; and V is the velocity. of propagation ‘of sleetromagnelis sotian : —$—$—3 313] - ELECTRICAL WAVES AND OSCILLATIONS. 371 throngh the dielectric. Close to the surfaco of the sphere 8 = 0, Y=-/6, thus y and © differ in phase by +/6. At large distance from the sphere 9 — jy, ec enn sone pees. Sd ee Dye sinda — 2 vi —F) VK @=y= Oe 7 cos (+5) Mee cintncuie Pd wake cen pee ae 4 Wt Pa ee 8 EU ae win(p+o—%)- ‘Thus at a great distance from the sphere P varies as a*/r?, while © only varies as a/r, thus the electromotive intensity is ‘ory approximately tangential. The goneral character of the Tines of electrostatic induction is similar to that in tho case of the cylinder shown in Fig. 110. 313.) The time of vibration of the electricity about the dis- tribution represented by the second zonal harmonic is given by ® cubic equation, whose imaginary roots I find to be has—7 $18. The rate of these vibrations is more than twice as fast as those about the first harmonic distribution; the rate of decay of these vibrations, though absolutely greater than in that caso, is not increased in so great a ratio as the frequency, so that the system will make more vibrations before falling to a given fraction of its original value than before. The time of vibration of the electricity about the distribution reprosonted by the third zonal harmonic is given by a biquadratic equation whose roots are imaginary, and given by tha = — 8542-761, sha =— 2154-84 The quicker of these vibrations is more than three times faster than that about the firet zonal harmonic, and there will be many more vibrations before the disturbance sinks to a given fraction of its original value, The slower vibration is of nearly the same period as that about the first harmonic, but it fades TE a a ea Bb2 ‘The vibrations about distributions of electricity by the higher harmonics thus tend to get quicker as of the harmonic increases, and more vibrations the disturbance sinks into insignificance. 814.) We have seen in Art. 16 that a charged sphere when moving uniformly produces the same magnetic field as an element of current at its centre. If tho sphero is oscillating instead of moving uniformly, we may prove (J. J. Thomson, Phil. Mag. [5], 28, p. 1,1889) that if the period of its oscillations is large compared with that of a distribution of electricity over tho surface of the sphere, the vibrating sphere produces the snme magnetic field as an alternating current of the same period. Waves of electromotive intensity currying energy with them travel through the dielectric, so that in this case the of the sphere travels into space far away from the aphere. however, the period of vibration of the sphere is leas than that of the electricity over its surface, the electromotive in- tensity and the magnetic force diminish very rapidly as we recede from the sphere, the magnetic field being practically confined to the inside of the sphere, so that in this cage the energy of the moving sphere remains in its immediato neigh- bourhood. We may compare the behaviour of the electrified sphore with that of a string of particles of equal mass placed at equal intervals along a tightly stretched string; if ono of the particles, — sny one of the end ones, is agitated and made to vibrate more slowly than the natural period of the system, the disturbance will travel as a wave motion along the string of parti the energy given to the particle at the end will be away from that particle; if however the i agitated is made to vibrate more quickly period of vibration of the system, the disturl jacent particles will diminish in geometrical progression, tho energy will practically be confined to within a distance of the disturbed particle, This case possesses tional interest since it was used by Sir G. G. Stokes to fluorescence, $15.] To consider more closely the effect of reflection let us take the case of two concentric spherical conductors of radius : Fi He HE 315] ELECTRICAL WAVES AND OSCILLATIONS, 878 spheres, the components of magnetic induction are given by a=Y BR: (ar) 408; oon b=~ 2faey an)aok, arte ote, Wo may Picci Art. 511, that if the spheres are metallic and not excessively small the electromotive intensity parallel to the surface of tho spheres vanishes when =a and when r= b: thue we have mage BOWE {a k,*(A8)}, O= rie (bE, (Ab) 108, (PB (XB)}. Eliminating A and B, we =a SF. (oBs a) A toe, (ab); = 2 for, (na)) & [oes (av))- When 7 = 1, this becomes ae Ate ised at tan A {b—a} = we (110) GG) + A re. no decay of the vibrations apart from that arising from the resistance of the conductors, Ifa is very small compared with b, this equation reduces to Ab 1-eor The least root of this equation other than A = 0, I find by the method of trial and error to be Ab = 2-744. This case is that of the vibration of a spherical shell excited by some cause inside, here there is no radiation of the energy into space, the electrical waves keep passing backwards and forwards from one part of the surface of the sphere to another. The wave length in this case is 27b/2-744 or 2-29b, and is therefore less than tho wave length, 4xb//3, of the oscillations which would occur if the vibrations radiated off into space: this 5 an example of the general principle in the theory of vibrations tandb = that when dissipation of energy takes place cither from ficthon, e= canviea where p = VA, A, being equal to 2.744/b._ - hppa leat sagan bacco 7 a = 2 BS, (nrc b= Bie so that dp =—2Beos08, (Ayr) em ‘When 7 = b the normal displacement current = d —420 cos Op sinpt =— 2 Beoa8,(A,¥) ¢ Substituting this value of Be? in (111), we havo. a=Yarbpsingt? L | b=—S2sppsin to ——— 316.) ELECTRICAL WAVES AND OSCILLATIONS. 875 ! At the surfaces of the sphere the maximum intensity of the J magnetic force is 2abpCsind, ) or since bp = PA,b, and Ab = 2-744, the maximum magnetic force is 22x 2-744 VCain ge. For air at atmospheric pressure V0 may be as large as 25 without the electricity escaping; taking this value of VC, the maximum value of the magnetic force will be 431 sind; this indicates a very intense magnetic field, which however would be difficult to detect on account of its vory rapid rate of reversal. Electrical Oscillations on Two Concentric Spheres of nearly equal radius. 316.] When d, the difference between the radii a and b, is very small compared with a or b, equation (110) becomes Ad (1+A%a") — AMat—\0? +1 There will be one root of this equation corresponding to a vibration whose wave length is comparable with a, and other roota corresponding to wave lengths comparable with d. When the wave length is comparable with a, A is com; with 1/6, 0 that in this case Ad is very small; when this is the case (tan Ad)/Ad = 1, and equation (112) becomes approximately tan dd = (112) 11 téat rs & or Ag = 4/2. ‘The wayo longth 2/A is thus equal to #2 times the radius of the sphere. In this caso, sinco the distanco between the spheres is very small compared with the wave length, the tangential electro- motive intensity, since it vanishes at the surface of both spheres, will remain very small throughout the space between them; the electromotive intensity will thus be very neazly radial between the spheres, and the places nearest each other on the two spheres x. {BE} (ar) +CBy (Ar)} sin et? ‘ or substituting the preceding values of B and 0 only the lowest powers of 1/Ar, i ee Fp fethem pen AO— mh ain Get, or 2A ‘cond (r—a) sin de'P, ‘The tangential electromotive intensity is therefore, 24 Vsind(r—a) sin ve is thus, except just at the surface of the spheres, very small with the tangential electromotive intensity. The intensity changes sign as we go from r = a to r = b, 80 ‘that the electrification on the portions of the spheres opposite to ) each other is of the same sign. In this case the lines of electro- ‘motivo intensity arc approximately tangential; during the Vibrations they move backwards and forwards across the short “space botwoon the sphoros. The case of two parallol planes can ‘be regarded as the limit of that of the two spheres, and the work shows that the wave length of the vibrations will either be a sub-multiple of twice the distance between the planos, or else a length comparable with the dimensions of the plane at right angles to their common normal. | If we arrange two metal surfaces, say two silvered glass | plates, so that, as in the experiment for showing Newton's rings, | tho distance between the plates is comparable with the wave | length of the luminous rays, care being taken to insulate one | plate from the other, then ono of the possible modes of electrical vibration will have a wave length comparable with that of the luminous rays, and s0 might be expected to affect a photographic plate. These vibrations would doubtless be exceedingly difficult to excite, on account of the difficulty of getting any lines of induction to run down between the plates before discharge took place, but this would to some extent be counterbalanced by the fact that the photographic method would enable us to detect vibrations of exceedingly small intensity. On the Decay of Electric Currents in Conducting Spheres. 317.] The analysis we have used to determine the electrical ogcillations on spheres will also enable us to determine the rate at which a system of currents started in the sphere will decay if left to themselves. Let. us first consider the case when, us in the preceding investigation, the lines of magnetic force are circles with a diameter of the sphere for their common axis, ‘Using the same notation as before, when there is only a single sphere of radius a in the field, we have by equation (107) ot asa) Afar, (a)) hes) —Kp Fite): (113) 7] —-BLECTRICAL WAVES AND OSCILLATIONS. aT | 317] ELECTRICAL WAVES AND OSOILLATIONS. 879 on™ ‘the form « te The most persistent type of current will be that corresponding to the smallest value of 4’, Le. XN = 1-43032/a. ‘The timo required for a current of this typo to aink to 1/e of its original value in a copper sphere when ¢ = 1600 is -000379a" seconds; for an iron sphere when » = 1000, # = 10%, it is 06220" seconds, thus the currents will be much more persistent in the iron sphere than in the copper one, The persistence of the vibrations is proportional to the square of the radius of the sphere, thus for very large spheres the rate of decay will be ex~ ceedingly slow; for example, it would take nearly 5 million years for currents of this type to sink to 1/e of their original value in @ copper sphere as large as the earth. Since 5, (Aa) = 0, we see from (105) that B= 0, and there- fore that the magnetic force is zero everywhere outside the sphere. Hence, since theso currents produce no magnotic offect outside the sphere they cannot be excited by any external magnetic influence. The curront at right anglea to the radius inside the sphere is by Art. 310 sing d to eO: ae Taacan tare a os or in particular, when n = 1 UJ “r)} ef Be © r8,(tr)) om Now rs, ‘v)} vanishes when d’r = 2-744, hence the tangential current will vanish when 2744 143030 = -G0la; thus there is a concentric spherical surface over which the current of this type is entirely radial. The magnetic force vanishes at the surface and at the centre, and as we travel along a radius attains, when n =1,a maximum when r satisfies the equation £ s.r) =0. — — . 318.) ELBCTRICAL WAVES AND OSCILLATIONS. 881 | while in the dielectric surrounding the sphere, we have =Bt Qs cpt , P=BiE ange Re dQ. (118) Q=-BE E(u) et, | R= 0. | Since the electromotive intensity tangential to the sphere is continuous, we have, if a is the radius of the sphere, AS, (\'s) = BE, (Aa). (116) If » is the magnetic induction tangentially to a meridian, then, since the line integral of the electromotive intensity round a cireuit is equal to the rate of diminution of the number of lines of magnetic induction ene through it, ea cr (PQ. Pare else mali ci is continuous, we have at the surface @y . 5 5 (Zin the sphere = » in the dielectric, ah eee a = ga #5. (0'a)) = BE, (a8, (Aa)}. (117) Eliminating A and B from equations (116) and (117), we get 5,(’a) B.{da) (urs) "Eeh0s)) Hinkle) In this ease the currents and magnetic forces change so slowly that Aa or pa/V is an exceedingly small quantity, but when this is the case we have proved Art. 317, that approximately —F.0e)___t, Kehoe ® go that equation (118) becomes awS,(¥a)+ # fa8,(x'a)} =0. (ats) But by equation (100), Art. 309, a z 8, (Xa) +(n+1)8, (X's) =—X'0S, (0), while for iron, for which is very mates very closely tog, (xq) The smaller roots of the equation Bq (2) = 0, when 7 = 0, 1, 2, are given below; n=0, c=7, 27, Smeg a=1, r= 143037, 2.45907, N= 2, @= 183167, 28950, Thus for a copper sphere for which currents of the most pormanent type, i.e. those the root X’a = =, take to fall to 1/e of their -000775a7 seconds, which for copper sphere | earth is ten million years, These numbers | Horace Lamb in the paper on ‘ Electrical Motion. Conductor, Phil. Trans. 1883, Part I. u 319] As the magnetic force outside the they will, omitting the time factor, be given by ea = iesicre: | | ELECTRICAL WAVES AND OSCILLATIONS. 383 | form u=S,'r)¥, v= S,('r) ¥”, w= 8, (Ar) ¥", where ¥’, ¥”, ¥°” are surface harmonies of the n™ order, ‘The radial current is S,(N'r) (EP 4 2 P"4 =Y), at the surface of the sphere the radial current must vanish, i.e, §, (n'a) £ V4 Yy'4 crv}. Now the second factor is a function merely of the angular coordinates, and if it vanished thero would not be any radial currents at any point in the sphere, henee, on the hypothesis that thore are radial currents in the sphere, we must have S, (A’a) = 0, Le. ut, v, wall vanish on the surface of the sphere. But if there are no currents on the surface the electromotive intensity must vanish over the surface, and hence also the radial magnetic induc- tion ; for tho rate of change of the radial induction through asmall area on the surface of the sphere is equal to the electromotive force round that area. But neglecting tho displacement current in the dielectric the magnetic force outside the sphere will be derived from a potential ; hence, since the radial magnetic force vanishes over the sphere 7 = a, and over r=, and since the space between the two is acyclic, the magnetic forco must vanish everywhere in the region between them. Thus the presence of radial currents in the sphere requires the i force due to the currents to be entirely confined to the inside of the sphere. 820.] Returning to the case where the system is symmetrical about an axis, we sce from equation (120) that if the sphere is an iron one, \’ is given approximately by the equation S, (X’a) =0. Honce, by equation (114) the electromotive intensity, and therefore the currents, vanish over the surface of the sphere. Since the currents also vanish at the centre, they must attain a maximum at some intermediate position ; the distance v of Ushw Lorie b ag abe respectively, then rp = xa+yb +20, V3 (rp) = 2VVat+yVib+eVie+2 {2 Eres and Hence, by (97), rp = C009 08, (X'r) Nia nae i F200. (41) 8, (va) 405A) or by (120) pee oe . 321] ELEOTRICAL WAVES AND DSOILLATIONS. 385 lg we hence a = 2, where p is an integer. | aM aa hd ares adele | tion 8, (a) = or icrae ‘The roots of this equation are given in Art. 317. We shall for the present not make any assumption as to the magnitude of », but suppose that A,,A,... are the values of A’ which satisfy (122). The value of ip corresponding to A, is —1,/4np, henoo by (121) wo havo oat, aoe, 7p = 0028 (0,8, (A,r)e 4 40,8, (yr) FF tn}. To determine C,, C, ... we have the condition that when ¢ = 0, = alt oe aioe hence, for all values of r between 0 and a, we have ST = 0,8, (47)+ 08,0) + (123) Siew by But ro if A,, A, are different roots of (122) [P5098 der)ar=0, while SPrstonar = 5S fsayg fee} —a(2Re)'}. (124) But x FeAiye) +2 25.0) Hpe a) AOs8) =9, and GANA Gare ZACa= 0. Substituting in (124) the values of 2, 8,(hya) and FSi) given by these equations, we get JFPsiorsrite= sha! Oya) Inte (042)H—MN ce the beatin | 323) ELECTRICAL WAVES AND OSCILLATIONS. 887 ‘tho series converges very rapidly it is more convenient to leave it in ite present form. Since we neglect the polarization currents outside tho sphere, the magnetic foree in that region is derivable from a potential, __ hence we find that the radial magnetic force is | baeatceeees | aE CRC Es ee aie ae 8Hsin a* Pp =* 1 ome wl’, er pa pe Tho sphere produces the same effect at an external point as a small magnet whose moment is BH spe Mey 7 “poi p ; =e Hence, the normal magnetic force at the surface of the sphere is 6H 2a, —7 0080 Ee tre # Ontside the sphere the magnetic foree is the same as that due to a magnet whose moment is SHety tee pe om, placed at its centre. Sh et eae Thus tho magnetic offects of the currents induced in a aoft iron ephere are less than those which would be produced by copper sphere of the same size placed in the same field. Thia is due to the changes of magnetic foree proceeding more slowly in the iron sphere on account of its greater self-induction; as the changes in magnetic force are slower, the electromotive forces, and therefore the currents, will be smaller. Since S,(A,a) = = 0 when u is large, the currents on the surface of the sphere vanish, and the currents congregate towards the middle of the sphere. ce 525,] The clectrical vibrator which Herts vu experiments (Wied. Ann, 94, pp. 155, 551, 609, 188 sented in Figure 113. A and 8 are square zine plates whoae sides copper wires C and D each about 30 em. lor tho plates, those wires terminate in brass b EXPERIMENTS ON ELECTROMAGNETIC WAVES, 389 | | ensure the success of the experiments it is necessary that these balls should be exceedingly brightly and smoothly polished, i and inasmuch as the passage of the sparks from one ball to the other across the air space EF roughens tho balls by tearing | particles of metal from thom, it is necessary to keep repolishing | the balls at short intervals during the course of the experiment. Tt is also advisable to keep the air epace EF shaded from the light from any sparks that may be passing in the neigh- bourhood. In order to ex- cite electrical vibrations in [ s}ee EMD: "| this system the extremities Fig. 118, of an induction coil are con- nected with ¢ and D respectively. When the coil is in action it produces so great a difference of potential between the balls £ and F that tho electric strength of the air is overcome, sparke paas ncross the air gap which thus becomes a conductor; the two plates A and 8 are now connected by a conducting circuit, and the charges on the plates oscillate backwards and forwards from one plate to another just as in the case of the Leyden jar. 826] As these oscillations are exceedingly rapid they will not be exeited unless the electric strength of the air gap breaks down suddenly; if it breaks down so gradually that instead of a spark suddenly rushing across the gap we have an almost continuous glow or brush discharge, hardly any vibrations will be excited. A parallel case to this is that of the vibrations of a simple pendulum, if the bob of such a pendulum is pulled out from the vertical by a string and the string is suddenly cut the pondalum will oscillate; if howover the string instead of breaking suddenly gives way gradually, the bob of the pendulum will merely sink to its position of equilibrium and no vibrations will be excited. It is this which makes it necessary to keep the balls £ and F well polished, if they are rough there will in all like- lihood be sharp points upon them from which the electricity will gradually escape, the constraint of the system will then give way ly instead of suddenly and no vibrations will be excited. ‘The necessity of shielding the air gap from light coming from other sparks is due to asimilar reason. Ultra-violet light in which these sparks abound possesses, as we saw in Art, 39, the property of producing a gradual discharge of electricity from the negains = ‘The Kinetic energy of the currents is Ay + be+ Mya. ‘The potential energy is Lagrange’s tions a yt TEE = 0, D2’ +My'+ ue =0. Thus if @ and y each vary as ¢'P*, we have (gy —Ee*) +2 (GMa) = 0, #(f —Ept) +9 (4 —Mpt) =0. Eliminating y and 2 we get j-") (5-2?) =(G-Mv*) + or p= al ah —/6-ip- 329.] EXPERIMENTS ON ELECTROMAGNETIC Waves. 