A History of the Theories of Aether and Electricity (1910) — part 28 of 29

Now many indications point to the probability that the various types of forces which are observed in ponderable bodies — forces of cohesion, of chemical union, and so forth — are ultimately electric in their nature. Such an assumption

Closing Years of the Nineteenth Century. 447

would have the great advantage of explaining the contraction postulated by Fitz Gerald, since it would represent the con- traction as actually produced by the motion. But if this assumption be correct, the theory of electricity and aether is without doubt the fundamental theory of Natural Philosophy ; and the framework of space and time should be chosen with a view chiefly to the expression of electrical phenomena. This may most naturally be done by stipulating that the wave- fronts of disturbances generated in free aether shall, in the system of length and time adopted, be accounted spheres whose centres are at the origins of disturbance and whose radii are proportional to the times elapsed since their initiation. Eeferred to axes of (#,y,z,£) which satisfy these conditions, the fundamental equations of the electric field assume the form which has been taken as the basis of all our theoretical investigations.

Imagine now a distant star which is moving with a uniform velocity w or c tanh a relative to this framework (x, y, z, t). The theorem of transformation shows that there exists another framework (a?,, y,z, t^, with respect to which the star is at rest, and in which moreover the condition laid down regarding the wave-surface is satisfied. This framework is peculiarly fitted for the representation of the phenomena which happen on the star ; whose inhabitants would therefore naturally adopt it as their system of space and time. Beings, on the other hand, who dwell on a body which is at rest with respect to the axes (x, y, z, t) would prefer to use the latter system ; and from the point of view of the universe at large, either of these systems is as good as the other. The equations of motion of the aether are the same with respect to both sets of coordinates, and therefore neither can claim to possess the only property which could confer a primacy — namely, an absolute relation to the aether.*

To sum up, we may say that the phenomena whose study is the object of Natural Philosophy take place each at a definite

• This was first clearly expressed by Einstein, Ann. d. Phys. xvii (1905), p. 891.

448 The Theory of Aether a?id Electrons in the

location at a definite moment ; the whole constituting a four- dimensional world of space and time. To construct a set of axes of space and time is equivalent to projecting this four- dimensional world into a three-dimensional world of space and a one-dimensional world of time ; and this projection may be performed in an infinite number of ways, each of which is distinguished from the others only by characteristics merely arbitrary and accidental.*

In order to represent natural phenomena without introducing this contingent element, it would be necessary to abandon the customary three-dimensional system of coordinates, and to operate in four dimensions. Analysis of this kind has been devised, and has been applied to the theory of the aether ; but its development belongs to the twentieth century, and consequently falls outside the scope of the present work.

From what has been said, it will be evident that, in the closing years of the nineteenth century, electrical investigation was chiefly concerned with systems in motion. The theory of electrons was, however, applied with success in other directions, and notably to the explanation of a new experimental discovery.

The last recorded observation of Faradayf was an attempt to detect changes in the period, or in the state of polarization, of the light emitted by a sodium flame, when the flame was placed in a strong magnetic field. No result was obtained; but the conviction that an effect of this nature remained to be discovered was felt by many of his successors. TaitJ examined the influence of a magnetic field on the selective absorption of light ; impelled thereto, as he explained, by theoretical considera- tions. For from the phenomenon of magnetic rotation it may be inferred§ that rays circularly polarized in opposite senses are propagated with different velocities in the magnetized medium ; and therefore if only those rays are absorbed which have a

• Cf. H. Minkowski, Raum und Zeit. : Leipzig, 1909. t Bence Jones' Life of Faraday, ii, p. 449. J Proc. R.S. Edinb. ix (1875), p. 118. § Cf. pp. 174, 216.

Closing Years of the Nineteenth Century. 449'

certain definite wave-length in the medium, the period of the ray absorbed from a beam of circularly polarized white light will not be the same when the polarization is right-handed as when it is left-handed. "Thus," wrote Tait, "what was originally a single dark absorption-line might become a double line."

The effect anticipated under different forms by Faraday and Tait was discovered, towards the end of 1896, by P. Zeeman.* Eepeating Faraday's procedure, he placed a sodium flame between the poles of an electromagnet, and observed a widen- ing of the D -lines in the spectrum when the magnetizing current was applied.

