The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 33 of 39

It has been found by Jamin that formula (566) is not quite accurate at and near to the polarising angle. It appears from experiment that a certain small amount of light is reflected at all angles, and that instead of a sudden change of phase of 180° occurring at this angle there is a gradual change, beginning at a certain distance on one side of the polarising angle and not reaching 180° until a certain distance on the other side. Lord Rayleigh shewed that this discrepancy between theory and experiment can often be attributed largely to the presence of thin films of grease and other impurities on the reflecting surface. Drude found that the out- standing discrepancy could be accounted for by supposing the phenomena of reflection and refraction to occur, not actually at the surface between the two media, but throughout a small transition layer of which the thickness must be supposed finite, although small compared with the wave-length of the light.

Waves in Metallic and Conducting Media.

599. In  a  metallic  medium  of  specific  resistance  t,  equations  (A)  of  §  592, 
     

namely

KdX_dy dJ3

C dt ~dy dz {bb'>'

etc., must be replaced (cf. equation (546)) by

t C dt] dy dz etc.

For a plane wave of light, the time may be supposed to enter through the complex imaginary eipt and we may replace -j by ip- Thus the left-hand of

equation (567) becomes —~- X, while the left-hand of equation (568) becomes

[ h —pi) X. It accordingly appears that the conducting power of the

4nrC2 medium can be allowed for by replacing K by K + — — .

TT To

600. In a non-conducting medium, the equation -~ -^ = V2^, satisfied by

each of the quantities X, Y, Z, a, yS, 7 (cf. §577), reduces to

• p°-K[A _

i— V = V*

O

%=v"%

598-601] Metallic and Conducting Media 545

when the wave is of frequency p/2ir. The corresponding equation for a con- ducting medium must, by what has just been said, be

-'{%+%)*-"x ^

an equation which has already been obtained in § 583 a.

For a plane wave propagated in a direction which, for simplicity, we shall suppose to be the axis of x, the solution of this equation will be

^ = AeW e±(-Q+ir) x (570),

where (q + irf = - ^ + Zt* (571).

Clearly the solution (570) represents the propagation of waves with a velocity V equal to p/r, the amplitude of these waves falling off with a modulus of decay q per unit length.

On equating imaginary parts of equation (571) we obtain

qr=^ (572),

so that q is given by

2^p = 2ZLP>

t r t

601. For a good conductor t is small, so that q is large, shewing that good conductors are necessarily bad transmitters of light. For a wave of light in silver or copper we may take as approximate values in C.G.S. units (remembering that t as given on p. 342 is measured in practical units)

r = 1-6 x 10~6 ohms =16 x 103 (electromag.), /* = 1, V= 3 x 1010,

from which we obtain q = 1*2 x 108. It appears that, according to this theory, a ray of light in a good conductor ought to be almost extinguished before traversing more than a small portion of a wave-length. This prediction of the theory is not borne out by experiment.

We shall see below (§ 600) that the difficulty is to some extent removed on taking account of the presence of electrons in the metal. Before passing to the more general theory in which the electrons are taken into account we shall examine the phenomenon of metallic reflection according to our present simple theory, and shall again find that the simple theory fails to agree with the facts.

35

546

The Electromagnetic Theory of Light [ch. xvm

Metallic Reflection.

602. Let us suppose, as in fig. 138, that we have a wave of light inci- dent at an angle 6l upon the boundary between two media, and let us suppose medium 2 to be a conducting medium of inductive capacity K2. Then (cf. § 599) all the analysis which has been given in §§ 593 — 597 will still hold if we take K2 to be a complex quantity given by

K2 = K2' + ^- (574).

Since K2 is complex, it follows at once that V2 is complex, being given by

C2

K2 =

K.