391 But for a cireuit as short as a Hertzian vibrator L// and M/Z’ will be exceedingly small, so that we have as before P=, The Resonator. 328] When the olectrical oscillations are taking place in the vibrator the space around it will be the seat of oloctric and tmagnetic intensities, Hertz found that he could detect theso by means of an instrument which is called the Resonator. It consists of a piece of copper wire bent into a cirele ; the ends of the wire, which are placed very near together, are furnished with two balls or a ball and a point, these are connected by an insu- lating screw, so that the distance between them admits of very fine adjustment. A resonator without the serew adjustment is shown in Fig. 114. With o vibrator having the dimensions of the one in Art. 325, Hertz used a resonator 35 om. in radius. 329,] When the resonator was held near the vibrator Hertz found that sparks passed across the air space in the resonator and that the length of the air space across which Fig. 1 the sparks would pass varied with the poxi- tion of the resonator. This variation was found by Herts to be of the following kind : Let the vibrator be placed so that its axis, the line € F, Fig. 113, is horizontal; let the horizontal line which bisects this axis at right angles, i. o. which passes through the middle point of the air space E F, be called the base line. Thon, when the resonator is placed #0 that its centre is on the base line and ita plane at right angles to that line, Hertz found that sparks pass readily in the resonator when its air space is either vertically above or vertically below its centre, but that they cease entirely when the resonator is turned in its own plane round its centre until the air space is in the horizontal plane through that point. Thus the sparks are bright when the line joining the ends of the resonator is parallel to the axis of the vibrator and vanish when it is at right angles to this axis. In intermediate positions of the air gop faint sparks pass between tho terminals of the resousker. = Pcie! Hatisoual tha sparks exo eee sa eases ths vibestor nel inal See warcas the’ain yap by esloulalicg, by aceday aera motive force round tho circuit from the diminution number of lines of magnetic force passing through it, 831.] The Se oxplained if we consider the arrangement of the Faraday tubes radiating from the vibrator, ‘The tendency to spark ywill be proportional to the number of tubes which streteh air gap; these tubes may fall directly on the air gap} may be collected by the wire of the resonator and air gap, the resonator acting as a kind of trap for will be due only to those tubea which fall directly upon it, 331.) EXPERIMENTS ON ELECTROMAGNETIC Waves, 393° “the air gap is parallel to the tubes, ie, when it ia at the highest or lowest point of tho resonator, some of the tubos will be caught and will stretch across the gap and thus tend to produce a spark. When, however, the gap is at right angles to the tubes, Le. when it is in the horizontal plane through the centre of the resonator, the tubes will pass right through it. None of them will stretch across the gap and there will be con- | sequently no tendency to spark, When the plane of the resonator is at right angles to the axis of the vibrator, the tubes when they meet the wire of the resonator are, as in the last case, travelling at right angles ry 2 + Fig. 115. of the resonator will not collect the pass right through it, and none of them will stretch acrosa the gap. Thus in this case there is no tendency to spark whatever may be the position of the air space, Let us now consider the case when the centre of the resonator is on the base line and its plane horizontal. In this case, as we see by the figures Fig. 116, Faraday tubes will be caught by the wire of the resonator and thrown into the air gap wherever that may be; thus, whatever the position of the gap, Faraday tubes will stretch across it, and there will be a tendency to spark. When the gap is a3 near as possible to the vibrator the Faraday tubes which strike against the resonator will break and a yortion a the other side, (a); these portions bend and (c); then break | 4 ees up again, one 333] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 395 _ round the gap, so that there will be less tendency to spark, " thoogh this tendeney will still remain finite. Resonance, 882.] Hitherto we have said nothing as to the effect produced ‘by the size of the resonator on the brightness of the sparks, this effect is however often very great, especially when we are using condensers with fairly large capacities which can execute several vibrations before the radiation of their energy reduces the amplitude of the vibration to insignificance. The cauvee of this offect ia that the resonator is itself an electrical system with a definite period of vibration of its own, hence if we use a resonator the period of whose free vibration is equal to that of the vibrator, the efforts of the vibrator to produce spark in the resonator will accumulate, and we may be able as the result of this accumulation to get a spark which would not have been produced if the resonator had not been in tune with the vibrator. The case is analogous to the one in which a vibrating tuning fork sets another of the same pitch in vibration, though itdoes not produce any appreciable effect on another of slightly different pitch. @ ) ee a ry Fig. 117. 883.) Professor Oliver Lodge (Nature, Feb. 20, 1890, vol. 41, Pp. 368) has described an experiment which shows very beautifully the offect of electric resonance. A and B, Fig. 117, represent two Leyden jors whose inner and outer coatings are connected by wire bent so as to include a considerable area. The circuit connecting the coatings of one of these jars, A, contains an air break, Electrical oscillations are started in this jar by connect- ing the two coatings with the poles of an electrical machine. 4 ———— Meiad bare tho! aystoxa which 8 witendlegtl estecebla arity aires tack a tions have to take place before the radiation : the system has greatly diminished the amy tions. When the capacity is soa tbe eae quickly that only « small number of vibrations h ciable amplitude; there are thus only a small n pulses acting on the resonator, and even if the few conspire, the resonance cannot be expected | marked, In the case of the vibrating sphere wo saw ( that for vibrations about the distribution repr first harmonic the amplitude of the second vibration is « 1/35 of that of the first, in such a case as this the practically dead-beat, and there can be no appreciab or interference effects. these in the same way as they did from the cylinder; These closed tubes will move off from the vikrato velocity of light, and will carry the energy o ~~ a ——s 335-] EXPREIMENTS ON ELEeTROMAGNETIC WavES. 397 ‘with them. In consequence of this radiation the decay of the ‘oscillations in tho vibrator will be very rapid, indood wo should | expoct the rate of decay to be comparable with its value in the ease of the vibrations of electricity over the surfaces of spheres or cylinders, where the Faraday tubes which originally stretehed from one part to another of the electrified conductor emit closed tubes which radiate into space in the same way as the similar | tubes in the case of the Hertzian vibrator: we have seen, how= ever, that for spheres and cylinders the decay of vibration is so rapid that they may almost be regarded as dead-beat We should expect a somewhat similar result for the oscillations of the Hertzian vibrator. 334.) On the other hand, the disposition of the Faraday tubes shows us that the electrical vibrations of the resonator will be much more persistent. In this case the Faraday tubes will stretch from side to side across the inside of the 7 resonator asin Fig. 118, and these tubes will oscillate backwards and forwards inside the resonator; they will have no tendency to form closed curves, and consequently there will be little or no radiation of enorgy. In this caso the decay of the vibrations will be chiefly due to the resistance of the resonator, Fig, 118. as in the corresponding eases of oscilla- tions in the electrical distribution over spherical or cylindrical cavities in a mass of metal, which are discussed in Arts, 315 and 300, 835.) The rate at which the vibrations die away for o vibrator and resonator of dimensions not very different from thove used by Hertz has heen measured by Bjorknos (Wiad. Ann, 44, p. 74, 1891), who found that in the vibrator the oscillations died away to 1/e of their original value after a time 7'/-26, whero T ig the time of oscillation of the vibrator, This rate of decay, though not so rapid as for spheres and cylinders, is still very rapid, as the amplitude of the tenth swing is about 1/14 of that of the first. The amplitudes of the successive vibrations aro represented graphically in Fig. 119, which is taken from Bjerknes! ey 4 falls to 1/10 of its original of these oscillations confirms 338.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 399 = > vertically against the wall, its plane being thus at right angles to the base line of the vibrator. The zine plate was connected to earth by means of the gas and water pipes. In one sct of the centre of the resonator was on and its plane at Fight angles to the base line. When it is in this position the Faraday tubes from the vibrator strike the wire of the resonator at right angles; the resonator therefore does not eateh the tubes and throw them into the air gap, and the spark will be due to the tubes which fall direetly upon the air gap. Thus, aa might be expected, the sparks vanish when the gap is at the highest or lowest point of the resonator, when the tubes are at tight angles to the direction in which the sparks would pass, and the sparks are brightest when the air gap is in the horizontal plano through the base line, when the incident tubes are parallel ‘to the sparks, 837.] Let the air gap be kept in this plane, and the resonator moved about, its centre remaining on the base line, and ite plane at right angles to it. When the resonator is quite close to the zine plate no sparks pass across the air space; feeble sparks, however, begin to pass as soon as the resonator is moved a short distance away from the plate. They increase rapidly in brightness as the resonator is moved away from the plate until the distance between the two is about 1°8 m., when the brightness of the sparks is a maximum, When the distance is still further in- creased the brightness of the sparks diminishes, and vanishes again at a distance of about 4 metres from the zinc plate, after which it begins to increase, and attains anothor maximum, and so on. Thus the sparks exhibit a romarkable periodic character, similar to that which occurs when stationary vibra- tions are produced by the reflection of wave motion from a surface at right angles to the direction of propagation of the motion. 838] Let the resonator now be placed so that its plane is the vertical one through the base line, the air gap being at the highest or lowest point; in this position the Faraday tubes which fall directly on the air gap are ot right angles to the sparks, so that the latter are due entirely to the Faraday tubes collected by tho resonator and thrown into the air gap. When the resonator is in this position and elose to the reflect- ing plate sparks pass freely. As tho resonator recedes from the = other, 839.] If the zine reflecting plate is mounted on ; frame work, so that it ean be placed behind the reson eee following experiments :— Hold the resonator in the position it had in the! ment at some distance from the vibrator and observe | the zinc plate being placed on one side out of action : the reflector immediately behind the resonator, the increaso in brightness; now push the reflector back, ebout 2 eet Cee eke ee pushing it still further back the sparks will increase | { whon tho reflector is about 4 metres away they will brighter than when it was absent altogether. 340.] Hortz only used one size no selected s0 as to be in tune with the vibrator. Sarasin and De In Rive (Comptes Rendus, March 31, 1891), who repeated this experiment with vibrators and resonators of varions sizes, found however that the apparent wave length of the vibrations, that is twice the distance between two adjacent places — sparks vanish, depended entirely upon the size of the and not at all upon that of the vibrator, The | contains the results of their experiments; A denotes the length, a ‘loop’ meana a place where the sparks are maximum brightness when the resonator is held in tho position, a ‘node’ a place where the brightness is a ‘Tho line beginning * 1/4 A wire! relates to another series of ments which we shall consider subsequently. It is includ nae Ys pa Sake lectrieal vibrator gives out vibrations of i peri vibration ; tho impulso then travels up to tho reflected, tho electromotive intensity in the imp versed by reflection ; after reflection the impulse ag reflected impulses will conspire, so that if the reson in the first position a bright spark will be produced, J reflected impulse will strike the resonator the second ti ite vibration ia in the opposite phase to that which after the fit impact if the time which has elapred 340.) EXPERIMENTS ON BLECTROMAGNETIC WavEs, 403 | reflector, be no tendency to spark when the resonator is held in this position. Thus weeeo that on this view the distances from the reflecting plano of the places where the sparks have their maximum brightness will depend entirely upon the size of the resonator, and not upon that of the vibrator. This, as we have seen, was found by Sarasin and De la Rive to be a very marked feature in their experiments. We have assumed in this explanation that the vibrator does not vibrate. Bjerknes’ experiments (1. c.) show that though the vibrations die away very rapidly they are not absolutely dead-beat. The existence of a small number of oscilla- tiona in the vibrator will cause the effects to bo more vivid with a resonator in tune with it than with any other resonator. Since, howover, tho rato of decay of tho vibrator is infinitely rapid compared with that of the resonator, the positions in whieh the sparks are brightest will depend much more upon the time of oscillation of the resonator than upon that of the vibrator. 841.] We have still to explain why the places at which the sparks were a maximum when the resonator was in the first position (i.e. with its plane at right angles to the base line) were the places where the sparks vanished when the vibrator was in the second position (i.0. with its plane containing the base linc and tho axis of the vibrator). When the resonator is in the first position the sparks arc wholly duc to the Faraday tubes which fall directly upon the air gap, hence the sparke will be a maximum when the state of tho resonator corresponds to the imeidence upon it of Faradsy tubes from the vibrator of the same kind as those which reach it after reflection from the zine plate. When the resonator is in the second position, having of opposite signs, and thus do not produce any tendency to spark. When the resonator is in this position the maximum sparks will be produced when the positive tubes strike against one side of the resonator, the negative tubes agaist tae caer, pd2z iba sisson wee|squal Moitiaaaae dian theory would lead us to expect that resonator should be half a wave pass, the current in the resonator will vanish | yesonator, as Wwe may neglect the capacity of thero will be a node at each end of tho res expect the wave length to be 2m times the | 8 times, as found by Sarasin and De la Rive. — Parabolic Mirrors. $43.] If the vibrator is placed in tho focal cylinder, and if it is of such a kind that the 1 emits are parallel to the focal line, then the the vibrator will, if the laws of reflection of | same as for light, after reflection from the # parallel beam and will therefore not d ‘they recede from the mirror; if such a bea parabolic mirror whose axis (i.e, the axis of it parallel to the beam, ie will bo brought to 6 fo line of the second mirror, For these reasons the u mirrors facilitates very much many experiments magnetic waves. ‘The parabolic mirrors used by Hertz wero made of | and their foeal length was about 12-5 em. The vil was placed in the focal line of one of the mirrors co equal brass cylinders placed so that their axes with each other and with ertieeye oe the cylinders was 12 cm. and the d r sida Sahni sovaided an all GOAN “Thon Sfbivte-tatbefosal le 'of ax egal parabola two picoos of wire, each had a straight pice then bent round at right angles so as to pass me ee = 345] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 405 | of the mirror, the length of this bent piece being 15 em. The ends which came through the mirror were connected with a i ¥ig- 120, | spark micrometer and the sparks were observed from behind the mirror. The mirrors are represented in Fig. 120. | - Electric Screening. 844.) If the mirrors are placed about 6 or 7 feet apart in such away that they face each other and have their axes coincident, then when the vibrator is in action vigorous sparks will be observed in the resonator. If a screen of sheet zinc about 2m. high by 1 broad is placed between the mirrors the sparks in the resonator will immediately cease; they will also cease if a paste~ hoard screen covered with gold-loaf or tin-foil is placed between the mirrors; the interposition of a non-conductor, such se a wooden door, will not however produce any effect. We thus see that a very thin motallic plate acts as a perfect screen and is absolutely opaque to electrical oscillations, while on the other hand a non-conductor allows these radiations to pass througi quite freely. The human body is a suiliciently good conductor to produce considerable screening when interposed between the vibrator and resonator, g 845.) Jf wire be wound round a large rectangular framework in such a way that the turns of wire are parallel to one pair of ‘sides of tho frame, and if this is interposed botweon the mirrors, it will stop the sparks when the wires are vorticsl and thus parallel to the Faraday tubes emitted from the resonator; the sparks however will begin again if the framework is turned through « right angle so that the wires are at right angles to the Faraday tubes. a4 349.] EXPERIMENTS ON ELECTROMAGNETIC Wayes, 407. normal to the wave front makes with the normal to the surface an angle tan~"y, where ia the refractive index, all the light is refracted and none reflected. Trouton (Nature, February, 21, 1889) has observed a similar effect with these electrical vibrations, From a wall 3 feet thick reflections were obtained when the vibrator, and therefore the Faraday tubes, wero perpendicular to the plane of incidence, while there was no reflection when the vibrator was turned through a right angle so that the Faraday tubes were in the plane of incidence. This experiment proves that in the Electromagnetic Theory of Light the Faraday tubes and the electric polarization are at right angles to the plane of polar- ization. Before proceeding to deseribe some other interesting oxperi- ments of Mr. Trouton's on the reflection of these waves from slabs of dielectrics, we shall investigate the theory of these phenomena on Maxwell's Theory. 849.] Let us suppose that plane waves are ineident on a plate of dielectric bounded by parallel planes, let the plane of the paper be taken as that of incidence and of zy, let the plate be bounded by the parallel planes 2=0, « =— h, the wave being in- cident on the plane a=0, We shall first take the case when the jion and Faraday tubes are at right angles to the planc of incidence. Let tho electromotive intensity in tho incident wave be repreeented by the real part of Aciarrtysph ; if i is the angle of incidence, ’ the wave length of the vibrations, ¥ their velocity of propagation, aa x Let the intensity in the reflected wave be represented by the real part of A’e' (02+ by+p0, ‘The coefficient of y in the exponential in the reflected wave must be the same as that in the incident wave, otherwise the ratio of the reflected to the incident light would depend upon the portion of the plate on which tho light fell. The coofficient of zin the expression for the reflected wave can only differ in sign from that in the incident wave: for if Z is the clectxo- eae tan Qa I oe aa a= ct, b=— sing, P=>F 4 349.) EXPERIMENTS ON ELECTROMAGNETIC Waves. 