A theoretical explanation of the phenomenon was imme- diately furnished to Zeeman by Lorentz.f The radiation i& supposed to be emitted by electrons which describe orbits within the sodium atoms. If e denote the charge of an electron of mass ra, the ponderomotive force which acts on it by virtue of the external magnetic field is e [r . K], where K denotes the magnetic force and r denotes the displacement of the electron from its position of equilibrium; and therefore, if the force which restrains the electron in its orbit be &, the equation of motion of the electron is

mi? -t- K2r = e [f . K].

The motion of the electron may (as is shown in treatises on dynamics) be represented by the superposition of certain particular solutions called principal oscillations, whose distin- guishing property is that they are periodic in the time. In order to determine the principal oscillations, we write T^ent^- * for r, where r0 denotes a vector which is independent of the time, and n denotes the frequency of the principal oscillation : substitut- ing in the equation, we have

(K- - mn*) rc = en^/^~l [r, E].

• Zittingsverslagen der Akad. v. "Wet. te Amsterdam v (1896), pp. 181, 242 ; vi (1897), pp. 13, 99 ; Phil. Mag. (5) xliii (1897), p. 226. t Phil. Mag. xliii (1897), p. 232.

2 G

450 The Theory of Aether and Electrons in the

This equation may be satisfied either (1) if r0 is parallel to K,

in which case it reduces to

K* - mri* = 0,

so that n has the value Km"*, or (2) if r0 is at right angles to K, in which case by squaring both sides of the equation we obtain

the result

(V - mn*)z = #tfK\

which gives for n the approximate values KW~* ± el£/2m.

When there is no external magnetic field, so that K is zero, the three values of n which have been obtained all reduce to icwV which represents the frequency of vibration of the emitted light before the magnetic field is applied. When the field is applied, this single frequency is replaced by the three frequencies »cm£, »cm"i + eK/2m, icm'i - eK/2m ; that is to say, the single line in the spectrum is replaced by three lines close together. The apparatus used by Zeeman in his earliest experi- ments was not of sufficient power to exhibit this triplication distinctly, and the effect was therefore described at first as a widening of the spectral lines.*

We have seen above that the principal oscillation of the electron corresponding to the frequency *cra~£ is performed in a direction parallel to the magnetic force K. It will therefore give rise to radiation resembling that of a Hertzian vibrator, and the electric vector of the radiation will be parallel to the lines of force of the external magnetic field. It follows that when the light received in the spectroscope is that which has been emitted in a direction at right angles to the magnetic field, this constituent (which is represented by the middle line of the triplet in the spectrum) will appear polarized in a plane at right angles to the field ; but when the light received in the spectroscope is that which has been emitted in the direction of the magnetic force, this constituent will be absent.

We have also seen that the principal oscillations of the electron corresponding to the frequencies Km-* ± eJ£/2m are

• Later observations, with more powerful apparatus, have shown that the primitive spectral line is frequently replaced by more than three components.

Closing Years of the Nineteenth Century. 45 1

performed in a plane at right angles to the magnetic field K. In order to determine the nature of these two principal oscilla- tions, we observe that it is possible for the electron to describe a circular orbit in 'this plane, if the radius of the orbit be suitably chosen ; for in a circular motion the forces *2r and .-e[r . K] would be directed towards the centre of the circle ; and it would therefore be necessary only to adjust the radius so that these furnish the exact amount of centripetal force required. Such a motion, being periodic, would be a principal oscillation. Moreover, since the force e [r . K] changes sign when the sense of the movement in the circle is reversed, it is evident that there are two such [circular orbits, corresponding to the two senses in which the electron may circulate; these must, therefore, be no other than the two principal oscillations of frequencies K.m~^ ± el£/2m. When the light received in the spectroscope is that which has been emitted in a direction at right angles to the external magnetic field, the circles, are seen edgewise, and the light appears polarized in a plane parallel to the field ; but when the light examined is that which has been emitted in a direction parallel to the external magnetic force, ,the radiations of frequencies Km~i ± eK/2m are seen to be •circularly polarized in opposite senses. All these theoretical •conclusions have been verified by observation.