2^2

and hence that the angle 62 is complex, being given (cf. equation (557)) by

sin20o

sin2 ^ sin2 ^ C2 _.2ffKl/h

K2 'a K2 Etfh — vlK2fJhi The value of u is now given, from equation (562), by

.(575).

u

2 —

K2 ^ cos2 02

ix2 Kx cos2 0X

-^1sec2^1-^-1-tan26»1 (576)

(cf. equation (575)) for light polarised in the plane of incidence. For light polarised perpendicular to the plane of incidence, the value of u is found, as before, by interchanging electric and magnetic symbols.

On putting u = a + i/3, we have, as before (equation (564)),

2T _ 1-M _ l-a-i/3 Z' ~ 1 + u ~ 1 + a + i/3 '

If we put this fraction in the form pe% then the reflected wave is given by

% = 2!" eiKi (-xcos^+2/sin 6, - Vxt) _ %* pgi*, (-zcos 9x+y sin e^ Vit+),

Comparing this with the incident wave, for which

Z = Z' eiK1 {x cos e'+y sin 9l ~ Vl t]

we see that there is a change of phase k^x at reflection, and the amplitude is changed in the ratio 1 : p. The electric force in the refracted wave is accompanied by a system of currents, and these dissipate energy, so that the amplitude of the reflected wave must be less than that of the incident wave.

1 - a - i0

We have

pe'x =

1 + a + i/3 '

602-604] Metallic Reflection 547

so that n2 = — — — = 1 (577^

P (1 + «#■ + £» (l + a)2 + /32 l°";

shewing that p < 1, as it ought to be. Also

y = - tan"1 -£- - tan"1 =-£- = - tan"1 - — ^—- (578).

A 1 - a 1 + a l-a--/33

603. Experimental determinations of the values of p and ^ have been obtained, but only for light incident normally, the first medium being air. For this reason we shall only cany on the analysis for the case of 6 = 0. It is now a matter of indifference whether the light is polarised in or at right angles to the plane of incidence ; indeed it is easily verified that the values given for p and % by equations (577) and (578) are the same in either case.

Taking for simplicity the analysis appropriate to light polarised in the plane of incidence, and putting 0 = 0, /*i = l, -5^ = 1, we have from equation (576)

u2 = — = b - ,

fi2 fju2 ipr/x2

and, since u = a + i/3, this gives

a3-/Sa=— ' (579)

fa

a{3 = -2-^ (580).

prfx,

604. Let us consider the results as applied to light of great wave-length, for which p is very small. For such values of p, a/3 is clearly very large compared with a2 — /32, so that a and (3 are nearly equal numerically, and we may suppose as an approximation that (cf. equation (580))

a = -/3 = v/2^ (581).

When a and f3 are equal and large, equation (577) becomes

9 ' pr/j.

"!=1-i=1-2V^ <582>-

Let us suppose that an incident beam has intensity denoted by 100, and that of this a beam of intensity R is reflected from the surface of the metal, while a beam of intensity 100 — R enters the metal. Then R may be called the reflecting power of the metal.

The intensity of the absorbed beam is

100 -.8 =100(1 -p-)

=2°°y^ (583>-

35—2

548

The Electromagnetic Theory of Light [ch. xviii

We notice that for waves of very great wave-length (p very small) R approximates to 100, so that for waves of very great wave-length all metals become perfect reflectors. This is as it should be, for these waves of very long period may ultimately be treated as slowly-changing electrostatic fields, and the electrons at the surface of the metal screen its interior from the effects of the electric disturbances falling upon it (cf. § 114).

Equation (583) predicts the way in which 100 — R ought to increase as p increases, and an extremely important series of experiments have been conducted by Hagen and Rubens* to test the truth of the formula for light of great wave-length. The following table will illustrate the results obtained f :

100 -R for \ = 12fi

Mpfal

I.U. t_ I i L 1

observed

calculated

Silver

1-15

1-3

Copper

1-6

1-4

Gold

2-1

1-6

Platinum ...