409 Equations (3) and (4) are equivalent to the condition that the tangential ‘ic force is continuous, magnetic Solving equations (1), (2), (3), (4), we get A’= — A(K*V" 008" —V* cos? i) (e'@#—e 4) = A, B=2AV cos i(KV’ cog r+ V cosi) e®*+ a, B'=2AV cosi(K V’ cos r— Vcosi) e'* ~ A, O = 4AKVV' cosi cosre!* + A, where B= (K2V™ cos" 7 + V? cos i) (<9 18) +2KVV" cosi cos 1 (ett 4 1"), Thus, corresponding to the incident wave of electromotive intonsity (3) cos * (x cosi+ysiné+ V2), there will be a reflected wave represented by —(K*V" cost V* cos*i) sin (57 h eos r) x ca . re cil ¢ cos [2 (—meosi+ysini+ Vt)+ 2 —s] +0, where X’ is the wave length in the plate, Di= (K*V" cost r+ V* cos? i} sit (2 2 cos) 4+4K*V V2 cost i cost reost (27h cor): E2V cost r+ Vieosti, 2a Sed ten 9 SV reatecar (yr hover): ‘The waves in the plate will be V¥ cos i (KV" cos r+ V cos i)x 008 = (@+h) oon r+y ain e+ vi)—s] +D, and Veosi (KV cos7— V cos i) x cos [57 (leh) cosr+y sinr+¥')—-3]— Ds while the wave emerging from the plate will be 2K" cos i eos r eos | 57 ((e+Mcosi+ysini +72) 3] =D. we have darkness whenever 2h cos 7 is = length of the light in the plate. ‘There will be a critical angle in this en 1¢ solu sineion EPV* cos r—Vicosti = 0 areal If the plate is non-magnetic the magnetic unity, and we have Yee R= a=aay’ cot? r—cot?i=0, © an equation which cannot be satisfied, so that ‘angle in this case. This result would not ho wore possible to find a magnetic substance wl parent to electric waves; for if p’ is tho magnetic so equation (6) becomes of the aubstanee, we have al KK =p #0 that equation (6) becomes Sot = cots or oor = eoti. Since eR ing = sind 350.) EXPERIMENTS ON ELECTHOMAGNETIC W. 41 | swe may transform this equation to ow eR). dint = EX EO), | hence if i is ronl, w’ must be greater than K. No substanee is known whieh fulfils the conditions of being transparent and having the magnetic permoability greater than the specific inductive capacity, which are the conditions for the existence of a polarizing angle when the Faraday tubes are at right angles to the plane of incidence. When the plane is infinitely thick, we see that Aa — KY" cos r= V cost =— EV cos rt Veo? or if the magnetic permeability is unity, , sin (i—r) =~ Sar) which is analogous to the expression obtained by Freanel for the amplitude of the reflected ray when the incident light is polarized in the plane of incidence. 850] In the preceding investigation the Faraday tubes were at right angles to the plane of incidence, we shall now consider the case when they are in that plane: they are also of course in the planes at right angles to the direction of propagation of the several waves. Let the electromotive intensity at right angles to the incident ray bo Agilaxr byt p0, that at right angles to the reflected ray Alelmartly tp, Let the electromotive intensity at right angles to the my which travels in the same sense as the incident one through the plate of dielectric, i.e. in a direction in which « diminishes, be Belt=+ly+p, while that at right angles to the ray travelling in a direction in whieh x increases is represented by Bel-sa+by+y0, ‘The electromotive intensity at right angles to the ray emerging from the plate is Celensysi 350.) EXPERIMENTS ON BLECTROMAGNETIC WAVES. 413 respectively, while the emergent wave is 2K tani tanr cos ((a-+i)coni+y sini + vio] =D, where DP = (KC ton? r+ ton? if sin® (27 h 008 7) +4K° tant tani eos® (57 A eos r) J tan? + tan?é Qn and tan? = Se tanptani Sz heosr) From these expressions we see that, as before, there is no reflected wave when h is vory small compared with 4’ and when 4 eos vis a multiple of A‘/2; these results are the same whether the Faraday tubes are in or at right angles to the plane of incidence. We see now, however, that in addition to this the reflected wave vanishes, whatever the thickness of the plate, when K tanr = tan {_ or since v/K sin r = sini where p’ is the magnetic permeability, the reflected wave vanishes when tan’ = HE); vE-1 if the plate is non-magnetic »’= 1, and we have tani= VK. When X tanr = tani the reflected wave and one of the waves ‘in the plate vanish ; the electromotive intensity in the other wave in the plate is equal to ae ox P 4 cos 57 (weosr+y sin r+ V"t), and the emergent wave is con 5 ((2+h)oosi+y sini + Vi—"* os r)- The intensity of all these waves are independent of the thickness of the plate. Tf the plate is infinitely thick we must put B= 0 in equations (7); doing this we find from these equations that . ‘K tany—tan f) Ma A ane stant’ ar sin 26 sini cosr+ K costsinr” 351-] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 415 neglect 4? in equation (9), which thon becomes a? =—Anpip/e, or @ =4/2app/o(1-), thus a’ is exceedingly large compared with a. ‘Wo shall first consider the case when the Faraday tubes are at right angles to the plane of incidence as in Art, 349, The condition that the electromotive intensity parallel to the surface of the plate is continuous will still be true, but since there is no real angle of refraction in metals it is convenient to recognize the second condition of that article as expressing the condition that the tangential magnetic force is continuous, The tangential magnetic fores is parallel to y and is equal to 1 dE pep where y is the magnetic permeability, By means of this and the previous condition we find, using the notation of Art. (349), A= —A (a2 —a*) (4 —') + D, Ba2Aa(a‘/u+a)e*+D, Bi=2Aa(a/y—a) e~""" +D, C=Ata(a/p)e*+D, where D = (al?/p2 + at) (88 — -) + 20 (a'7p) (eM 4), Since <‘(¢*+#”+P4 represents 8 wave travelling in the plate in the direction of the incident wave, i... so that @ is increasingly negative; the real part of 1a’ must be positive, otherwise the amplitude of the wave would continually increase as the wave travelled onwards; hence if ha’ is very large, equations (10) become approximately, remembering that a’/a is also very large, A'=—A, Bath 4, (10) C=B=0. Hence in this case there is complete reflection from the metal plate, and since A’+A =0 we see that the electromotive in- tensity vanishes at the surface of the plate, and since 0=0 there is no electromotive intensity on the far side of the plate. Ae for the emergent wave becomes asza] TV Ta 8 eth MO _ ‘The thickness of the conducting material which, when inter- in the path of the wave, produces a given diminution in the electric intensity is thus proportional to the specific resist “Bnce of the material ; this result has been applied to measure the ‘Specific resistance of electrolytes under very rapidly alternating ‘currents (soo J. J. Thomson, Proc, Roy. Soc. 45, p. 269, 1889), The preeeding investigation applies to the case when tho Faraday tubes are at right angles to the plane of incidence, the same results will apply when the Faraday tubes are in the plane of incidence: the proof of these results for this caso we shall however leave as an exercise for the student. Reflection of Light from Metals. 852.] The assumption that a’/a ia very largo is logitimate when we are dealing with waves as long as those produced hy Hertz's apparatus, it ceases however to be so when the length of the wave is as small as it is in the electrical vibrations we call Tight. We shall therefore consider separately the theory of the reflection of such waves from metallic surfaces. With the view of making our equations more general we shall not in this case neglect the effects of the polarization currents in the metal; when wwe include theso, the components of tho magnetic force and electromotive intensity in the metal satisfy differential equations of the form ee ee ee See Maxwell's Electricity and Magnetism, Art. 783; here K’ is the specific inductive capacity of the metal. 853.) Let us first consider the case when the incident wave is polarized in the plane of incidence, which we take ns the plane of ay, the reflecting surface being given by the equation # = 0. In this caze the electromotive intensity Z is parallel to the axis of =; let the incident wave be Za elerrty+e, se ss Za Acl—s+ty+P9, acces | 4-] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 419° | Now for transparent substances the relation between a’ and @ | whore y’ is the rofractive index of the substance, In the case of metals however the relation between a’ and a is oat = ee Rega, say, (4) | which is of exactly the same form as the preceding, with Re written instead of »’, the refractive index of the transparent substance. ‘Thus, if in Fresnel's formula for the reflected light we suppose ‘that the refractive index is complex and equal to Re", where R Biber ie tin by: equation (a\icwe shall arrive at the remales given by the preceding theory of the reflection of light by metals. 54.) Let us now consider the case when the plane of polariza- tion is perpendicular to the plane of incidence; in this case the electromotive intensity is in the plane of incidence and the magnetic fores y ab right angles to it. If the incident wave is expressed by the equation y= elartlyset , then in the dielectric we may put y= ellartty tnt 4 Atet(manelys pe, while in the metal we have y= Betles+hy+20, Since the magnetic force parallel to the surface is continuous, pro tses 144, (3) The other boundary condition we shall employ is that Q, the tangential electromotive intensity parallel to the axis of y, is continuous. Now if g is the electric polarization parallel to y, and v the conduction current in the same direction, then in the dielectric above the metal nei ey, at da SA or since =Ko +t 9= (50 aap @ hea will be seon that for all these metals without exception the jue of & is greater than unity, so that the real part of Re" a2 (1—/*) is negative. Equation (4), Art. 353, shows, however, that tha real part of R+e?* is equal to xA’/K, an. essentially positive quantity. This shows that the electromagnetic theory of metallic reflection is not general enough to cover the facts, In this respect, however, it ia in no worse position than any existing theory of light, while it possesses the advantage other thvorics of explaining why metals are opaque, 856.] Tho direction in which to look for an improvement of theory seems pretty obvious, The preceding table shows bow rapidly the effects vary with the frequency of the light vibrations; they are in this respect analogous to the effects of “anomalons dispersion’ (soe Glazebrook, Report on Optical B.A. Report, 1885), which have been accounted for by ‘ing that the molecules of the substance through which the seal have free periods of vibration comparable with the of the light vibrations. The energy absorbed by such is then a function of the frequency of the light ‘vibrations, and tho optical character of the medium cannot be fixed by one ar two constants, such as the specific inductive (3%), Art. 353, We see from that equation thatif a! in iron were very large, the intensity of the light 1 iron would be very nearly the same as that light, in other words iron would have a very power. The reverse, however, seems to be trae; (Wied. Ann, $9, p. 649, 1890) gives the follow: 953 851 132 861 535 Rubens (Wied. Ann. 37, p. 265, 1889) gives for the the following numbers :— 90-3 TLL 70-0 .] ‘EXPERIMENTS ON ELECTROMAGNETIO WAvES. 428 with respect to the optical constants of motals in the table in Art, 353. The theory of metallic reflection is however fur from aecounting for the facts that we cannot attach much to considerations based on it. The only conclusion we come to is the negative one, that there is no evidence to that iron does retain its magnetic properties for the light ‘brati The change in Phase produced by the Transmission of Light through thin Films of Metal. 858.) Quincke (Pogg., Ann. 120, p, 699, 1863) investigated the change in phase produced when light passed through thin silver ~ plates, and found that in many cases the phase was accelerated, the effect being the samo as if the velocity of light through silver was greater than that through air. Kundt (Phil. Mag. 15], 25, p. 1, 1888), in a most beautiful series of experiments, measured the deviation of a ray passing through a small metal prism, and found that when the prism wax made of silver, gold, or copper, the deviation was towards the thin end. With platinum, nickel, bismuth, and iron prisms the deviation was, on the other hand, towards the thick end. We can readily find on the electromagnetic theory of light the change in phnxe pro- duced when the light passes through thin film of metal. The equation (11) of Art, 361 shows, that if the incident wave (sup- posed for simplicity to be travelling at right angles to the filin) is represented by et(ax tpt) the emergent wave will be 4a (a'/y) ear p) ; (jee a) — $20 (a) (em) or if the film is so thin that ha’ is 9 smal] quantity, the emergent wave is equal to chia sons pl pay 14a Pte) Now, since in this case b= 0, we have by equation(4)of Art. 353 g = R*e2, hence the emergent wave is equal to eile «(ax +36) 1+4iipa ee ie yf he ecru mamoniorene wares #85 Rerirctiox or Evrcrnowaonerio Waves From Wrnes. Reflection from aw Grating. 859.] We shall now consider the reflection of electromagnetic ‘Waves from a grating consisting of similar and parallel metallic ires, whose crogs-sections we leave for the present indotermi- “nate, arranged at equal intervals, the axes of all the wires being one plane, which we shall take as the plane of ys, the axis of being parallel to the wires: the distance between the axes of ‘two adjacent wires is a, We shall suppose that a wave in whieh electromotive intensity is parallel to the wires, and whose is parallel to the plane of the grating, falls upon the wires. ‘The electromotive intensity in the incident wave may be repre- ae, " sented by the real part of Ae a ""*”, » being measured from the plane of tho grating towards the advancing wave. The incidence of this wave will induce currents in the wires, and ‘these currents will themselves produce electromotive intensities to s in the regiun surrounding them ; these intensities will evidently be expressed by a periodic function of y of such a character that when y is increased by a the value of the function remains unchanged. If we make the axis of > coincide with the axis of one of the wires, the electromotive intensity will evidently be an even function of y. Thus Ey, the electro- motive intensity due to the currents in the wire, will be given by an equation of the form E,= A, cos ——* where m is an integer. Sinee the electromotive intensity satisfies the equation PE @E 1 PE de tay vi ae’ wo have axtm? | xt Ee omen, ve ‘We shall assume that the distance betweon the wires of the grating is very small compared with the length of the wave; thus, unless m is zero, the first term on the right-hand side of the above equation will be very large compared with ‘ne seccmh,wo 2 changed b where a is given by (2) and deponds upon tho sizo of oe ‘The other part of the ‘s Qe: 4 Clog (2c # con 22Y 4 «a )e @ Eis inspprecale 6 6 cients ee the distance hetween tho wires; henes the retacHon aiieniaan distance from tho grating, is the aa in phase as from a continuous metallic surface. 860.] If tho electromotive intensity bad been at right an the wires the reflection would have been very small grating of this kind will act like a polariscope, by reflection or transmission an unpolarised set of ¢ vibrations into a polarised one. When used to produce polarisa- tion by transmission we may regard it as the electrical analogue of a plate of tourmaline crystal. Scattering of Electromagnetic Waves by a Metallic Wire, 861.] Tho seattoring produced when a train of plane electro magnetic waves impinges on an infinitely long metal cylinder, whose axis ia at right angles to the direction of propagation of — the waves and whose diameter is small compared with the wave — Tength, can easily be found as follows :— We shall begin with the ease where the electromotive intensity in the incident wave is parallel to the axis of theeylinder, which we take as the axis of 2; the axis of x being at right angles to the fronts of the incident waves. Let A be the wave length, then Hy, the electromotive intensity in tho incident waves, may be represented by the equation gt Tey where the real part of ite right-hand side fs 16 be taken. The positive direction of x is opposite to that in which the waves are travelling, In the neighbourhood of the cylinder @/A is emall, so that we may put 120, Baek (14) : —— | re ——_—__—— — 362.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 429 approximately, or if 7 and @ are the polar coordinates of the point where the intensity is 2, age, —Fe Ey=e* Q +25 reose): Let E, be the electromotive intensity due to the currents indueed in the cylinder, then Z, satisfies the differential equa- tion PE, dE, UME, _ 1 PE, “de + hd * 8 dot ~ ae 4c =—"* i Sirs hte Saas ‘The solution of which outside the cylinder is ae) fre EB, = A, cos nd K, S7)e ny where K, represents the ‘external’ Beasel’s function of the n™ order, Thus B,={40Ky (Fr) +A, cos 0x, (2r) Qa sn +A,c08 20, (Fr) tu. fe® Now since the cylinder is a good conductor, the total tangential electromotive intensity must vanish over its surface, see Arts, 300 and 301. Hence ifcis the radius of the cylinder, #,+ 2, = 0 when r=c; from this condition we get 1 4j=-— 3 4,=-—"S_ =Ay=...=0. K, (=) AK, (0 Th a K. 22 K. 2a zo AK) se | em n=] - — ase : KG) * Go 362.] Let us first consider the effect of the cylinder on the lines of magnetic force in its neighbourhood. If a, 8 we Yas 364.) EXPERIMENTS ON ELECTROMAGNETIC Waves. 431. ‘or in Cartesian coordinates a{e—@+y)} =C e+"); these curves are shown in Fig. 121. 863,] Since the direetion of motion of the Faraday tubes is at right angles to themselyes and to the magnetic foree, when the lines of magnetic force near the cylinder are circles, these tubes will, in the neighbourhood of the cylinder, move radially, the positive tubes (i.c. those parallel to tho tubes in the incident ware) moving inwards, the negative ones outwards, In the special easo where the clectromotive intensity vanishes at the SG a a Fig.on © nxis of the cylinder, the incident wave throws tubes of one sign into the half of the cylinder in front, where x is positive, and tubes of opposite sign into the half in the rear, where 2 is negative; in this case, if the positive tubes in the neighbourhood of the eylinder are moving radially inwards in front, they are moving radially outwards in the rear and vice versed ; there are in this case but fow tubes near tho equatorial plano, and the motion of these is no longer radial. 364.] When tho distance from the cylinder is lange compared with the wave length, we have #22) =n, Gay —ser fh (n/a)! a x, (2r)=-14* — aa J _uxrunmrenrs oN BtEoTnomsoNeTo waves. 433 the preceding article were due to the cylindrical shape of the ‘The only case to which the results of this article would be applicable without further investigation is that in which As the cloctromotive intensity is at right angles to the axis of the cylinder, the magnetic force will be parallel to the axis. Ist the magnetic force H, in the incident wave be expressed by the equation 6) Hy=e* : I When x which is equal to cos @ is small compared with A, this , ‘is approximately Srey tg, ian ® is 2 Hy=«* {i Sak F eiheosae = tease}. Since H, the magnetic force, satisfies the differential equation CH PH _ 1 dH a * dy ~ Vide" the magnetic force H, duo to the currents induced in the cylinder may be expressed by the equation B.