It was found by Cornu* and by C. G-. W. Konigf that the more refrangible component (i.e., the one whose period is shorter than that of the original radiation) has its circular vibration in the same sense as the current in the electromagnet. From this it may be inferred that the vibration must be due to a resinously charged electron; for let the magnetizing current and the electron be supposed to circulate round the axis of z in the direction in which a right-handed screw must turn in order to progress along the positive direction of the axis of z ; then the magnetic force is directed positively along the axis of z, jand, in order that the force on the electron may be directed

• Comptes Rendus, cxxv (1897), p. 555.

• Ann. d. Phys. Ixii (1897), p. 240.

2G2

452 The Theory of Aether and Electrons in the

inward to the axis of z (so as to shorten the period), the charge on the electron must be negative.

The value of e/m for this negative electron may be determined by measurement of the separation between the components of the triplet in a magnetic field of known strength ; for, as we have seen, the difference of the frequencies of the outer com- ponents is eKjm. The values of e/m thus determined agree well with the estimations* of e/m for the corpuscles of cathode rays.

The phenomenon discovered by Zeeman is closely related to the magnetic rotation of the plane of polarization of light. f Both effects may be explained by supposing that the molecules of material bodies contain electric systems which possess natural periods of vibration, the simplest example of such a system being an electron which is attracted to a fixed centre with a force proportional to the distance. Zeeman's effect represents the influence of an external magnetic field on the free oscillations of these electric systems, while Faraday's effect represents the influence of the external magnetic field on the forced oscillations which the systems perform under the stimulus of incident light. The latter phenomenon may be analysed without difficulty on these principles, the equation of motion of one of the electrons being taken in the form

mr + K2r = eE + e[r.H],

where m denotes the mass and e the charge of the electron,, r its distance from the centre of force, K2r the restitutive force, E and H the electric and magnetic forces. When the electron performs forced oscillations under the influence of light of frequency n, this equation becomes

(K2-m?i2)r = eE + e[r.H].

The influence of the magnetic force on the motion of the electron is small compared with the influence of the electric force, i.e. the second term on the right is small compared with the first term ; so in the second term we may replace r by its

• Cf. p. 405. t Cf. pp. 213-216, 307-309, 367-370.

Closing Years of the Nineteenth Century. 453

value as found from the first term, namely, eE/(icz - run*). The •equation thus becomes

r = K2 - mn* + (i'-wm1)1^'^' If P denote* the electric moment" per unit volume, we have

P = ei x the number of such systems in unit volume of the medium ;

so P must be of the form

where e evidently represents the dielectric constant of the medium, and o- is the coefficient which measures the magnetic rotatory power. In the magneto-optic term we may replace H by K, the external magnetic force, since this is large com- pared with the magnetic force of the luminous vibrations. Thus if D denote the electric induction, we have

D = fE/47rc2 + <r [E . K].

'Combining this with the usual electromagnetic equations,

curl H = 47ri>,

curl E = - H, we have

• curl curl E = *E/c2 + 4:r<r [E . K].

When a plane wave of light is propagated through the medium in the direction of the lines of magnetic force, and the axis of x is taken parallel to this direction, the equation gives

(VEy

.and these equations, as we have seen,f are competent to explain the rotation of the plane of polarization.

*Cf. p. 428. t Cf. p 215.

454 The Theory of Aether and Electrons in the

From the occurrence of the factor (KT - mw) in the denomi- nator of the expression for the magneto-optic constant <r, it may be inferred that the magnetic rotation will be very large for light whose period is nearly the same as a free period of vibration of the electrons. A large rotation is in fact observed* when plane-polarized light, whose frequency differs but little from the frequencies of the D-lines. is passed through sodium vapour in a direction parallel to the lines of magnetic force.

The optical properties of metals may be explained, according to the theory of electrons, by a slight extension of the analysis which applies to the propagation of light in transparent sub- stances. It is, in fact, only necessary to suppose that some of the electrons in metals are free instead of being bound to the molecules : a supposition which may be embodied in the equations by assuming that an electric force E gives rise to a polarization. P, where

E = aP + /3P + 7P ;

the term in a represents the effect of the inertia of the electrons ;• the term in ]3 represents their ohmic drift ; and the term in y represents the effect of the restitutive forces where these exist. This equation is to be combined with the customary electro- magnetic equations

curl H = E/c2 + 47rP, - curl E = H.