35

3-5

Nickel

4-1

36

Steel

4-9

4-7

Bismuth

17-8

11-5

Patent Nickel P

57

5-4

„ „ M

7-0

6-2

Constantin

6-0

7-4

Posse's alloy

7-1

7 3

Brande's and Schijnemann's alloy

9-1

8-6

In the calculated values, the value of K is assumed to be unity, and an error is of course introduced from the fact that the wave-length dealt with, X = 12/i, although large is still finite.

It will be seen that the agreement between the calculated and the observed values is surprisingly good, when allowance is made for the extreme difficulty of the experiments and for the roughness of some of the approxi- mations which have to be made.

605. Hagen and Rubens also conducted experiments for light of wave-lengths A, = 25*5 p, 8yu,, and i/x. On comparing the whole series it is found that the differences between observed and calculated values become progressively greater on passing to light of shorter wave-length. Drude has conducted a series of experiments on visible light, from which it appears that the simple theory so far given fails entirely to agree with observation for wave-lengths as short as those of visible light.

• Annalen der Physik, 11, p. 873; Phil. Mag. 7, p. 157. t Phil. Mag. 7, p. 168.

604-607] Electron Theory 549

Electron Theory.

606. We have now reached a stage in the development of electromagnetic theory in which it is clear that the simple conceptions which have so far been employed are no longer adequate to give a complete explanation of the phenomena. The conceptions on which the preceding analysis has been based have been the original conceptions of Maxwell's theory: it is natural now to examine in what way the theory can be modified or improved by the intro- duction of the more modern conceptions of the electron theory. Instead of regarding a current as a continuous flow of electricity, we shall take definite account of the presence of electrons. We shall have to consider two sets of electrons, the "free" and "bound" electrons of § 345 a, these being the mechanisms respectively of conduction and of inductive capacity.

The application of an electric force X will result in a motion of free electrons similar to that investigated in § 345 a, and in a motion of the bound electrons similar to that discussed in § 151. But if X is variable with the time, the inertia of the electrons will come into play and the resulting motions will be different from those given by Ohm's law and Faraday's law. We shall suppose that at any instant the current produced by the motion of the free electrons is «/, and that that produced by the motion of the bound electrons is u^.

607. We may consider first the evaluation of Uf. Taking N to be the number of free electrons per unit volume, and allowing for change of notation, equation (c) of § 345 a may be re-written in the form

cx=™'+l?t <584>-

in which, as throughout this chapter, X is expressed in electrostatic units, while Uf is in electromagnetic units, and r' stands for 7/iVe2, so that t becomes identical with the specific resistance t when the currents are steady.

This equation is applicable to our present investigation if we suppose X to be periodic in the time of frequency p/2tt. Taking X = XQ eipt, the solution of equation (584) is

CTogft* ,,„_,

u'= , ra . (58D)-

The quantity r here may depend on p, and without a full knowledge of the structure of matter it is impossible to decide how important the dependence of t on p may be. We are therefore compelled to retain it as an unknown quantity in our equations, remembering that it becomes identical with t when p = 0, and is probably numerically comparable with r for all values of p.

550 The Electromagnetic Theory of Light [ch. xviii

We may note that the real part of the current, corresponding to the force X = X0 cos pt, is

CX0 . N

— — cos {pt — e) cos e,

T

in which tan e= „ „ ,, shewing that the inertia of the electrons, as repre- sented in the last term of equation (584), results in a lag e in the phase of the current, accompanied by a change in amplitude. The rate of generation of heat by the current Uf, being equal to the average value of UfX0 cos pt,

is found to be A — r^cos2e or A -, where

T Tp

Ti, = T'sec2e=T' + -p^:/ (586).

It is worth noticing that for light of short wave-length the last term in rp may be more important than the first term r. Thus tp may be largest for good conductors, and smallest for bad conductors.

608. We turn to the evaluation of i(b, the current produced by the small excursions of the bound electrons, as they oscillate under the periodic electric forces.