=en" fay (FE r) + Ayeo00 K, (3) +-4,c0020K, (77h, where A,, A, and A, aro arbitrary constants. ‘The condition to be satisfied at the boundary of the cylinder is that the tangential cloctromotive intensity at its surface should vanish. In this ease we have, however, (if, 1h) = 4 (intonsity of eurrent at vight angles to 7). ‘The current in the dielectric is a polarization current, and if Z is the tangential electromotive intensity, the intensity of this current at right angles to r is Kae tn dt’ which is equal to K typ tx Thus the condition that # should vanish at the surtere is rf 366.) EXPERIMENTS ON ELECTROMAGNETIC WAVES, 435 When 7/4 is small, this condition lends to the equation “Bia eens ( 7) +2 cosa fr re PK, (2S Fh — Foose fot 22 x, (2 22)t] =0, where C is a constant. Substituting the approximate values of X,, K, and K, this ae i [1 ty Se Jog (r/yA) relied ~ cos 9? SS ee Fig. 199. Bia Except when « * is wholly real, ie. except when the rate of variation of the magnetic force in the incident wave at the axis of the eylinder vanishes, by far the most important term is that which contains cos , so that the equations to the lines of electro- motive intensity are SP e000 = a constant = C’, say. Peeler isha ihiveiciooasiy sv oad tec At the times when «2°! ix wholly real, the lines are ap- proximately circles concentric with the cross-section of the cylinder, since in this case the term involving the logarithm is the most important of the variable terms. Ffa = 369.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. 437 of the reflector measured in the direction of the electromotive _ intensity is considerable, whatever may be the breadth of the Teflector at right angles to the electromotive intensity. On the other hand, when the electromotive intensity is at right angles to the axis of the cylinder, the electromotive intensity in the scattered wave increases as the square of the radius of the cylinder, so that in this case the size of the reflector is all im- portant. These results are confirmed by Trouton’s experiments on ‘The Influence the Size of the Reflector exerts in Hertz’e Experiment,’ Phil. Mag. [5], 32, p. 80,1891. On the Scattering of Electric Waves by Metallic Spheres. 869.] We shall proceed to discuss in some detai) the problem of the incidence of a plane electric wave upon a motal sphero *. Tf o, 8, y; f, g, h are respectively the components of the tmagnetic force and of the polarization in the dielectric which are radiated from the sphere, then if y stands for any one of these quantities it satisfies # differential equation of tho form ay dy dy _ 1 dy @) where V is the velocity with which electric action is propagated through the diclectric surrounding the sphere. If A is the wave length of the disturbance incident upon the sphere, thon the components of thagnotio induction and of electric polarization ia will all vary as es; thos V-2dp/at? may be replaced by —47n'y/M, so that writing k for 27/A, equation (1) may be written dy ds a Br. 2Y sy ao, a solution of which is by Art. 308, y= tM Z (kr), whore r is the distance from the centre of the sphere. Since the waves of magnetic force and dielectric polarization are radiating outwards from the sphere enh Salter) = Gee * Te? * The sonttoring by an insulating sphere is disoussed by Lord Rayleigh (PAil. Mag. 1, 98, 1881), ae ees rersincs ars wet BS Ne eee Hoasent in io Tvaity College, Cambridge, by Profesor Michell in 1800, Ido not know ‘of any papers which disoam the special problezs of the acattering by metal spheres. —_ 370.) EXPERIMENTS ON SRACEROMAGHETID WAVES. 439 Now Sle) =(, ~ a3 iu have cig Fi (kr) = kr f(r). (3) Falke) + LEY ¢ (hr) +f (br) = 0, which ply written as {lerg’, (hor) + (21 + 1) f,(ler)} = — kerf, (ler) ate) =—J'-a(la) by (3); hence, since the constant of integration must vanish since all the fs involve e~'*Y, Senate “Ter f a (ker) + (20 + 1) fuller) =f, (hr), (4) and by (101), Art. 308, (2m+ 1) fuller) = — {fa (kr) + r*fuss(hr)}. (8) Writing (n+ 1) for m in (4), we have Hef n+ (kr) + (20+ 3) fara (hr) =—halhr) =-feal. From this equation we sce that To prove that equation (2) gives the most general expressions for a, 8, y, we notice that the values of a, § may be written a = 3fa(br{ {on +2) ets (nay ener 2 ean} +(y a Ab B= Sf(lr)[ fia +2) Set (a —1yeene aa i + @d-24)»\] The most general expressions for a, 8, when they represent radiation outwards from the sphere, may however, Art. 308, be expressed in the form a= Xf, (lr) U,, } (8) B=2f, (lr) Vay where U,, V, are solid spherical harmonics of degree n, Since 372] | EXPERDIENTS oN reapers 4 Thus if a, 8, y are given by (2), then we have = | aw Ms tenet) fans oy Me alte fan (tr) eed +2 (24-1) fall) (y See — 2 , a Bm af (mt tf) SR nba fay br) FS +2 (2nt1) Bfa(br) (04-24), seas {inti healer) St ctutetha te enh +23(2n+1) Pf, (kr) («ey Se oe). 872.) In the plane electrical wave sich to ace us suppose that the electric polarization hy in the wave front is parallel to s and expressed by the equation igpzer TO) = akties), where the axis of « is at right angles to the wave front, We have to expand /, in the form that is, at places separated by the intervals rat /§-3 along the wire the rotation of the resonator will produce no effect upon the sparks, while on one side of one of these posi- tions it will increase, on the other side diminish the brightness: of the sparks, If A’ were very large compared with A, that is, if tho velocity of the waves travelling freely through the air ‘were very much greater than that of those travelling along the wire, the distance between the places where rotation produces no effect would be 4A, which is the distance between the nodes observed in the experiments described in Art. 380, Herts, how- Peter ty aoa rama A= 56, ife-2)=75; hence 4’ = 8-94. Thus from these experiments the the free air waves would appear to be greater than the wire in the proportion of 894 to 56 or 16 to velocity of the air waves is about half as large again the wire waves. We have, however, in the preceding rivwdipdeas et ances winds hoe es ae ee { practice ; we have assumed, for example, that in hood of the resonator the front of the air waves was at | ; angles to the wire. Since the resonator was close to wha | the vibrator this assumption would be justifiable if there had een no reflection of the air waves from the walls or floors of the room. Bineo the (hlckness af the all ous atl Sea with the wave length it is not likely, unless they were very damp, that there would be much refleetion from them ; the case of the Boor is howevor very different, and 3¢ is difcullte/ses how reflection from it could have been entirely avoided. Re flection from the floor would however introduce waves, the normals to whose fronts would make o finito angle with the wire. The cloctromotive intensity in tho spark gap due to such ‘waves would no longer be represented by a term of the form cos (2n(V"t—=)/n’), but by one of the form ad cos (2% (V"t—= con d)/2’), where @ is the angle totwoon the norinal to the wav froat a | the wire. Thus in the preceding investigation we must, for such waves, replace \’ by X’sec0, and their apparent wave Jength along the wire would be A’seo@ and not \’, 80 that the refles- tion would havo the effect of increasing the apparent wave length of the air wayes, Tho result then of Herte’s experiments — that the wave length of the air waves, measurod parallel to the mY a 385.) EXPERIMENTS ON ELECTROMAGNETIC WAVES. 459 wire, was greater than that of the wire waves, may perhaps be explained by the reflection of the waves from the floor of the room, without supposing that the velocity of the free air waves is different from that of those guided by the wire, 885.] The experiments of Sarasin and De ln Rive (Archives des Sciences Physiques et Natwrelles Gentve, 1890, t. xxiii, p. 113) on the distance between the nodes (1) along » wire, (2) when pro- duced by interference between direct air waves and waves re- flected from a large metallic plate, scem to prove conclusively that the velocity of the waves guided by a wire is the same as that of free air waves. The experiments on the air waves have already beon deseribed in Art. 339; those on the wire waves were made in a slightly difforent way from Horts’s oxperimenta. Fig. 125. ‘The method used by Sarasin and De la Rive is indicated in Fig. 125. Two metallic plates placed in front of the plates of the vibrator have parallel wires /’, / soldered to them, the wires being of equal Jength and insulated. The plane of the resonator is at right angles to the wires, and the sir gap is at the highest point, so that the air gap is parallel to the shortest distance between the wires. The resonator is mounted on a wagon by means of which it can be moved to and fro along the wires, while a scale on the bench along which the wagon slides enables the position of the latter to be determined. The resonator with its mounting is shown in Fig. 126, Sarasin and De la Rive found that as long ns the sume resonator was used the distance between the nodes as determined by this apparatus was the 386.] BXPRRIMENTS ON ELECTROMAGNETIC WAvEs. 461 the vibrator are of the same sign as those arriving. Thus, when the end is free, the current vanishes and the electromotive in- tensity is a maximum, while when the end is attached to a large plate the electromotive intensity vanishes and the current is a maximum, Since the sparks in the resonator, when used as in Sarasin and De Ja Rive's experiments, are duc to the tubes falling directly on the air gap, the aparks will be brightest when the electromotive intensity is a maximum, and will vanish whon it vanishes; thus the loops when the ends are free will coincide with the nodes when the wires are attached to large plates. ‘This was found by Sarasin and De la Rive to be the case. A similar point arises in connection with the experiments with wires to that which was mentioned in Art. 342 in connee- tion with the experiments on the air waves. The distance between the nodes, which is half the wave length of the vibra- tion of the resonator, is, as is seen from the table in Art, 340, very approximately four times the diameter; if the resonator were a straight wire the half wave length would be equal to the length of the wire, and we should expect that bending the wire into a circle would tond to shorten the period, we should there- fore have expected the distance between the nodes to have been littlo less than the cireumference of the resonator. Sarasin and De Ia Rive’s experiments show however that it was 80 per cont. greater than this: it is remarkable however that the dis- tance of the first node from the end of the wire, which is a loop, was always equal to half the circumference of the resonator, which is the value it would have had if the wave length of the vibration emitted by the resonator had been equal to twice its circumference, 386.] The experiments of Sarasin and De la Rive show that when vibrators of the kind shown in Fig, 113 are uscd, tho oscillations which are detected by a circular resonator ara thore in the resonator rather than the vibrator. Rubens, Paalzow, Ritter, and Arons (Wied, Ann. 37, p. 529, 1889; 40, p.55, 1890; 42, pp. 154, 581, 1891) have used another method of measuring wave lengths, which though it certainly requires great care and labour, yet when used in a particular way would seem to give very accurate results. The method depends upon the change which takes place in the resistance of a wire when it is heated by the passage of a current through 88.] EXPERIMENTS ON ELECTROMAGNETIC WAVES. ares series with Z was bent round two picees of glass tubing ugh which Sie ioe 2. DI pan hs laf a ‘Leyden jar, and the electricity which flowed through the wire Z and disturbed the balance in the Bridge was due to the charging and discharging of these jars. ‘The pieces of glasa tube were attached to a frame work, see Fig. 128, which was moved along the wire, and the deflection of tho galvanometer observed as it moved along the wire. The AH Big. 129, relation between the galvanometer deflection and the position of the tubes is shown in Fig. 129, where the ordinates represent the deflection of the galvanometer and the abscissae, the distance of the turns in tip bolometer cirenit from the point F in the wire. The curve shows very clearly the harmonic character of the disturbance along the wire, 888.] The results however of experiments of this kind were not very accordant, and in the majority of his experiments Rubens used another method which had previously been used by Lecher, who instead of a bolometer employed the brightness of the discharge through an exhausted tube as a measure of the intensity of the waves. 4 ‘EXPERIMENTS ON BLECTROMAGNETIC WAVES. , if there were no capacity at the ee be an ‘an: 0G ‘hamsber of quarter wave the i Reacace or tay Saat Rouaraees ar ee flow through it, produce no diminution in the electro- ’ intensity at the ends J, H; in other positions of the ee ee ee current, which in its absence would go tothe ends, would be diverted by the bridge, so that tho electromotive intensity at the ends would be weakened. Thus, when the Moflection of the bolometer was a maximum, the distances of ‘the bridge from the ends J, H would be an odd multiple of a quarter of the wave length of the vibration travelling along the wire; thus, if these vibrations were ‘forced’ by the vibrator, | the positions of the bridge which give a maximum deflection in the bolometer would depend upon the period of the vibrator. | Rubens’ experiments show that this was not the ease. We may therefore, as the result of these experimenta, assume that the effect of the sparks in the vibrator is to give an electrical impulse to the wires and start the ‘free’ vibrations proper to them. Tho capacity of the plates at the ends of the wire makes the investigation of the free periods troublesome ; we may how- ‘evor avail ourselves of the resulta of some experiments of Lecher’s (Wied. Ann. 41, p. 250, 1890), who found that the addition of ‘eapacity to the ends might be represented by supposing the wires to an extent depending upon this additional capacity. 889.] Let AB, CD, Fig. 131, be the original wires, Aa, BS, Qy, D2 the amount by which they have to be prolonged to represent the capacity at the ends, we shall call the wires of, 72 the equivalent’ wires. Let FQ represent the position of the bridgo. 4 P B 7 € @ D " Fig. 181. ‘The electrical disturbance produeed by the coil may start ‘several systems of currents in the wires «8, yd. Then there may ‘be a system of longitudinal currents along a8, yi determined by ‘the condition that the currents must vanish at a, 8, and at y, & Another system might flow round a PQy, their wave length being determined by the condition that the currents along the wire must vanish at a and y, and that by symmetry the electrification uh 4 should be as small as possible they would the shortest course, i.o, that round the cirouit aPQy: t rents would induce currents round the ein experiments (Wied. Ann. 41, p. 880, 1890) show t circulating round aPQy, 8PQd ure much more ef producing the electrical disturbance at the end longitudinal ones along of, yd. Asia tone copa the disturbance at the ends, Lecher used an luminosity in the tube served as an indication | tude of the electromotive intensity acrosa 83, In in parallel, and moved this about until the Iu tube was a maximum; he then eut the wires almost entirely destroyed by it. qi position of the bridge in which the luminosity of the tube maximum depended upon the length of the bridge; i were lengthened it had to be pushed towards, tha scironite aPQy, BPQS which chiefly cause in the tube, Since the currents in the circuit duced by those in the circuit a?@Qy, they will the time of the electrical vibration of the a EXPERIMENTS ON ELECTROMAGNETIC WAVES. 467 same aa that of 8PQs The periods of vibration of these are determinod by the conditions that the current must at their extremities and that these must be in opposite conditions ; these conditions entail that the wave e must be odd submultiples of the lengths of the cireuit. the two circuits are in unison the wave lengths must be the ‘same, hence the ratio of the lengths of the two circuits must be of the form (2n—1)/(2m—1), where n and m are integers, ‘This conclusion is verified in a remarkable way by Rubens’ experiments with the bolometer. Tho relation between the deflections of the bolometer (the ordinates) and the distances _ of the bridge from @ in Fig. 127 (the abeciasae) is represented in Fig. 130. Tho length of the bridge in these experimenta | was 140m. that of the curved piece of the wire HG was 83 em., and that of the straight portion @J was 570em. The lengths | Aa, Bp which had to be added to the wires to represent the effects of the capacity at the ends were assumed to be 85 em, for the end of the wire next the coil, and 60 cm. for the end next the bolometer. These two lengths were chosen so as best to fit in with the observations, and were thus really determined by the measurements given in the following table ; in spite of this, 80 many maxima were observed that the observations furnish satisfactory evidence of the truth of the theory just described, Distance a/ point of nila ta an-1, me Crrereatins pak Calewlated,| Observed, gaj/ajsajil] a 50. 4 Baha h ts ao 86 B a) 2/5) 8] us 1s c 4} 3] 7/5 | ag: | x89 D a}a/sa/3] as | a4 zg a)a) 5/7] m | a5 r a}a]a]6] a3 | a0 @ a} ¢]a]7] aa | see az apof;ils| a | ae a a} a} 1] 5} sos | sos x aiapa]7 | om 523 L L 391 a] EXPERIMENTS ON ELECTROMAGNETIO WAVES. 469 Substance. Observer. x, | Temper Pa 2-050 B 2526 D 241 D 238 D 2.68 D i ae 5 2.70D Boryl along axis. : “4 624 248 D or papontizdlay bea ” 7.58 2.50 D Topaz .. oe wee » 6-56 261D Gypsum : ” 6.38 282 D Alom 4 64 22 D Rock Salt i 5.85 236 D Petroleum Bpirit . Hopkinson* | 1.92 1.992 Petroleum Oil, Field’ ” 2.07 2.075 0 > 2.10 2.078 co 9 218 2.086 co % 2.98 2123 2 RY 478 2-158 2 (j 8.02 2-185 co il oe ” 3.16 2-181 Neat's-foot Oil; . é 8.07 2.125 0 Benzene G,H,...-....| Hopkinson? | 288 2.2614 D » ‘Negreano® 2.2988 2.2434 D paar * 2.2921 2.2686 D Toluene GHy. .-- +... if 2.242 2.224 D » on » 28018 2.945 D ee eeeeeess| ‘Hopkinson? | 242 2.2470 D Stetwet Bo 24288 D » » logreano' 2.2679 219 Metaxylene C,H. o 2.9781 2.248 D Preudocumene C, Hy, 2.4810 2.201 D Hoptinmoas | 2ai0* 2201 lopkinson 2.25 roy 2.2254 Ne * 22618] 20 |] 2168 Hopkinwon* | 267 |] 2678D (at 10°) » 475 1.8055 ” 2.05 1.9044 D vt ts. | Cohn and Arona? | 76. 1779D aa Rosa® 75-7 + Klemendit, Wien. Berichte 96, 2nd abth. p. 807, 1887. » Boltzmann, Wien, Berichte 70, 2nd abth. p. 842, 1874. * Curie, Annales de Chimie ot de Physique, 6, 17, p. 885, 1889. * Hopkinson, Phil. Trans. 1878, Part I, p.17, and PAil. Trans. 1881, Part 11, p. 855. * Hopkinson, Proc, Roy. Soo. 48, p. 161, 1887. * Negreano, Compt. rend. 104, p. 425, 1887. 7 Qohn and Arons, Wied. Ann. 88, p. 18, 1888, * Ross, Phil, Mog. [5), 81, p. 188, 1891. ——] 393-] EXPERIMENTS ON KLECTROMAGNETIO Waves, 471. Ayrton and Perry (Practical Electricity, p, $10) found that ‘the specific inductive capacity of a vacuum in which they estimated the pressure to be .001 mm. was about -994. ‘This would make X for air referred to this vacuum aa the unit about 1-006, whilo ,* from o vacuum to air is about 1-000588, there is thus a serious discrepancy between these valucs. 392.] Wo see from the above table that for somo substances, such as sulphur, paraffin, liquid hydrocarbons, and the permanent gases, the relation K =:* is very approximately fulfilled; while for most other substances the divergence between K and p* is con- siderable. When, however, we remember (1) that even when is estimated for infinitely long waves this is done by Cauchy's formula, and that the values so deduced would be completely invalidated if there were any anomalous dispersion below the visible rays, (2) that Maxwell's equations do not profess to con- tain any terms which would account for dispersion, the marvel is not that there should be substances for which tho relation K=y* | does not hold, but that there should be any for which it docs. | ‘Po give the theory a fair trial we ought to moasuro the specific inductive capacity for electrical waves whose wave length is the same as the luminous waves we use to determine the refractive index. 