In discussing the propagation of light through the metal, we may for convenience suppose that the beam is plane-polarized

• The phenomenon was first observed by D. Macaluso and 0. M. Corbino,. Comptes Rendus, cxxvii (1898), p. 548, Rend. Lincei (5) vii (2) (1898), p. 293. The theoretical explanation was supplied by AV. Voigt, Gott. Nach., 1898, p. 349, Ann. d. Phys. Ixvii (1899), p. 345. Cf. also P. Zeeman, Proc. Amst. Acad. v (1902), p. 41, and J. J. Hallo, Arch. N6erl. (2) x (1905), p. 148.

Voigt also predicted that if plane-polarized light, of period nearly tbe same as that of the D radiation, were passed through sodium vapour in a magnetic field, in a direction perpendicular to the lines of magnetic force, the velocity of propa- gation would be found to depend on the orientation of the plane of polarization, so that the sodium vapour would behave as a uniaxal crystal. This prediction was confirmed experimentally by Voigt and Wiechert : cf . Voigt, Gott. Nach., 1898, p. 355: Ann. d. Phys. Ixvii. (1899), p. 345. Cf. also A. Cotton, Cornpte* Rendus, cxxviii (1899), p. 294, and J. Geest, Arch. Neerl. (2), x (1905), p. 291.

Closing Years of the Nineteenth Century. 455

and propagated parallel to the axis of 2, the electric vector being- parallel to the axis of x. Thus the equations of motion reduce to

— = -r -=-^- + 4n-

For Ex and P* we may substitute exponential functions of

where n denotes the frequency of the light, and /* the quasi-index of refraction of the metal : the equations then give at once

<y _ i) (_ a?lt + pnS~^i + y) = 47TC2.

Writing v (1 - K v/ - 1) for p, so that v is inversely proportional to the velocity of light in the medium, and « denotes the coefficient of absorption, and equating separately the real and imaginary parts of the equation, we obtain

40 (y- an)

f?nz + (y - aw2)2

When the wave-length of the light is very large, the inertia represented by the constant a has but little influence, and the equations reduce to those of Maxwell's original theory* of the propagation of light in metals. The formulae were experi- mentally confirmed for this case by the researches of E. Hagen and H. Kubensf with infra-red light ; a relation being thus established between the ohmic conductivity of a metal .and its optical properties with respect to light of great wave- length.

When, however, the luminous vibrations are performed more rapidly, the effect of the inertia becomes predominant; and

• Cf. p. 290.

t Berlin Sitzungsber., 1903, pp. 269, 410; Ann. d. Phys. xi (1903), p. 873 ; Phil. Mag. vii (1904), p. 157.

456 The Theory of Aether and Electrons in 'the

If the constants of the metal are such that, for a certain range of values of n, VZK is small, while v~ (I - K2) is negative, it is evident that, for this range of values of n, v will be small and K large, i.e., the properties of the metal will approach those of ideal .silver.* Finally, for indefinitely great values of n, V~K is small .and v2 (1 - K2) is nearly unity, so that v tends to unity and K to zero : an approximation to these conditions is realized in the X-rays.f

In the last years of the nineteenth century, attempts were made to form more definite conceptions regarding the behaviour of electrons within metals. It will be remembered that the •original theory of electrons had been proposed by WeberJ for the purpose of explaining the phenomena of electric currents in metallic wires. Weber, however, made but little progress towards an electric theory of metals ; for being concerned chiefly with magneto-electric induction and electromagnetic ponder omotive force, he scarcely brought the metal into the discussion at all, except in the assumption that electrons of opposite signs travel with equal and opposite velocities relative to its substance. The more comprehensive scheme of his successors half a century afterwards aimed at connecting in a unified theory all the known electrical properties of metals, such as the conduction of currents according to Ohm's law, the thermo-electric effects of Seebeck, Peltier, and W. Thomson, the gal vano- magnetic effect of Hall, and other phenomena which will be mentioned subsequently.

The later investigators, indeed, ranged beyond the group of purely electrical properties, and sought by aid of the theory of electrons to explain the conduction of heat. The principal ground on which this extension was justified was an experimental result obtained in 1853 by G. Wiedemann and K. FranzJ who found

• Cf. p. 179.

t Models illustrating the selective reflexion and absorption of light by metallic bodies and by gases were discussed by H. Lamb, Mem. and Proc. Manchester Lit. .and Phil. Soc. xlii (1898), p. 1 ; Proc. Lond. Math. Soc. xxxii (1900), p.. 11 ; Trans. 'Camb. Phil. Soc. xviii (1900), p. 348.

• Cf. p. 226. § Ann. d. Phys. Ixxxix (1853), p. 497.

Closing Years of the Nineteenth Century. 457

that at any temperature the ratio of the thermal conductivity of a body to its ohmic conductivity is approximately the same for all metals, and that the value of this ratio is proportional to the absolute temperature. In fact, the conductivity of a pure metal for heat is almost independent of the temperature; while the electric conductivity varies in inverse proportion to the absolute temperature, so that a pure metal as it approaches the absolute zero of temperature tends to assume the character of a perfect conductor. That the two conductivities are closely related was shown to be highly probable by the experiments of Tait^ in which pieces of the same metal were found to exhibit variations in ohmic conductivity exactly parallel to variations in their thermal conductivity.

The attempt to explain the electrical and thermal properties •of metals by aid of the theory of electrons rests on the assump- tion that conduction in metals is more or less similar to conduction in electrolytes ; at any rate, that positive and negative charges drift in opposite directions through the sub- stance of the conductor under the influence of an electric field. It was remarked in 1888 by J. J. Thomson,* who must be regarded as the founder of the modern theory, that the differences which are perceived between metallic and electro- lytic conduction may be referred to special features in the two cases, which do not affect their general resemblance. In electrolytes the carriers are provided only by the salt, which is dispersed throughout a large inert mass of solvent ; whereas in metals it may be supposed that every molecule is capable of furnishing carriers. Thomson, therefore, proposed to regard the current in metals as a series of intermittent discharges, caused by the rearrangement of the constituents of molecular systems — a conception similar to that by which Grothussf had pictured conduction in electrolytes. This view would, as he showed, lead to a general explanation of the connexion between thermal and electrical conductivities.

• J. J. Thomson, Applications of Dynamics to Physics and Chemistry, 1888, p. 296. Cf. also Giese, Ann. d. Phys. xxxvii (1889), p. 576. t Cf. p. 78.

458 The Theory of Aether and Electrons in the

Most of the later writers on metallic conduction have pre- ferred to take the hypothesis of Arrhenius* rather than that of Grothuss as a pattern ; and have therefore supposed the interstices between the molecules of the metal to be at all times swarming with electric charges in rapid motion. In 1898 E. Eieckef effected an important advance by examining the consequences of the assumption that the average velocity of this random motion of the charges is nearly proportional to the square root of the absolute temperature T. P. DrudeJ in 1900 replaced this by the more definite assumption that the kinetic energy of each moving charge is equal to the average kinetic energy of a molecule of a perfect gas at the same temperature , and may therefore be expressed in the form qT, where q denotes a universal constant.

In the same year J. J. Thomson § remarked that it would accord with the conclusions drawn from the study of ionization in gases to suppose that the vitreous and resinous charges play different parts in the process of conduction : the resinous charges may be conceived of as carried by simple negative corpuscles or electrons, such as constitute the cathode rays : they may be supposed to move about freely in the interstices between the atoms of the metal. The vitreous charges, on the other hand, may be regarded as more or less fixed in attachment to the metallic atoms. According to this view the transport of electricity is due almost entirely to the motion of the negative charges.

An experiment which was performed at this time by Eiecke|| lent some support to Thomson's hypothesis. A cylinder of aluminium was inserted between two cylinders of copper in a circuit, and a current was passed for such a time that the amount of copper deposited in an electrolytic arrangement

• Cf. p. 384.

t G-ott. Nach., 1898, pp. 48, 137. Ann. d. Phys. Ixvi (1898), pp. 353, 545, 1199; ii. (1900), p. 835.