393,] Though we are as yet unable to construct an electrical system which emits electrical waves whose lengths approach those of the luminous rays, it is still interesting to measure the values of the specific inductive capacity for the shortest electrical ‘wares we can produce, We can do this by a method used by von Bezold (Pogg. Ann. 140, p. 541, 1870) twenty years ago to prove that the volocity with which an electric pulse travels along a wire is independent of tho matorial of wire, it was also used by Hertz in his expori- ments on electric waves. This method is as follows. Let ABCD be a rectangle of wires with an air space at EF in the middle of CD; this rectangle is connected to one of the poles of an induction coil by a wire attached to a point X in AB, then if X is at the middle of AB the pulse coming along the wire from the induction coil will divide at K and will travel to # and F, reaching these points simultaneously; thus # and J’ will be in similar oloctric states and there will be no tendency to epark across the air gap EF, i | 472 —- EXPERIMENTS ON EI Tf now we move K toa position which is respect to # and F, then, when a pulse | angle, it will reach one of these points they do through ain, then which goes round AZ will Trefave. the pate yas tem arrives ab J; thus # and F will not be move K towards B or elee keop K fixed EF and, as the waves travel more slowly Fig, 182, through the dielectric than through air, lengthen the side AD of the figure. If wo do this until the sparks disappear we may conclude that FE and F are in similar electric states, and therefore that the time tuken by the pulse to travel round one arm of the cireuit is the same as that round the other, By seeing how much the length of the one arm exceeds that of the other we ean com- Best the velocity. a lanoppasrnoho An eee in which BC is immersed with that 894.) Lhave used (Phil. Mag. [6], 30, p. 129, 0, 1800) this method to dotermine the velocity of propagation of clectromagnetic action: through paraffin and sulphur. This wag done by leading one of the wires, say BO, through a long motal tube filled with either paraffin or sulphur, the wire being insulated from the tube which was connected to earth. By measuring the length of wire it was necessary to insert in AD to stop the sparks, I found that the velocities with which electromagnetic action travels thro sulphur and paraffin are respectively 1/1-7 and 1/1-35 velocity through air, The corresponding values of the specific inductive capacities would be about 2-9 and 1-8, 895.) Rubens and Arona (Wied. Ann, 42, p. 681; 44, p, 208), while employing a mothod based on tho same principles, | L — el EXPERIMENTS ON BLECTROMAGNETIO WAVES. 473 ory much moro sensitive by using a bolometer in- | observing the sparks and by using two quadrilaterals: of one, The arrangement they used is represented in 133 (Wied. Ann. 42, p. 584). = Fig. 138, The poles P and Q of an induction coil aro connected to the balls of a spark gap S, to each of these balls a metal plate, 40 cm. square, was attached by vertical brass rods 18 em. long. Two small tin plates x, y, 8 em, square, were placed at a dix- tance of between 3 and 4 em. from the large plates. Then wires connected to these plates made sliding contacts at u and v with the wire rectangles ABCD, EL FGH -230 em. by 350m. Oneof these rect- angles was placed vertically over the other, the distance between them being 8 em. The points u,v were connected with each other by a vertical wooden rod, ending in a pointer which moved over a millimetre scale, The direct action of the coil on the rectangles was scrooned off by interpesing a wire grating through which the i — The two wires connecting she platen: werw attache tueter real atllar to ibat deatetbed la APL 3BY: ‘By means sliding coil attached to the bolometer circuit, Arons and Rubens: investigated the electrical condition of the cireuits wADJ, wBCK, &c., and found that approximately there was a node in the middle and a loop at each end; these circuits then may be regarded as excouting clectrical vibrations whose wave longths are twice the lengths of the circuits. If the times of tho cirenits on tho left of u,v aro the same as those om the right, the plates Jand X will be in similar electrical states, as will, J and M, and there will be no deflection of the galvanometer in the bolometer cireuit. When the wires are surrounded this will be when wu, v are at the middle points of AB, practice Arons and Rubens found that the deflection galvanometer never actually vanished, but attained a 1 decided minimum when u,v were in the middle, and | effect produced by sliding w, v through 1 em. could easily be detected. To determine the velocity of propagation of electromagnetic action through different dielectrics, one of the short sides of the rectangles was made so that the wires passed through a zine box, 18 cm. long, 13 em. broad, and 14 cm, high; Ea haa fully insulated from the box; the wires outside the box were straight, but the part inside was sometimes straight and some- times zigzag. This box could be filled with the dielectric under observation, and the velocity of propagation of the electro- alteration made in the null position (i.e. the position in which the deflection of the galyvanomoeter in the bolometer cireuil was a minimum) of wy by filling the box with the diolectric. Let p, and p, be the roadings of the pointer attached to uv when a straight wire of length D, and a zigzag of length D, are respectively insorted in the box, the box in this case empty. Then sinea in each case the lengths of the cireuits on the right and left of wv must be the same, the difference in the lll 396.) EXPERIMENTS ON RIECTROMAGNETIO WAvES. 475 lengths of tho circuits on the left, when the straight wire and the zigzag respectively are inserted, must be equal to the differ- ence in the lengths of the cireuits on the right. The longth of tho cireuit on the left when the zigzag is in exceeds that when the straight wire is in by (D,—pa)—(Po—Pr)s whilo tho difference in the length of tho circuits on tho right is Pi Pri aaa} D,—D,~ (PsP) = P= Prs or D,—D, = 2(p,—P,)- When the wires are surrounded by the dielectric, Arons and Rubens regard them as equivalent to wires in air, whose lengths are nD, and nD,, where n is the ratio of the velocity of trans- mission of electromagnetic action through air to that through the dielectric; for the time taken by a pulse to travel over a wire of length nD, in air, is the same as that required for the pulse to travel over the length D, in the dielectric. We shall return to this point after describing the results of these experiments, If p, and p, are the readings for the null positions of uv when the box is filled with tho dielectric, then we have, on Arons and Rubens’ hypothesis, %(D,—D,) = 2(r—Ps)5 or, climinating D,—D,, = Bebe, Pt honco, if p,, P., Py, P, ate determined, the value of n follows immediately. Tn this way Arons and Rubens found as the values of for the following substances :— cs VE. Caster Oil... 205 216 Olive OU... Leta 175 Xyll 2... 160 153 Petroleum. . . 1-40 144 ‘The values of X, the specific inductive capacity in a slowly varying ficld, wore determined by Arons and Rubens for the same samples as they used in their bolometer experiments. 896.] The method used by Arona and Rubens to reduce their observations leads to values of the specific inductive capacity 397-] EXPERIMENTS ON ELECTROMAGNETIC Waves. 477 then the condition that there should be no sparks is that the at Zand F should be the same. We can deduce the _ expressions for the potentials at Z and F from that at K when B _ and Farenodesorloops. Let us consider the ease when the capacity of the knobs Zand Fis so small that the current at 2 and F vanishes. Then we can easily show by the method of Art. 298 that if there is no discontinuity in the current along the wire, and if the self-induction per unit length of the wire is the same at all points in KAD, and if the portions AX, DF ore in air while AD is immersed in a dielectric in which the velocity of propagation of olectromagnotic action is V’, that through air being V, then if the potential at K is ¢,cos pt, that at F is pauls 008 pt a where A = 005 (,AD) cos {> (KA +DF)—sin($, AD) x fusin (DF) cos( # KA) + isin (7 HA) con(f DF)|. and » = V/V". The potential at Eis 4, 60s nt cos KE if KE represents the total length KB+BO+C2, the whole of which is supposed to be surrounded by sir. Hence, if the potentials at # and ¥ are the same, we havo cos (F; AD) cos F (KA +DF)—sin( 2, AD) x fasin (fp DF) 00 (KA) + sin(F HA) con (F Dah =cosPKE. (1) To make the interpretation of this equation as simple as possible, suppose KA = DF, equation (1) then becomes 008 (Fj AD) 00s (PA) — (n+ * 3) bein (AD) sin (5 KA) = oo ( > KE)- (2) 398.) EXPERIMENTS ON ELECTROMAGNETIC Waves. 479 Let us take the caso when HA=DF, then this equation = 1 (BA) cot (fr AD) = (1 + aR EA) = 4 (u+2) tan( PKA), or cot (PKA) = 3(n+ 2) tan FAD. (4) Let us consider the special caso when p.AD/V" is small, tho solution of (4) is then 2 1 PRA =5-4(ut 2) ie Pp wad wa or Ffexa+*S* splat reel tial Hicks & wie vad be a node at Z, this wire being entirely surrounded by air, then F(KE)=35 honee if p= p, 2KA+"** ap = KE, dKE_ +1 eo that p= 3 Arons and Rubens when reducing their observations took the ratio 3K E/AD to be always equal to x. The above investigation shows that this is not the case when pAD/V" is small. We might show that 3KE/3AD is equal to » when KA/AD is small, ‘The results given on the third view of the cloctrical vi- eee Seca wien seen Deseo eee hold for vibrating strings and bars. Thus if we have three strings of different materials stretched in series between two points, the time of longitudinal vibration of this system is not proportional to the sum of the times a pulse would take to travel over the strings separately (see Routh’s Advanced Rigid Dy- namics, p. 397), but is given by an equation eomewhat re- sembling (3). 401.) EXPERIMENTS ON ELECTROMAGNETIC Waves. 481 is connected to the gas pipes. When the coil is working, electrical oscillations take place in the condonser, the period of which is of the order 1/25,000,000 of a second. There ia thus on the side of AA’ a periodic electric ficld which has aw aa the plane of symmetry. Two squaro plates, CD, C’D’, are placed in ‘this field parallel to AA’ and sym- e i Fa soldered at DD’ to tho middle points of the sides of these plates. ‘The wires are connected at KE’ to two carbon points kept facing each other at a very small distance apart. When the coil is working no sparks are observed between ZH and 4’, this is due to the sym- metry of the apparatus. When, however, a glass plate is placed between AA’ and CD sparks im- modiately pass between Z and £’; ‘these are caused by the induction received by CD differing from that received by O’D’. By inter~ posing between AA’ and (” ‘ a sheet of sulphur of suitable thick~ ness the sparks can be made to disappear again. We can thus find the relative thicknesses of plates of glass and sulphur which produce the came effect on the electromagnetic waves passing through them, and wo can therefore compare the specific inductive capacity of glass and sulphur under similar electriesl conditions. M. Blondlot found the specific inductive capacity of tho sulphur he employed by Curie’s method (Annales de Chimie et de Phy- aique, [6], 17, p. 385, 1889), and assuming that its inductive capacity was the same in rapidly alternating fields as in steady ones, he found the specific inductive capacity of the glass to be 284, which is considerably less than its value in steady fields. 401.] I had provioualy (Proc. Roy. Soo, 45, p. 292) arrived at the same conclusion by measuring the lengths of the electrical ti S| 403.) BXPERIMENTS ON ELECTROMAGNETIC WAVES, 483 _ Thus Kerr (Phil. Mag. [5], 3, p. 321, 1877), whose have been verified and extended by Righi (Annales de Chimie et de Physique, (6), 4, p. 433, 1885; 9, p, 65, 1886; 10, p. 200, 1887), Kundt (Wied. Ann. 23, p, 228, 1884), Du Bois (Wied. Ann. 39, p. 25, 1890}, and Sissingh (Wied. Ann. 42, p.115,1893) found that when plane polarized light is incident on the pole of ‘an electromagnet, 80 as to act like a mirror, the plane of polarization of the reflected light is not the same when the ‘magnet is ‘on’ as when it is ‘off’ ‘The simplest case is when the incident plano polarized light | falls normally on the pole of an electromagnet. In this case, when the magnet is not excited, the reflected ray is plane polarized, and can be completely stopped by on analyser placed in 4 suitable position. If the analyser is kept in this position and the electromagnet excited, the field, as scen through the analyser, is no longer quite dark, but becomes so, or very nearly so, when the analyser is turned through a small angle, showing that the plane of polarization has been twisted through a small angle by reflection from the magnetized iron. Righi (1.¢.) has shown that tho reflected light is not quite plane polarized, but ‘that it is elliptically polarized, the axes of the ellipse being of very unequal magnitude. These axes are not respectively in and at right angles to the plane of incidence. If we regard for ‘ moment the reflected elliptically polarised light as approxi- mately plane polarized, the plane of polarization being that through the major axis of the ellipse, the direction of rotation of the plane of polarization depends upon whether the pole from which the light is reflected is a north or south pole. Kerr found that the direction of rotation was opposite to that of the currents oxeciting tho pole from which the light was rofleoted. ‘The rotation produced is small Kerr, who used a small electromagnet, had to concentrate the lines of magnetic force in the neighbourhood of the mirror by placing near to this a large mass of soft iron, before he could get any appreciable effects. By the use of more powerful magnets Gordon and Righi have succeeded in getting a difference of about half a degree between the positions of the analyser for maximum darkness with the magnetizing current flowing first in one direction and then in the opposite. A piece of gold-leaf placed over the pole entirely stops the riz a 406.) EXPERIMENTS ON ELZCTROMAGNETIO WAVES. 485 was polarized at right angles to the plane of incidence, obtained a reversal of the sign of the rotation of the plane of polarization: Rejlaction from Tangentially Magnetized Tron. 405.) In the preceding experiments the lines of magnetic foree ‘wore at right angles to tho roflocting surface ; somewhat similar effects are however produced when the mirror is magnetized tangentially. In this case Kerr (Phil. Mag. [5], 5, p. 161, 1878) found :-— 1. That when the plane of incidence is perpendicular to the Tines of magnetic force no change is produced by the magnetiza- tion on the reflected light. | 2. No change is produced at normal incidence. 3- When the incidence is oblique, the lines of magnetic force being in the plane of incidence, the reflected light is elliptically polarized after reflection from tho magnotized surface, and the axes of the ellipse are not in and at right angles to the plane of incidence. When the light is polarized in the plane of incidence, the rotation of the plane of polarization (that is the rotation from the original plane to tho plano through the major exis of the ellipse) is for all angles of incidence in the opposite direction to that of currents which would produce a magnetic field of the same sign as the magnet. When the light is polarized at right angles to tho plane of incidence, the rotation is in the same direction as these currents when the angle of incidence is between 0” and 75° according to Kerr, between 0° and 80° according to Kundt,and between 0° and 78°54’ according to Righi. When the incidence is more oblique than this, the rotation of the plane of polarization is in the opposite direction to the electric ourrents which would produce a magnetie field of the same sign. 406.] Kerr's experiments were confined to the case of light reflected from metallic surfaces. Kundt (Phil. Mag. [5], 18, p. $08, 1884) has made a most interesting series of observations of the effect of thin plates of the magnetic metals iron, nickel and cobalt, on the plane of polarization of light passing through these plates in a strong magnotic field where the lines of force are at right angles to the surface of the plates. Kundt found that in theee circumstances the magnetic metals possess to an oxtraordinary degree the power of rotating the 4 408] SXPERIMENTS ON ELRCTROMAGNETIC Waves. 487 and a steady current sont through the film from two electrodes. magnet galvanometer (G) was pro- Fig. 135.7 duced, and this continued as long aa the electromagnet was ‘on, showing that the distribution of current in the film was altered by the magnetic field. The method used by Hall to measure this effect is described in tho following extract taken from one of his papers on this subject (Phil. Mag, [5], 19, p. 419, 1885), ‘In most cases, when possible, the metal was used in the form of a thin strip about 1-1 centim. wide and about 3 centim. long between the two pieces of brass B,B (Fig, 135), which, soldered to the ends of tho strip, served as electrodes for the entrance ‘and eseape of the main cur- rent. To the arms a, a, about 2 millim, wide and perhaps 7 millim. long, were soldered the wires w, w, which led to a A Thomson galvanometer. The notches ¢, ¢ show how adjust- P ment was secured. The strip thus prepared was fastened to a plate of glass by means of a coment of beeswax and rosin, all the parts shown in the Pig. 186. figure being imbedded in and covered by this cement, which was so hard and stiff as to be quite brittle at the ordinary temperature of the air. * The plate of glass bearing the strip of metal so embedded was, when about to be tested, placed with B, B vertical in the narrow _ force measured was the normal mngnotic force outside the iron, ‘Bince the plate was very thin the normal magnetic foree outside the iron would bo Jarge compared with that inside; the normal _ magnetic induction inside would however be equal to the normal magnetic foree outside, so that Hall in this case measured the Felation between the electromotive intensity produced and the magnetic induction produeing it, ‘Hall has thus established for steady currents the existence of an effect of the same nature as that which Kerr's experiments proved (nssuming the electromagnetic theory of light) to exixt for the rapidly alternating currents which constitute light. Here however the resemblance ends; the values of the coefficient C deduced by Hall from his experiments on steady currents do nob apply to rapidly alternating light currents, Thus Hall found that for stondy currents the sign of C wns positive for iron, negative for niekel; the magneto-optical properties of these bodies are however quite similar. Again, both Hall and Righi found that the C for bismuth was enormously larger than that for iron or nickel. Righi, however, was unable to find any traces of magneto-optical effects in bismuth. ‘The optical experiments previously described show that there is an electromotive intensity at right angles both to the magnetic foree and to the electromotive intensity; they do not however show without further investigation on what function of the electromotive intensity the magnitude of the transverse intensity depends. Thus, for example, the complete current in the metal is tho sum of the polarization and conduction currents, Thus, if the electromotive intensity is X, the total current u is given by the oquation eats a) 4ixdt “o"" or if the effects are periodic and proportional to «'?