I Ann. d. Phys. (4) i (1900), p. 566 ; iii (1900), p. 369 ; vii (1902), p. 687. § Rapports pres. au Congres de Physique, Paris, 1900, iii, p. 138. || Phys. Zeitsch. iii (1901), p. 639.'

Closing Years of the Nineteenth Centwy. 459*

would have amounted to over a kilogramme. The weight of each of the three cylinders, however, showed no measurable change; from which it appeared unlikely that metallic con- duction is accompanied by the transport of metallic ions.

The ideas of Thomson, Kiecke, and Drude were combined by Lorentz* in an investigation which, as it is the most complete r will here be given as the representative of all of them.

It is supposed that the atoms of the metal are fixed, and that in the interstices between them a large number of resinous- electrons are in rapid motion. The mutual collisions of the electrons are disregarded, so that their collisions with the fixed atoms alone come under consideration ; these are regarded as analogous to collisions between moving and fixed elastic spheres.

The flow of heat and electricity in the metal is supposed to take place in a direction parallel to the axis of xt so that the metal is in the same condition at all points of any plane perpendicular to this direction ; and the flow is supposed to be steady, so that the state of the system is independent of the time.

Consider a slab of thickness dx and of unit area ; and suppose that the number of electrons in this slab whose ^-components of velocity lie between u and u + du, whose ^-components of velocity lie between v and v + dv, and whose ^-components of velocity lie between w and w + dw, is

/ (ut v, w, x) dx du dv dw.

One of these electrons, supposing it to escape collision, will in the interval of time dt travel from (x, y, z) to (x + u dt, y + vdt, z + wdt) : and its ^-component of velocity will at the end of the interval be increased by an amount e Edtjm^ if ra and e denote its mass and charge, and E denotes the electric force. Suppose that the number of electrons lost to this group by collisions in the interval dt is a dx du dv dw dt, and that the

• Amsterdam Proceedings (English edition) vii (1904-1905), pp. 438, 585, 684

460 The Theory of Aether and Electrons in the

number added to the group by collisions in the same interval is b dx du dv dw dt. Then w^e have

f (u, v, w, x) + (b — a) dt = f (u + eE dt/m, v, w, x + u dt), .and therefore

7JT ^\ J? r\ J?

i &fi of oT

b - a = — — + u — •

m du dx

Now, the law of distribution of velocities which Maxwell postulated for the molecules of a perfect gas at rest is expressed by the equation

rz /= TT~^ a'3 Ne"*,

where N denotes the number of moving corpuscles in unit volume, r denotes the resultant velocity of a corpuscle (so that r2 = u* + v~ + w*), and a denotes a constant which specifies the -average intensity of agitation, and consequently the temperature. It is assumed that the law of distribution of velocities among the electrons in a metal is nearly of this form; but a term must be added in order to represent the general drifting of the electrons parallel to the axis of x. The simplest assumption that can be made regarding this term is that it is of the form

u x a function of r only ; we shall, therefore, write

a

/ = NTT~* a'3 e «2 + u^ (r). The value of ^ (r) may now be determined from the equation

eE'df df b - a = — ~- + u —-; m du dx

for on the left-hand side^ the Maxwellian term

would give a zero result, since b is equal to a in Maxwell's .system ; thus b - a must depend solely on the term u-% (r) ; and

Closing Years cf the^ Nineteenth Century. 46 T

an examination of the circumstances of a collision, in the manner of the kinetic theory of gases, shows that (b - a) must have the form - ur^ (r)/l, where I denotes a constant which is closely related to the mean free path of the electrons. In the terms on the right-hand side of the equation, on the other hand, Maxwell's term gives a result different from zero; and in comparison with this we may neglect the terms which arise from u\ (r). Thus we have

urv(r) leE d 8\ N --,

/ \m

or

lu --, fieNE d (N\ 2M* da) .

and thus the law of distribution of velocities is determined. The electric current i is determined by the equation

i = e Jj'J uf (it, v, w) du dv dw,

where the integration is extended over all possible values of the components of velocity of the electrons. The Maxwellian term in f (u, v, w) furnishes no contribution to this integral, so we

have

i = e JJJ v? x (r) du dv dw.