*, KS 1 u=(Ge+ 3% where K" is the specific inductive capacity of the metal and « its specific reaistance. We do not know from the experiments, without further dis- i may without appreciable error "field is uniform, so that a,, },, cy are independent of a, y, + By equation (2) of Art. 256 FG Mnge thee +e) (1) since J* + S¢ 4 $2 = 0 on Maxwell's hypothesis that all the currents are closed. Now since u is the component of the total eurrent parallel to x, itis cqual to the sum of the components of the polarization and conduction currents in that direction. ‘The polarization current is equal to K'dP 4x di’ the conduction current to P/, hence ey ae dt + IPERS cition onc’ tention to rtodlo ccrrinte std aeppeg that tho variables are proportional to «'?'; in this case the pre- ceding equation becomes 4ru=(Kip+4z/o) P, =F. dy dp but ssu= 7 ae? hence we have ap. (Kip+4x/o)P = 2% - ; similarly (K'up+4z/0)Q = 4 — x, (Hip+4a/o) n= 48 — ae; and therefore since . +a e= (Kep+ax/e)(48 )= ae a+ it +an a= Agel me tl 4 A tems tp, Ba Byetlatmere 4 pele—aere, Lg ite mes ph, mm A aa ba etomernt where A and B are constants, ‘We shall suppose that the metal is so thick that there is no reflection except from the face = 0; in this case the waves in the metal will travel in the negative direction of =. ‘Thus in the motal we may put a = A'etmetp), fa Beletms-20, yah Aeerateryo, where if m’ is complex the real part must be positive in order that tho equations should represent a wave travelling in the negative direction of =; the imaginary part of m’ must be nogative, otherwise the amplitude of the wave of magnetic foree -would increase indefinitely as the wave travelled along. Substituting these values of a, 8, y in equations (2), we get Al (—pP WK’ + Anp'p/o+P +m?) =—E(wiptss/a)m' (ley+me)B, (8) B (—piu K' + Aapip/o+P +m) & P+m’? te) AP = 7, (Et sa/e)— 7 (loot me) a (4) Eliminating A’ and B’ from these equations, wo get ap KK’ +4 auip/o +P +m™ = Laipstaje\@rmr(lasme). (8) ‘There are only two values of m’ which satisfy this equation and which have their real parts positive and their imaginary parts nogative. We shall denote these two roots by m, ,m™,5 ee the root when the plus sign is taken in the ambiguity in in equation (4), m, the ae rien ie ainnetga ance le =i a 10.) EXPERIMENTS ON ELECTROMAGNETIC WAVES, 495. ‘electromotive intensity discontinuous, for the total electromotive intensity is made up of two parts, one due to elec induction, the other due to the causes which produce the Hall "effect ; it is only the first of these parts which we assume to be continuous, If P ip the component parallel to of the total electromotive intensity, P’ the part of it due to electromagnetic induction, then P=P’+k(b, nae 1 dy ‘a Rca we) | =-Kptiyet since in the present case y does not depend upon y. Hence, substituting the values of w and v in terma of the magnetic force, the condition that P” is continuous is equivalent to that of SEiprinjewe ~ae(bas (ea) iptinje ds 4 dz ~ dx being continuous. We shall suppose that in the air k = 0, The condition that a is continuous gives Apt A=4, +4; (6) the condition that 8 is continuous gives Byt B=— To A + Aas (7) the condition that the normal magnetic induction is continuous gives Edy A) = (oa + hy = (do ak Ait ae J ‘or dividing by J, 1 “th 1 — jy (40-4) =H & A\+ ao): (8) ‘We can casily provo independently that this equation is true when /=0, though in that ease it cannot be legitimately deduced from the preceding equation. (i pt Aa/e/ Ep = Rete, see Art 353, the last. wh hl a the Tas A Fag timed _ The rotation observed is small, we shell therefore neglect the ‘Squares and highor powors of (m,—m,); doing this we find from the procoding equations that i . 4. te Ge) w . «(ee — a) 0485) where M is tho value of re, or m,, when & = 0, and w* = [*+ 2. ee ere ast PUR +4ewip/ot lms = Spe 4/0) (?+m,%) lay, —PuUK +4alipfe+ tm! = — Lips taje)(t+mg)lay ‘Hence, when m,—m, is small, we have approximately (m—m,) M =“ (Rip+42/2) lay - ZEB pV, olay, (10) ae <8 ImV,—Fay (in) ate car ery ir Fa etess where 0 and ¢ are real quantities, then if the reflected light perpendicularly to the plane of incidence is B= cos (pt+lo—ms), the reflected light polarized in the plane of incidenes will be represented by a = 000s (pt-+12—miz)—¢ sin (pt +/a—me); Kk _ As Kerr's and Kundt's experiments were made with magnetic it sooms desirable to consider the results of supposing métals to retain their magnetic propertics. When »’ is not equal to unity, 6 is proportional to cos? sin isin @(w’sints + 24° cos a c08:); this does not change sign for any value of i between 0 and +/2, so that the preceding hypothcsis cannot be made to agree with the facts by supposing the metals to retain their magnetic ies. 412.) Let us now consider the consequence of supposing that the transverse electromotive intensity is proportional not to the total current but to the polarization current; we can do this by putting pa HPs yp | Kipjiattjo"’* | where isa real quantity. This equation may be written ie: mk é Substituting this value of & in equation (11) we find AL iKK’pa, _sinicos*i | Bo 4mRe'® sinti—Re'* cost If we write this in the form 4 =H +.¢, where 0’ and ¢’ are real, we find v= KK'pay sin’ costé (sin a sin? — —Rsin 2acosi) (12) 4nR sinti—2R sin’ i cos i cosa + KR? costi The angle through which the analyser has to be twisted in order to produce the greatest darkness is, as we have seen, equal to @ the real part of A/B. Equation (12) shows that this Kka _ 414.) EXPERIMENTS oN EnxcTROMAGNETIO Waves. 501 With the notation of Art. 355 this may be written nein? = n*(1—K#) 008 From tho table in Art, 355 we seo that 1—/ is negative, hence, since 7 ie positive there is no real value of 7 less than +/2 which patisfies this equation, so that if this hypothesis were correet there would be no reversal of the direction of rotation of the Hence of the three hypotheses, (1) that the transverse olectro- motive intensity concerned in these magnetic optical effects ix proportional to the total current, (2) that it is proportional to the polarization current, (3) that it is proportional to the conduction current, we see that (1) and (3) are inconsistent with Kerr's experiments on the reflection from tangentially qmagnetized mirrors, while (2) is completely in accordance with them. 413.] The transverse electromotive intensity indicated by hypothesis (2) is of a totally different charactor from that discovered by Hall. In Hall's experiments the electromotive intensities, and therefore the currents through tho motallie plates, were constant; when however this is the case the ‘ polarization * current vanishes. Thus in Hall's experiments there could have ‘been no electromotive intensity of the kind assumed in hypo- thesis (2); there is therefore no reason to expect that the order of the metals with respect to Kerr's effect should be the same as that with respect to Hall's. It is worth noting that reflection from a transparent body placed in a magnetic field can be deduced from the proceding equations by putting a=0, since this makes the refractive index real. In this case we see, by equation (12), that the real part of A/B vanishes, so that the reflected light is elliptically polarized, with the major axis of the ellipse in the plane of incidence; any small rotation of the analyser would therefore in this case increase the brightness of the field. 414.) We now proceed to consider the case of reflection from a normally magnetized mirror. We shall confine ourselves to the case of normal incidence. If the incident light is plane polarized we may (using the notation of Art. 409) put B, = 0; we have also = 0, w, =m, @,=m,, and since the mirror is magnetized normally, a,= 0, = 414.) EXPERIMENTS ON ELMOTROMAGNETIC Waves, 503 but Mf = Rem, so that B_ wpK¥Reo A 45 (BPe*—1)' or, since the modulus of R*e*** is large compared with unity, B_pK'K a pak = eee W (sin ati0onc) approximately. Hence, if the magnotic force in the reflected wave, which is polarized in the same plane as the incident wave, is represented by 008 (pt + me), the magnetic force in the reflected wave polarized in the plane ee ea BIE «, in aos (pt+-ms) — 2 pM &, cos. sin (pt +m). ‘Thus in the expression for the light polarized in this plane one term represents a component in the same phase as the constiluent in the original plane, while the phase of the component repre- sented by the other term differs from this by quarter of a wave length. The resultant reflected light will thus be slightly elliptically polarized. As in Art, (411) however, we may show that the field can be darkoned by twisting the analyser through small angle from the position in which it completely quenched the light when the mirror was not magnetized. The angle for which the darkening is as great as possible is equal to the real term in the expression for B/A, i.e. to pK. Tar sine Thus though the reflected light cannot be completely quenched by rotating the analyeer, its intensity can be very considerably reduced ; this agrees with the results of Righi’s experiments, see Art. 403, We can deduce from this case that of reflection from » trans- parent substance by putting a=0, a8 this assumption makes the refractive index wholly real; in this ease the reflected light is ellip- tically polarized, but as the axes of the ellipse are respectively rial at sight angtee to the plane of the original polarization i of the field, We can solve by similar means the case of oblique reflection from a normally magnetized mirror; the results agree with Kerr's experiments; want of space compels us however to puss oa | apply the same principles to the case where light, as in Kundt’s experiments Art. 406, passes through thin metallic films placed — in a magnetic field. On the Effect produced by a thin Magnetized Plate on Light passing through it, 415.] We shall assume that the plate is bounded by the planes =0, s=~—h, the incident light falling normally on the plane 2=0, The external magnetic field is supposed to be parallel to tho axis of z, Lot the incident light be plane polarized, the magnetic force in it being parallel to the axis of z. The reflected light will consist of two portions, one polarized in the same plane as the incident light, the other polarized in the plane at right angles to this: the magnetic force in the latter part of the light will therefore be parallel to the axis of y. Ifo, 6 are the components of the magnetic foree parallel to the axes of w and y respectively, then in the region for which zis positive we have a= Agee) 4 Ae—meeed, p=Be'—™tP0, where A,e'("*+? represents the magnetic force in the icles waye and A and B are constants, In the plate we have = Ay ett +79 4 A Jemma +P) 4 Ay etme +74 Asai meep, and therefore, as in Art. 409, as 2 = 0, Bid elt FPO Avett +20 4 Agel (OOEt BO 4 A fete tm where m,,m, are the roots of equation (5) and A,, Ay’, A,, Ay ‘fare constants. After the light has passed through the plate, the components of the magnetic force will be given by equations of the form a=Cet(merPh, | p= Delme+P9, = 416.] EXPERIMENTS ON ELECTROMAGNETIC Waves. 509 Now if is the electric polarization parallel to 2, the transverse electromotive intensity is equal to Wey = Kipef K = hwo, where X is the electromotive intensity parallel to 2. Hence KR’ pe, /4 is the ratio of the magnitude of the transverse intensity to that producing the eurrent ; this ratio is for iron therefore equal to 1,6x10-"%p for magnetic fields of the strength used by Kundt, The factor multiplying p is so small as to make it probable that the effects of this transverse force are insensible except when the clectro- motive intensity is changing with a rapidity comparable with tho rate of change in light waves, in other words, that it is only in optical phenomena that this transverse electromotive intensity produces any measurable offect. DISTRIBUTION OP RAPIDLY ALTERNATING CURRENTS. 611 | When, however, the currents are variable these equations are no longer true ; we have instead of them the equations da? dF _av_, a da, * dé, ~ de, where 7'is the Kinetic Energy due to the Self and Mutual in- duction of the circuits, F as before is the Dissipation Function, and F is the Potential Energy arising from the charges that may be in any condensers in the system. If the currents are periodic and proportional to e‘?, the pre- ceding equation may be written as a? dP av "Pda, * da, dz, = °3 and thus when p increases indefinitely the preceding equation Spproximates “ a 4, da,” we have eimilarly afar de, = da, =~ Thus in this case the distribution of currents is independent of the resistances, and is determined by the condition that the Kinctic Energy and not the Dissipation Function is a minimum. 419.) We have already considered several instances of this effect. Thus, when a rapidly alternating current travels along a wire, the currents fly to the outside of the wire, since by doing this the mean distance between the parts of the current is a maximum and the Kinetic Energy therefore a minimum, i when two currents in opposite directions flow through two parallel plates the currents congregate on the adjacent surfaces ‘of the plates, since by so doing the average distance between the opposite currents, and therefore the Kinetic Energy, is a minimum. Mr. G. F.C. Searle has devised an experiment which shows this tendency of the currents in a very striking way. AB, Fig. 123, is an exhausted tube through which the periodic currents produced by the discharge of a Leyden jar are sent. When none of the wires leading from the jar to the tube passes parallel to it ya | DISTRIBUTION OP RAPIDLY ALTERNATING CURRENTS. 513 meviahle variation in the intensity of the eurrent frum one of AOB or ADB to the other, in other words, that the wave corresponding to the rate of alternation of the current is large compared with the length ACB or ADB; the case when this wave length is comparable with the length of the _ Gireuit is considered separately in Art. 298. Let the current flowing in along 0A and out along BP be denoted by ¢; wo shall assumo that # varies as ¢‘#, Let tho current in ACB be |g, that in ADB will bo #j. Then 7, the Kinetic Energy in | ‘tho branch ACDB of tho cireuit, is expressed by the oquation T=} (Ly +2M @—9)§ +N (2-9). | ‘The dissipation function F is given by P= U{RP+S(@—9¥, | and we have adr aF_, dt dy dy or (4 N—24 8 (2489-1) —s6 = 0. Let # = «'™, then from this equation we have _ (Ws, I= UeN—3M) pt Erw | or, taking the real part of this, corresponding to the current cos pt along OA, we find be {StS +L +N—-2M) (N-M) p*} coe pt—p {R(N—-M)—S (L—M)} sin gt w (L+N—2M) pte (+ 8 . be {R84 8) + (L+N-2M) (LM) p'} con pt +p [R(N—M)-S(L—M)} sinpt @ (L+ X= p+ he sy ? ‘These expressions may be written in the forms 4 (=a : $= lee ware TRF met = A cmp), oy eae fi tli deat coe t+) = Beon(pt+¢), R(V—M)-S(L—M)} where) fane= 577 ares ay ates or : p{R(W—M)—8 (LM) and ne RR 8) + (Le aM Vai) L =| | 420.] DISTRINUTION OF RAPIDLY ALTERNATING CURRENTS. 515 as unity; thus by introducing an alternating current of small intensity into a divided circuit, we can produce in the arms of this cirenit currents of very much greater intensity. The reason of this becomes clear when we consider the energy in the loop, when the rate of alternation is exceedingly mpid. The effects of the inertia of the system become all important, and the distri- bution of currents is that which would result if we considered merely the Kinetic Energy of the system. In this case, in accordanco with dynamical principles, the actual solution is that which makes the Kinetic Energy as small as possible consistent with the condition that the algebraical sum of the currents in ACB, ADB shall be equal to #. ‘Thus, as the Kinetic Energy is to be as small as possible, and this energy is in the field around the loop and proportional at each place to the square of the magnetic force, the currents will distribute themselves in the wires so as to neutralize as much as possible each other's magnetic effect. Thus if the wires are wound close together the currents will flow in opposite directions, the branch having the smallest number of turns having the largest current, so as to be on equal terms as far as magnetic force is concerned with the branch with the larger number of turns, In fact we see from equations (5) and (6) that the current in each branch is inversely proportional to the number of turns. If the toyo branches are exactly equal in all respects the current in each will be in the same direction, but this distribution will be ‘unstable, the slightest difference of the coefficients of induction in the two branches being sufficiont to make the current in the branch of least inductanee flow in the direction of that in the leads, and the current in the other branch in the opposite direc- tion, the intensity in either branch at the same time increasing largely. When the currents are distributed in accordance with equa- tions (3) and (4), the Kinetic Energy in the loop is IN-—M* Vaal" cos Pt. We notice that (LN —M*) /(L4N—2M) is always less than Lor N, £+N—2M is always positive, since it is proportional to the Kinotic Energy in the loop when the currents are equal and opposite. ula | 422:] DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS, 517 ' ‘The impedance of the loop is RS(R+8) +p" (R(N-MP+S(L— (R+SP +p (L+N—2 _ | which is equal to RS + Pp i{R(N—M)—S(L-M)y | RS OS SPUN | _ We see from the expression for the self-induction of the oop that it is greatest when p = 0, when its value is NRE42MRS+LS* (R+5) . | and least when p is infinite when it is equal to IN-M* L+N-20 If R(N-M)=S(L—M), the self-induction of the loop is independent of the period. From the expreasion for the impedance of the loop we see that it is least when p = 0 when ita value is RS RyS* and greatest when p is infinite when it is equal to R(N-Mf+S(L—MP | D+ ) - and if R(N-—M) =8(L—M), the impedance is independent of the period. Thus in this case the self-induction and the impedance are unaltered, whatever the Frequency of the currents. In oll other cases the self-induction diminishes and tho impedance increases as the frequency of the eurronts increases. 422.] We shall now proceed to investigate the general case when there are any number of wires in parallel. Let @, be the eurrent in the leads, #,, #,,...#, the currents in the » wires in parallel ; we shall assume, as before, that there is no induction between these wires and the leads. Let-a,, be the self-induction and 7, the resistance of the wire through which the eurrent is @,, a,, the coefficient of mutual induction between this wire and the wire through which the current is 2. Let a, be the self- induction, 7) the resistance of the leads, /, the electromotive foree in the external circuit; we shall suppose that this varies as From (10); the expressions for them are however in general = we 2 (As pi dp Peat ‘atten Aga) tone Bins Fomy Gun | and A’,, is the minor of D corresponding to the constituent a,,, while BS Ay t At AF 2A + 2A $ 2A He ‘Thus the self-induction of the wires in parallel is in this case - while the impedance is dr (Ant dat AP +1 (A tA nt 1 MP te} JS 1 oe = “Fed atk When there is no induction between the wires in parallel, 4 .] DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS. 521 the values of @,, 4,,... in the first equation, we tpt) 4+ z {(4ua4P +712) Brat (ua? +79) By + --} 4 =X, which may be written ma eee (12) If 4/B,, be written in the form Lip +R, whore Z and R are real quantities, then Z is the effective self-induction of the eireuit land R the impedance. By equation (11) we have A. Rea ke If an electromotive force X, of the same period as X, acted on the second circuit, then the current #, induced in the first cirenit would be given by cee eee, Comparing these results we get Lord Rayleigh’s theorem, that when a periodic electromotive force F' acts on a circuit A the current induced in another cireuit B is the same in amplitude and phase as the current induced in A when an electromotive fores equal in amplitude and phase to F’ sets on the circuit B. When there are only two circuits in the field, i. (cng ip + tieh By PE asin” ao. ee = git far ee tnt io - = (au- Hy) P+ tt ot ery ‘Thus a presence ot a second circuit Ee the self induction of the first by while it increases the impedance by Pre thy Parte _ ‘The coefficient of ip in the first line is the coefficient of self- “induction of the first cireuit,—wo seo that it is diminished by of 5/B,, to the form Jip+R without any limitation as to the value of p would usually lead to very complicated expressions ; ‘wo can, however, obtain without difficulty the values of Z and 2, (1) when p is very large, (2) when it is very small. When ip is very lange we sce that Figr Cams Onn | 9 and A,, is the minor of D corresponding to the constituent a). If A,, denotes the minor of D corresponding to the constituent @,,, then we have by (11) ee be 7 Saag a (13) Substituting these values of 2,, ¢,, &e, in terms of #,, in the Function, we find that 3 Feats rey! Fonte Arlt Pty Ay Ay t2ty Ap Arg t veel i we might of course have deduced this value directly from that of 5/Byy. When ¢p is very small, wo seo by putting p= 0 in 4/B,, that | Vyas Tan vee Tee and Ry is the minor of C corresponding to the constituent 7,5 if Zt,, denotes the minor of C corresponding to the constituent ees (2) ayia n= R Rn RR, ei | 427.] DISTRIBUTION OF RAPIDLY ALTERNATING CURRENTS, 525 in metallic connection. We shall suppose that a is the radius of the firet solenoid, > that of the second, c that of the third, and #0 on, a, b,c being in ascending order of magnitude; and that ‘7, Mz, My... are the numbers of turns of wire per unit length ‘of tho first, socond, and third circuits, hen if J is the length of the solenoids, we have Oy Hs nla, ay=4z°n2lb*, a= 42*n,*le’, | @,,=47'n,n, la", a, = 47? nynll*, ay = aN | Gy =Ax" nr, n,la*, ay, = 4n*ngn le, . . . . and = A,=7>— = (4a2U)"- ng? rasta. B® (ce? —b*) (2 c2) a Now the coofficient of sclf-induction of the first circuit for very rapidly alternating current ia D Ay sed pong gl dea find that the self-induction equals 4alnza* (1-%3 —iz) ‘Thus the only one of the circuits ae affecta the self- induction of the first is the ono immediately adjacent to it. We can at once see the reason for this if we notice that Gy yy and therefore 4y=4y=4y Now when the rate of alternation is very rapid, 05, #,, #5...» the currents in the third, fourth, and fifth cireuits, &e. are by equation (13) Art. (425) proportional to A,s, Ay, Ayy...3 hence wo seo that in this case these currents all yanish, in other words J] pisrampvrion oF RarioLY ALTERNATING connENTS. 627 specific resistance ¢, then considering the tube as a solenoid with wire of square section a packed close together, © see that for the tube #= nbin,= = Intlng<- Now «/n,* for the tube is equal to 4/n,* for the secondary | of A, which may be an ordinary solenoid. We thus have aa =2rbla/r, Pp relation by: whlch: wo ean deduce 'e, Tn order that this method should be sensitive the interposition of the secondary ought to produce a considerable effect on the currents induced in the tertiary. If the resistance of the secondary is large this will not happen unless the frequency of the electromotive force is very great; for ordinary metals a frequency of about a thousand is sufficient, but this would be useless if the specific resistance of the tube were comparable with that of electrolytes. On the other hand, if the frequency is infinite, there will not be any current in the tertiaries whatever the resistance of the secondaries may be. Wheatstone's Bridge with Self-Induction in the Arms. 429.] The preceding investigation ean be applied to find the effect of self-induction in the arms of a Wheatstone’s Bridge. Let ABCO represent the bridge, let an electro- motive force X proportional to ¢‘P act in A the arm CB. Let « be the current in CA, y that in BA, 2 that in AO, then the currents along BO, AC, OC are respectively «—y, y—s, and w—y +s. Let the self-induction in CB, BA, AC, AO, BO, CO bo rmspectively A, C, B, L, M, N, Fig. 140. while the resistance in these arms are re- spectively a,c, 4, a, 8, y. We suppose, moreover, that there is no mutual induetion between the various arms of the Bridge. Then the Kinetic Energy 7 of the system of currents is ex- pressed by the equation 2T = Ax*+Cy'+ B (y—z)' + Le?+M (xy) + V(e—y+sP. | DISTRIBUTION OF RAPIDLY ALTERNATING CoBRENTS, 520 Combination of Self-Induction and Capacity. 430.] We have supposed in the preceding investigations that ‘the circuits wore closed and devoid of capacity; very interesting results, however, occur when some or all of the circuits are cut and their freo ends connected to condensers of suitable capacity. ‘We can by properly adjusting the capacity inserted in a cireuit in relation to the frequency of the electromotive force and the self-induction of the circuit, wake the circuit behave under the action of an electromotive force of given frequency as if it pos- sessed no apparent zelf-induction. The explanation of this will, perhaps, be clear if we consider the behaviour of a simple mechanical system under the action of | periodic force. Tho system we shall take is that of the rectilinear motion of a mass attached to a spring and resisted by |g frictional fore proportional to ita velocity. Suppose that an external periodic force X acts on the aystem, then at any instant X must be in equilibrium with the resultant ‘of (1) minus the rte of change of momentum of the system, (2) the force due to the compression or oxtension of the spring, £3) the resistance. If the frequency of X is very great, then for a given momentum (1) will be very large, so that unless it is counterbalanced by (2) a finite foree of infinite frequency would produce an infinitely small momentum. Let us, however, sup- pose that the frequency of the force is the same as that of the free vibrations of the system when the friction is zero. When the mass vibrates with this frequency (1) and (2) will balance each other, thus all the external force has to do is to balance the resistance. The system will thus behave like one without either mass or stiffness resisted by a frictional force. Tn the corresponding electrical system, self-induction corre- sponds to mass, the reciprocal of the capacity to the stiffness of the spring, and the electric resistance to the frictional resistance. Tf now we choose the capacity so that the period of the electrical vibrations, calculated on the supposition that the resistance of the circuit vanishes, is the samo as that of the external electro- motive force, the system will behave as if it had neither self- induction nor capacity but only resistance, Henco, if Z is the self-induction of a circuit whose ends are connected to the plates of a condensor whose capacity in electromagnetic measure is C, 4 432.) DISTRIBUTION OF RAPIDIY ALTERNATING CURBENTS. 531 “this wil vanish i Lp §?+€ | R—(L— Mp) a +(R4+8){L(R+8)-2R(L-M)) =0. (18) | If the roots of this quadratic are real, then it is possible to choose © 80 that the self-induction of the loop vanishes. An i special case is when S=0, M=0, when the quadratic reduces to Lpté? + €(R*— Lap") LR = 0; 2 this last value of 1/C makes ¢=, so that none of the current goes through ACB. When ¢ satisfies (15) the self-induction of the loop vanishes, If im that equation we substitute L+A for L and M+b the medium is at rest, here we haye =-% Kim ee =_%¢ v7) = ae £=-73' and V"¢ = 0. The boundary conditions satisfied by ¢ and its differential coefficients will depend upon whether the sphere is a conductor oran insulator. We shall first consider the case when it is an insulated conductor. In this case, when the system is in a steady state, the radial currenta in the sphere must vanish, other- wise the electrical condition of the surface of the sphere could not be constant. ‘Thus at any point on the surface of the sphere aX+y¥+2Z=0, this is equivalent to 1) where 4, is the value of ¢ inside the rotating sphere; hence we have ¢, =k, where X is « constant. Fiz SES UU 436,] ELECTROMOTIVE INTENSITY IN MOVENG BODIES, 539 ‘Thus in the outer fixed sphere the components of the electro- motive intensity are equal to the differential coefficients with respect to a, y, 2 of the function deus. ‘Thus the radial electromotive intensity close to the surface of the rotating sphere is ~ewaQs, while the tangential intensity is —Cwasing co 0. These results show that the effects produced by rotating uncharged spheres in a strong magnetic field ought to be quite Jarge enough to be measurable, Thus if the sphere is rotating so fast that « point on its equator moves with the velocity 3x 10°, which is about 100 feet per second, and if c=10*, then the maximum radial intensity is about 1/33 of a volt per centimetre, and the maximum tangential intensity about 1/2 of this: these are quite measurable quantities, and if it were necessary to in- crease the effect both ¢ and w might be made considerably greater than the values we have assumed. The surface density of tho eloctricity on the rotating sphers when (b—a)/a is small is - aad owas. 436,] If the outer fixed sphere is a conductor, the electromotive intensity must vanish when r > b, henco P=0, so that N=0, while M, L, K have the same yalues as before. In this case the surfuee density of the electricity on the surface of the rotating sphere is EK, reriire Q, and when b—a is small, this is equal to Ky - 275008 Since this expression is proportional to 1/8, the surface density con be increased to any extent by diminishing the distanes between the rotating and fixed surfaces. In the general ease, whon b—a is not necessarily small, the —— 438.) ELnorRomorive urrenstry iN Movine Aopres. 541 From thoes conditions we see from equations (1) that V3 (a+ Fut Ge+ Hw) = Gloom bn H (aw—eu) + F(ou—ae, a [Cf tm stn F) (bat Fut Gos tw) =[K (2 (edu) +m (aw—cu) + (bu-ar))f- From these equations ¢,, is uniquely determined, for we sce that ou+Fu+Gv+Hw is the potential due to a distribution of electricity whose volume density is = ge (ov —bw) + jqlae—ou) + ou—an)} together with a distribution whose surface density is - ii {(L(ce— bw) +m (aw—cu) +n (bu—av)}]. Having thus determined $, and deducing ¢ by the process exemplified in the preceding examples we can determine +,. 488.] The question as to whether or not the equations (1) are true for moving insulators as well as for moving conductors, ut, v, w being the components of the velocity of the insulator, is ‘a very important one. The truth of these equations for con- ductors has been firmly established by experiment, but we have, 50 far as I am aware, no experimental verification of them for insulators. The following considerations suggest, I think, that some further evidence is required before we can feel assured of the validity of the application of these equations to insulators. We may regard a steady magnetic field as one in which Faraday tubes are moving about according to definite laws, the positive tubes moving in one direction, the negative ones in the opposite, the tubes being arranged so that as many positive aa negative tubes pass through any ares. When a conductor is moved about in such a magnetic field it disturbs the motion of the tubes, so that at some parts of the field the positive tubes no longer balance the negative and an electromotive intensity ix produced in such regions. To assume the truth of equations (1), whatever the nature of the moving body may be, is, from this point of view, to assume that the effect on these tubes is the same whether the moving body be a conductor or an insulator of 440.] BLECTROMOTIVE INTENSITY IN MovING noDIES, 543 to the rotation we ought to be able to distinguish between the two effets, since the rotational one is reversed when the direction of rotation is reversed as well as when the magnetic field is reversed. | In deducing equations (2) of Art. 434, we assumed that equa- tions (1) held in the medium between tho fixed and moving RN ie ey tne cues tn nee . In the special case, however, when the layer of eee eancy ats the results will be the same whether this medium is an insulator or conductor, so that the results in this special case would not throw any light on whether equations (1) do or do not hold for a moving dielectric, Propagation of Light through a Moving Dielectric. 440,] We might expect that some light would be thrown on the cloctromotive intensity developed in a dielectric moving in a magnetic field by the consideration of the effect which the motion of the diclectrie would have on the velocity of light passing through it. We shall therefore investigate the laws of propaga~ tion of light through u dielectric moving uniformly with the velocity components u, v, w. In this case, since we have only to deal with insulators, all the carrents in the field are polarization currents due to altera- tions in the intensity of the polarization. When the dielectric is moving we are confronted with a question which we have not hind to consider previously, and that is whether the equivalent current is to be taken as equal to the time rate of variation of tho polarization at a point fixed in space or at a point fixed in the dielectric and moving with it; ive. if f is tho dielectric polarization parallel to x, is the current parallel to x a dt or Fruit ry ote tw'hr In the first case we should have, if a, 8, y are the components of the magnetic force, artf _dy_ ap, (3) dt ~ dy ds? in the second, d., d; dj d, dy dps ax(L aud y +0 49h) = 2. (4) SITY IN MOVING BODIES, es Rare at ae, hones from (6) we a iy mea tuted swtye (7) P Hoon ter band egeton ive, wo gt HM (fe tHE HG twp) ‘with similar equations for « and 6. _ Let us apply these equations to a wave of plane polarized "light travelling along the axis of x, the diclectrie moving with ‘velocity u in that direction. In this case equation (7) bocomes 1 d*o_dto |, dtc Kedar ~ a deat ® Lot ¢ = 00s (pt—ma); then if Vis the velocity of light through the dielectric when at rest, equation (9) gives Vim! = p'—upm, « Ber. Since w is small compared with V, we have approximately Faiutv. ‘Thus the velocity of light through the moving dielectric is increased by half tho yelocity of the dielectric. If we take equation (8), then (8) vie =(4 +udy 2 or putting as before, ¢ = cos(pt—mz), Vim! =(p—mu), hence Fe Veu; so that in this case the velocity of the light is increased by that of the dielectric. If we suppose that the condition (3) is the true one, viz, that de in rud nead 2-2 where t,, ¥), 1%) are the components of the velocity of the ether, Nn ce — (a +e at eae dZ aX a¥ d& 4 2-= ds dy” ene sen let «soe es u=—oYy, V=0x, w= 0; a iiieoes ox a aw (of -i)- a) | ee cits soictane of 2 ae 5 , B, a. x the components of Tee Gi = 28 in lea (12) Ba or= 2 (E— = If we substitute these values for X and Y, equation (11) becomes < vip=w (2% -v), ime dy Gvr=a(2F -y2)—wa, (13) vane (eh —vit) +b. From these equations we find by the aid of (12) vte= (#92 -vE) ea, di d m= 0 (23 Yaa) —wp: a: rae (0S -y F)- yaa Ga 441.) BLROTROMOTIVE INTENSITY IN MOVING BODIES, 549 Now the magnetic force may be regarded as made up of two parts, one due to the currents induced in the sphere, the other to the external magnotie field; the latter part will be derived from a potential. Let 2, be the value of this potential in the sphere; we may regard ©, as a solid spherical harmonic of degree m, since the most general expression for the potential is the sum of terms of this type. If a,, 8;, 7, are the components of the magnetic fores due to the currents, a), By, yy those due to the magnetic field, then eH ate =e — aoe "gags 91 = Et aye M+ dona ed Se = alte tfays (br) facta) (17) with cimilar expressions for 8, and y,. Ontside the sphere the magnetic force due to the currents will {neglecting the displacement currents in the dielectric) be de- rivable from a potential which sntisfies Laplace's equation ; hence Outside the sphere we may put, if w,’ represents a solid harmonic, d da oo : with similar expressions for 8, and y,, where a is the radius of the sphere. The magnetic force tangential to the sphere due to these currents is continuous, as is also the normal magnetic induction; hence, » being tho magnotic permeability of the sphere, we have Dat ap TyR "+ Mfr ba) ona (ke) 4} =-«,', a, =~ ae wn(nt jaw #(n2,) + anthe (Fas (a) 0, + Ka", (Fa) ,} = (R+1) wy Solving these equations, we find at the surface of the sphere (2n+1)(un¢n+1)ieQ, a are on rey n(2n+1) uke? f,, (ka) 2, (3) ~ (#1) Temtnt TFs (he) +m (x1) a*f,,, Ea)” i Thus the magnetic force due to currents consists of a radial force proportional to yr, together with a force parallel to y proportional to 27*—(4+ 4) a*/(u +2). Outside the sphere the total magnetic (02+ Be)(1- DS = som ee ‘Thus the magnetic effect of the currents at a point outside | the sphere is the same as that of a small magnet at the centre, with its axis at right angles to the axis of rotation and the external magnetic field. and whose moment is 67B ptwat S(u+2h 0 * 443.) Let us now consider the case when ka is largo, since, when s=1 pn — ee we havo ko aK *, where Re He, thus the real part of ia is positive and large; hence we have anpproximatel, 7 ete fille) = eta fits)= foe Ae =— spe Hence we find dao, =—3:k9 ae *Br singe’, + : on = tf Brainge't, — 444.) ELZCTROMOTIVE INTENSITY IN MovING BODIES. 553 ternal magnetic field which is not symmetrical about the axis of rotation, Thus the rotating sphere vorcens its intorior from all | bat symmetrical distributions of magnetic foreo if {4xaw/c}*a is large. A very interesting case of the rotating sphere ia that of the earth; in this case B= 637% 10', w@ = 2x/(24 x 60 x 60), so that approximately y {4zpox/o}da = 2x 10%}, ‘Thus if ¢ is comparable with 10*, which is of the order of the specific resistance of electrolytes, ka will be about 2000, and this will be large enough to keep the earth a few miles below its surface practically free from the effects of an external un- symmetrical magnetic field. Again, we have seen, Art. 84, that rarefied gases have con- siderable conductivity for discharges travelling along closed curves inside them. For gases in the normal state this con- ductivity only manifests itself under large electromotive in- tensities, but when the gas is in the state similar to that produced by the passage of a provious discharge, it has consider- able conductivity even for small electromotive intensities. We #00 from the preceding results that if there were a belt of gas in this condition in the upper regions of the earth’s atmosphere, and if the part of the solar system traversed by the earth were ® magnetic field, this gas would screen off from the earth all magnetic effects which ware not symmetrical about the axis of rotation. Thus the magnetic field at the earth’s surface would, on this hypothesis, resemble that which actually oxists in being roughly symmetrical about the earth's axis. The thickness of ‘a shell required to reduce the magnotic field to 1/« of its value at the outer surface of the shell is {4mw/o)-}, or if ¢ = 10%, about two miles. Tho result mentioned in Art, 470 of Maxwell's Electricity and Magnetiem, that by far the greater part of the mean value of the magnetic elements arises from some cause inside the earth, shows, however, that we cannot assign the earth's permanent magnetic field to this cause. 444.) The total magnetic potential outside the sphere is, when ka is large, by equation (24), Similarly the couple round & is equal to Pf forraeyas rent =! rr qhrdadyds. From ‘alae! (16) we see that Rr= = ey +Y) (aca hr) +R fF) Now by (4), Art. 370, Fan lr) + 1Pr? fa (lor) = — (2041) f(r), #0 that dup Rr=- Se t(m+1)f (hr) w., (25) or by (15) Rr =—S,n(n+l)r. ‘Thus the couple around = is = Sonn) ff fdedyas. When o is small we find, by substituting the value of r given in equation (20), that when the sphere is rotating in a uniform magnetic field the couple tending to stop it is apis et 446.] We see by equation (25) that the normal component of the magnetic force is proportional to f, (kr), while by (16) the other components contain terms proportional to /,-; (kr), but when ka is very large we have approximately ke Su-a(ka) = 1" ape oe Julbo) = 807s Thus when ka is very large f,(éa), and near the surface of the ek eap is very small compared with /,_, (ta), 80 that by (a6) the magnetic force along the normal to the sphore vanishes in comparizon with the tangential force, in other words the magnetic force is tangential to the surface. ‘This result can be shown to be true, whatever the shape of the body, provided it is rotating with very great velocity. If - | 448.] ELECTROMOTIVE INTENSITY IN MOVING BODIES, 557 R vanishes, and the force on the rotating sphere is that due to & pressure eset: this pressure will tend to make the sphere move from the strong | to the weak places of the field. We see, therefore, that not only does the rotating sphere disturb the magnetic field in the same way as 8 diamagnetic body, but that it tends to move as such a body would move, ie. from the strong to the weak parts of the field. 