When the integration is performed, this formula becomes

mu dx ' dxf

or

STT^W a . m /a2 dN da\

'±&Nl + 2e(N~fa'*adx)'

The coefficient of i in this equation must evidently represent the ohmic specific resistance of the metal ; so if y denote the specific conductivity, we have

4/r N

Let the equation be next applied to the case of two metals A and B in contact at the . same . temperature T, forming an

462 The Theory of Aether and Electrons in the

•open circuit in which there is no conduction of heat or electricity .(so that i and da/dx are zero). Integrating the equation

= m a2 clN " 2eNdx

.across the junction of the metals, we have

Discontinuity of potential at junction = -^— log -— ;

or since f ma2, which represents the average kinetic energy of an electron, is by Drude's assumption equal to q/T, where gr denotes a universal constant, we have

2 q N

Discontinuity of potential at junction = ^ - T log ~- •

O €> J\ A

This may be interpreted as the difference of potential con- nected with the Peltier* effect at the junction of two metals ; the product of the difference of potential and the current measures the evolution of heat at the junction. The Peltier discontinuity of potential is of the order of a thousandth of a volt, and must be distinguished from Volta's contact-difference of potential, which is generally much larger, and which, as it presumably depends on the relation of the metals to the medium in which they are immersed, is beyond the scope of the present investigation.

Eeturning to the general equations, we observe that the flux of energy JFis parallel to the axis of x, and is given by the equation

W = \m HI urif(u, v, iv) du dv dw,

where the integration is again extended over all possible values of the components of velocity ; performing the integration, we have

or, substituting for E from the equation already found,

TT_ ma2 . 4ml da

W = ^ - -r—r Naz -7- •

e 871-2 dx

• Cf. p. 264.

Closing Years of the Nineteenth Century. 463

Consider now the case in which there is conduction of heat without conduction of electricity. The flux of energy will in this case be given by the equation

W--***

K

where K denotes the thermal conductivity of the metal expressed in suitable units ; or

3ma da W = - K.-^— -j--

2q dx

If it be assumed that the conduction of heat in metals is effected by motion of the electrons, this expression may be compared with the preceding; thus we have

and comparing this with the formula already found for the electric conductivity, we have

7 W '

an equation which shows that the ratio of the thermal to the electric conductivity is of the form T x a constant which is the same for all metals. This result accords with the law of Wiedernann and Franz.

Moreover, the value of q is known from the kinetic theory of gases; and the value of e has been determined by J. J. Thomson* and his followers ; substituting these values in the formula for K/y, a fair agreement is obtained with the values of K/y determined experimentally.

It was remarked by J. J. Thomson that if, as is postulated in the above theory, a metal contains a great number of free electrons in temperature equilibrium with the atoms, the specific heat of the metal must depend largely on the energy required in order to raise the temperature of the electrons. Thomson considered that the observed specific heats of metals are smaller than is compatible with the theory, and was thus

• Cf. p. 407.

464 The Theory of Aether and Electrons in the

led to investigate* the consequences of his original hypothesis^ regarding the motion of the electrons, which differs from the one just described in much the same way as Grothuss' theory. of electrolysis differs from Arrhenius'. Each electron was now supposed to be free only for a very short time, from the moment when it is liberated by the dissociation of an atom to the moment when it collides with, and is absorbed by, a different atom. The atoms were conceived to be paired in doublets, one pole of each doublet being negatively, and the other positively, electrified. Under the influence of an external electric field the doublets orient themselves parallel to the electric force, and the electrons which are ejected from their negative poles give rise to a current predominantly in this direction. The electric conductivity of the metal may thus be calculated. In order to comprise the conduction of heat in his theory, Thomson assumed that the kinetic energy with which an electron leaves an atom is pro- portional to the absolute temperature ; so that if one part of the metal is hotter than another, the temperature will be equalized by the interchange of corpuscles. This theory, like the other, leads to a rational explanation of the law of Wiedemann and Franz.