448.] If instead of a rotating sphere in a steady magnetic field wo have a fixed sphere in a variable field, varying as <#, tho preceding results will apply if instead of putting 1 =—4rpwss/e wo put MY = — 4ryrp/c,and neglect the polariza- tion currents in the dielectric. We can prove this at onco by seeing that the equations for a, b, ¢ in the two cases become identical if we make this change, ‘The results we have already obtained in this chapter, when applied to the case of alternating currents, show that in a variable field when ka is large the currents and magnetic foree will be confined to » thin layer near the surface, and that a con= ductor will act like a diamagnetic body both in the way it disturbs the field and the way it tends to move under the ‘influence of that field. The movernent of currenta from the strong to the weak parts of the ficld has been demonstrated in some very striking experiments made by Professor Elihu Thom- son, Electrical World, 1887, p. 25% (see also Professor J. A. Fleming on ‘Electromagnetic Repulsion,’ Electrician, 1891, pp. 567 and 601, and Mr.G. T. Walker, Phil. Trans. A. p. 279, 1892). ‘The correspondence of the magnetic foree to the velocity of an incompressible fluid, flowing round the eonduetors, is more com- plete in this case than in that of the rotating sphere, inasmuch as we have not to except any part of the magnetic potential, whereas in the case of the rotating sphere we have to except that part of the magnetic potential which is symmetrical about the axis of rotation. al —e APPENDIX. a ‘Ix Art. 201 of the text there is n description of Porrot’s experiments ‘on the electrolysis of steam. As these experiments throw n grent deal of light on the way in which electrical discharges pass through guses I have, while this work has been passing through the press, made a series of experiments on the same subject. ‘The apparatus I used was the same in principle as Perrot's, T made some changes, however, in order to avoid some inconveniences to which it seemed to mo Perrot's form was liable, One source of doubt im Porrot’s experiments arose from the proximity of the tubes surrounding the electrodes to the surface of the water, and their lability to get damp in consequence, ‘These tubes were narrow, and if they got damp the sparks instead of passing directly through the steam might con ceivably have passed from one platinum electrode to the film of moisture on the adjacent tube, then through the steam to the film of moisture on the other tube and thence to the other electrode. If anything of thi kind happened it might be urged that since the discharge passed through water in its passage from one terminal to the other, some of the gases collected in the tubes gg (Fig. 84) might have been due to the decomposition of the water and not to that of the steam. To overcome this objection 1 (1) removed the terminals to w very much greater distance from the surface of the water ond placed them in & region surrounded by a ring-burner by means of which the etesm was heated to a temperature of 140°C to 150°C. (2) I got rid of the narrow tubes surrounding the electrodes altogether by making the tubes through which the steam escaped partly of metal aod using the metallic port of these tubes as the electrodes, ‘Instead of following Perrot’s plan of removing the mixed gases from the callecting tubes e¢ (Fig. $4) and then exploding them in a separate vessel, I collected the gases on their escape from the discharge tubes in rial tube about -76 om. in diameter and 36 om, in length ix joined on to Fic nd tie top af tis tube & fared on wo the decharge tubs 0D; this tube is blown out into a bulb in the region where the sparka pasa, SE sintgettin tants seats Wy as oxcvee © eee tube. This part of the tube in encircled by the ring-burner K by means a of which the steam can be superheated. ‘The electrodes between which the sparks pass are shown in detail in Fig. 143; A, B aro metal tubes, these must be made of a metal which does not oxidise, Tn the following experiment A,B are cithor brass tubes Fig. 143. thickly plated with gold, or tubes made by winding thick platinum wire into a coil, These tubes are placed in pieces of glass tubing to hold ‘them in position. These tubes stop short of the places F, @ where the delivery tubes joi the dischorge tube. The dischorge tube is closed at the ends by the glass tubes p and Q, and wires connected to the electrodes Aand 8 are fused through these tabes. ‘The delivery tubes which terminate in fine openings were fused on to the discharge tube at F and G. To get rid of the air which ia in the apparatus or which is absorbed by the water, the apparatus is filled so full of water at the beginning of the experiment that when the water is heated it expands sufficiently ‘to Gill the discharge tube and overflow through the delivory tubes. The water ix boiled vigorously for 6 or 7 hours with the ends of the de- livery tubes open to the atmosphere, Tho eudiometer tubes filled with ‘mercury are then placed over the ends of the delivery tubes, so that if any sir is mixed with the steam it will be collected in these tubes, Tho sparking ie not commenced until after the steam has run into the delivery tubes for about an hour without carrying with it a quantity of air large enough to be detected. The sparks are produced by @ large induction ooil giving sparks wbout 6 em. long whon the current from five large storage cells is pont throngh the primary. When a condenser of about 6 or 7 micro-farads enpacity ia added to that supplied with the instrument a current was produced which, when the distance between the electrodes A and B in oo ‘Short Sparks. a ‘These sparks were from 1-5 mm, to 4 mm, long. ‘The appearance of 1, That within the limits of error of the experiments the volumes of _ the excesses of hydrogen in one tube and of oxygen in the other which ‘Femain after the explosion of the mixed gasos aro rospectively equal to ‘the volumes of the hy@rogon and oxygen liberated in the water voltameter placed in series with the steam tube. 2. The excess of hydrogen appears in the tube which is in connection \ with the positive electrode, the excess of oxygen in the tube which is in | connection with the negative electrode. Tt thus appears that with theso short sparks or arcs the hydrogen appears at the poritice electrode instead of an in ordinary electrelysis at tho negative. ‘Tho following table contains the results of pome measurements of the elation between the excesses of hydrogen and oxygen in the eudiometar tubes attached to the stenm tube and the quantity of hydrogen liberated jn a water voltameter placed in series with the discharge tube. The ordinary vibrating break supplied with induction coils was used unless if i < Bae lal eeeeeeeey 2 2 2 2 2 a 8 + ‘ ‘ 4 ‘ |)" Tn this experioest slow mervury break, making aboot four beeaks & second, _ © To these experiments Leyden jure wore attached to the electrodes. 002 placed in series with that tube. | It will be seen that the results when the spark length is greater than {dhe ertioal langth agree with those obtained by Perrot (Art. 201) and ‘Ladeking (Art. 210), as both these observers found that the hydrogen ‘appeared at the negative, the oxygen nt the positive electrode. Lude= King worked with long sparks, so that his results are quite in accordance ‘with mine. In Perrot’s experiments the spark length was 6 mm. T bare never been able to reduce the critical length quite so low nm this, though I diminished the current to the magnitude of that used by Perrot; T have, however, got it as low as 8 mm., and it is probable that the critical length may not be governed entirely by the current, Twas not able to detect any decided change in the appearance of the spark as tho spark length possed through tho critical value. My observations on the connection between the appearance of the discharge and the electrode at which the hydrogen appears may be expressed by the statement that whon the discharge is plainly an aro the hydrogen appears at the positive electrode, and that when the hydrogen appears at the negative clectrode the discharge shows all the characteristics of a spark. It however looks much more like a spark than an are long before the spark length reaches the critical value. With regard to the mtio of the quantities of hydrogen liberated from ‘the steam tube and from the water voltameter, I found that when the spark length was a few millimetres greater than the critical length the amount of hydrogen from the steam was the same as that from the volta- meter. The following table contains a few measurements on this point :— é Sek not. pm hg md el 10 mm. 7 8 12 om? 78 0 14mm, 8 1a When the sparks were longer than 14 mm. the amount of hydrogen from the steam was no longer equal to that from the voltameter, The ‘results became irregular, and there was a further reversal of the clec- trode at which the hydrogen appeared when the epark length exceeded 22mm, In this cave, howover, tho current was so amall that it took several hours to berate 1 ¢,c. of hydrogen in the voltameter, With these very long sparks the proportion between the hydrogen from the steam and that from the yoltameter was too irregular to allow of any conclusions being drawn, 1 Tn this experiment there ona an air break 9 mm, long in series with the stenra habe. ‘are attached to the terminals of the steam tube or if an _ | sent throngh the primary. A current of the gas under examination entered the ditcharge tube through a glass tube C and blow the gas in, “the neighbourhood of the are ngninst the platinum electrode €, which was connected to one quadrant of an electrometer, the other quadrant of ‘which was connected to earth. To soreen € from external electrical in- fluences it was enclosed in o platinam tube D, which was cloved in by fine platinum wire gauge, which though it ecreened from externul electrostatic action, yet allowed the gases in the neighbourhood of the are to pass through it, This tube was connected to eurth, The electrode € after passing out of thie tube was attached to one end of gutta-percha covered ‘wire wound round with tin-foil connected to earth. The experiments were of the following kind. ‘The quadrants of the electrometer were charged up by @ battery, the connection with the battery was then broken and the rate of leak observed. When the orc ‘wos not passing the insulation was practically porfect. As soon, howover, as the are was started, and for as long as it continued, the insulation of the gus in many cases completely gave way. There are, however, many remarkable exceptions to this which we proceed to consider. Oxygen, ‘We shall begin by considering the case when a well-developed arc paszed through the oxygen. If the electrode & was charged negutively, it lost its charge very rapidly; it did not however remain uncharged, but acquired a positive charge, this churge increasing until € acquired a potential 7; V dee pendel greatly upon the size of the arc and the proximity to it of the electrode E, in many of my experiments it wae as large oo 10 or 12 volts. When € was charged positively to a high potential the electricity Ieaked from it until the potential fell to V; after reaching this potential the leak stopped and the gas seemed to insulate a well os when no dis charge posed through it, If the potential to which & was initially raised ‘was less than V (a particular case being when it was without charge to begin with) the positive charge increased until the potentisl of © was equal to V, after which it remained constent, Thus we seo (1) that an doctrodo immersed in the oxygen of the are can invulate a emall positive charge perfectly, while it very rapidly loses a negative one; (2) that an uncharged electrode immersed in this gas acquires a positive charge. When the distance between the electrodes A, B was increased until the discharge passcd a5 a ypark then the electrode E leaked slowly, whether charged positively or negatively. ‘The rate of loak in this case was how- APPENDIX, 569 Passed, and this ring was connected with ono of the quadrants of an electrometer, Asa further precaution against the creeping of the elec- tricity over the surface of the tube two thin rings of tin-foil connected to the carth were placed round the ends of the tube. When the are passed through oxygen the quadrants of the electrometer connected with the ring of tin-foil were positively electrified by induction, when the are passed through hydrogen they were negatively charged. ‘There experiments show that the oxygen in the arc behaves as if it had positice charge of electricity, while the hydrogen in the are behaves as if it had a negative charge. Tn all the above experiments the electrodes were so large that they were not heated sufficiently by the discharge to become luminous. Elster and Geitel found (Art, 43) that a metal plate placed near a red- hot plotinum wire becime positively electrified if the wire and the plate were surrounded by oxygen, and negatively electrified if they were sur rounded by hydrogen. If we suppose that the effect of the hot wire isto put the gas around it in a condition resembling the gas in the arc, Elster and Gcitel’s results would be explained by the preceding experiments, for ‘these have shown that when this gae is oxygon it ie positively electrified, while when it Is hydrogen it is negatively electrified. ‘These experiments suggest the following exphmation of the results of the investigation on the electrolysis of steam. We have seen (Art, 242) that when an eloctric dischargo passes through a gas the propertica of the gas in the neighbourhood of the line of discharge are modified, and (Art. 84) that this modified ns possesses very considerable conduetivity. ‘When the discharge stops, this modified gas goes back to its original con= dition. If now the discharges through the gaa follow one another so rapidly that the modified yas produced by one dischange has not time to Terert to its original condition before the next discharge passes, the suc~ cessive discharges will pass through the modified gax. If, on the other hand, the gas has time to return to its original condition before the next discharge passes, each discharge will have to make ite way through the ‘unmodified gas, We rgani the are discharge ae corresponding to the first of the preceding cases when the discharge passes through the modified gas, the sparke discharge as corresponding to the second caso when the discharge passes through the gas in its unmodified condition, From this point of view the explanation of the results observed in the electrolysis of steam are very simple. ‘The modified gus produced by the poseage of the discharge through the steam consists of a mixture of hydrogen and oxygen, these gases being in the same condition as when the are discharge passes through hydrogen and oxygen respectively, when, - Se passed, and this ring wns connected with one of the quadrants of an electrometer. Ass further precaution against the creeping of the elec~ | tricity over the surface of the tube two thin rings of tin-foil connected to the carth were placed round the ends of the tube, When the are passed through oxygen the quadrants of the electrometer connected with the ring of tin-foil were positively electrified Wy induction, when the arc | Passed through hydrogen they were negatively charged. ‘There experiments show that the oxygen in the arc behaves as if it ‘had @ positice chorge of electricity, while the hydrogen in the arc behaves we if it had a argative charge In all the above experiments the electrodes were so large that they were not hented sufficiently by the discharge to become luminous. Elster and Geitel found (Art. 49) that a metal plate placed near a red- hot platinum wire became positively electrified if the wire and the plate ‘were surrounded by oxygen, and negatively electrified if they wero sur- rounded by hydrogen. If we suppose that the effect of the hot wire isto ut the gas around it in # condition resembling the gas in the are, Elster and Geitel’s results would be explained by the preceding experimente, for these have shown that when this gas ix oxygen it ie positively electrified, while when it is hydrogon it is negatively electrified. These experiments suggest the following explanation of the results of ‘the investigation en the electrolysis of steam. We have seen (Art, 212) that when an cloctric discharge passes through a gas the properties of the gas in the neighbourhood of the Kine of discharge are modified, and (Art, 84) thot this modified gas possesses very considerable conductivity. When the diecharge stops, this modified gas goes back to its original cou- dition, If now tho discharges through the gas follow one another #0 rapidly that the modified gas produced by one diecharge has not time to overt to Its original condition before the next discharge passes, the suc~ cessive discharges will pass through the modified gas. Tf, on the other hand, the gas has time to return to its original condition before the next discharge pasees, each discharge will have to make ite way through the unmodified ge We regard the are discharge as corresponding to the first of the Preceding cates when the discharge passes through the modified gna, the spark discharge as corresponding to the second case when the discharge pastes throngh the gas in ite unmodified condition. From this point of view the explanation of the results observed in the electrolysis of steam are very simple, ‘The modified gas produced by the paseage of the discharge through the steam consists of a mixture of Iydrogen and oxygen, these gases being in the amo condition as when the are discharge passes through hydrogen and oxygen respectively, when, _ - INDEX, The unrabera refer lo the pages, Anensarion Are dinchange, Aisin te fon elec casio pase Ne pra 5 eee sats eter tion tn a8 itnet-work of conitugtary, 610. | Arons'snd Cohn, specific indactve capa city of water, 45, — | ph aalgbleeeton qxpresion for aryntems of wires, 617, for“ Linpedance ’ of sys — —counoolion between lom of ‘of electrodes and quantity of electri asing, 156. a8. period, rate of decay slong” isch 71,74, aero ange 09-714 86 8 Besa! eninge it rei harge, 180, ‘Beexol’s functions, values of, when vari: ‘ble, is stall ot lange, 263. 179. jerknes, decay of vibrations, 397, aes oh conductivity of bet gum 65. eet pepe hated non Bata, pedo Inductive capa pe etal discharge, 171, 188, ty of ae-infateplate paral! to an infinite one, 233, — iio plate beten tw tal pals —of one ibe inside another, 222. — of two infinite strips, 237. = ofa pile of plates, 280, P| —dexay of currents in spheres, $78, $82, +E", 363. — om the functions ‘Sand ep — chetical action of the 179. Tenant and Wolf, dust ‘wader iat Kerik ‘Faraday tobos UTivng dlcharge of 38 Teva eet of, 551 et aeq. ‘Licbig, spark potential, 72, 91. ¢ of tron in rapidly alterna eng’ tele, 828, =e — field, effect of on “foe 00, == a aya ats 134. ie 132 ot seq. lek dis —— striations, 142. = refleion of light from, 483 et x0q, fetion of light parsing through shin Matlock! tre dieskarge, 16. ‘Mechandoal eifecte due to negative rays, ~ alternating current Melee expanson doe to dicharge, 178, Moreury vapour, ruiry vapours dicharge films, field on, 482 — aotion of maynet on light through thin Mecha! oes a eeAtreee — Fy Modioed ty eeetio ducharge, alt ). as ep dot acre, 178, Meroury va throug, 210. ry vapour, lacharge Select Works PUBLISHED BY THE CLARENDON PRESS. ALDIS. A pres Algebra: with Answers to the ‘Examples, W.3, Annis, M.A. Crown Svo, ye. 6d. BAYNES. _ Seas By BR, E, Bayses, M.A, Crown 8y0, 70. iif. . A Handbook of Descriptive Astronomy. By G, F. Chamsens, FKAS. 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