We shall regard a molecule (or atom), as in § 151, as a cluster of electrons, and these electrons will be supposed capable of performing small excursions about their positions of equilibrium. As has already been said (§ 192) it is probable that this conception of the structure of the molecule represents only a half-way house towards the truth, but it provides a picture or model of the structure with the help of which many properties may be explained.

Let 01, $2, ... be generalised coordinates (cf. § 548) determining the positions of the electrons in the molecule, these being chosen so as to be measured from the position of equilibrium. So long as we consider only small vibrations, the kinetic energy T and the potential energy W of the molecule can be expressed in the forms

2W= and,2 + Zand A + a2A2 + (587),

2T= bj? + 2b J A + bj? + (588),

in which the coefficients an, a12, a™, ..., bn, ... may be treated as constants. By a known algebraic process, new variables <£1; <f>2, ... can be found, such that equations (587), (588) when expressed in terms of these variables assume the forms

2 IF = a4? + a,c£22 + (589),

2T = yS1012 + /S2^2+ (590),

these equations involving only squares of the new coordinates <f>1} (f>.2, .... The coordinates found in this way for any dynamical system are spoken of as the "principal coordinates" of the system.

607-609] Electron Theory 551

The equation of motion of the molecule, when acted on by no external forces, is readily found to be (cf. equations (500))

fis<j>s=-«s<f>s, (5=1,2,...) (591).

These equations are known to represent simply periodic changes in <f>1} </>2, ... of frequencies nJ27r, n2/2ir,... given by

"."-J; (592).

It is possible that we have evidence of the frequencies of molecular vibra- tion in certain of the lines of the spectrum emitted by the substance under consideration; if so equations (592) connect the frequencies of these spectral lines with the coefficients of the principal coordinates of the molecule.

609. If the molecule is now supposed to vibrate under the influence of externally applied forces (such, for instance, as would occur during the passage of a wave of light through the medium), equation (591) must be replaced (cf. equation (508)) by

0,4>, = -<*,<!),+ ®, (593),

where <&g is that part of the "generalised force" corresponding to the coordinate <ps, which originates in the externally applied forces.

If X is the electromotive force in the wave of light at any instant, each electron will experience a force Xe, and there will be a contribution of the form %sXe to ^>g.

Again the electrostatic field created by the displacements of the electrons in the various neighbouring molecules will contribute a further term to <E>g. The displacement of any electron through a distance f will produce the same field as the creation of a doublet of strength e|. Thus if there are M molecules per unit volume, the total strength of the doublets per unit volume, say V, may be supposed to be of the form

r = ife(7l01 + 7202+...) (594),

and these will produce an electric intensity of which the average value may be taken to be (cf. § 145) kY, which must be added to the original intensity X of the wave.

The total value of <&s is therefore £se (X + kF), so that on replacing as by its value from equation (592), equation (593) becomes

^s(4>s + ns2<f>s) = ^e(X + KV) (595).

If we suppose X to depend on the time through the factor eipt, then <f> will clearly depend on the time through the same factor, and we may replace §8 by — p2(ps. Equation (595) now becomes

•- A («■■-■) (596)'

552 The Electromagnetic Theory of Light [ch. xviii

whence, by equation (594),

T = M*% ^l AX + kT) (597),

and if we write

e-M'lfjr+0 (598),

this gives, as the value of T,

r=T^ex (599>

The current produced by the motion of the bound electrons is ub in electromagnetic, and therefore Cub in electrostatic units. Its value in

electrostatic units is also (cf. § 345 a) Neu or Se jr. , where the summation

is taken through a unit volume, and this in turn is equal to Y. Thus

T ip0 X

UbCl-K0 C

The total current, expressed in electromagnetic units, is

ldf

In calculating f we must remember that the polarisation produced by the motion of the bound electrons is already allowed for in the presence of the term ub. We accordingly take / equal simply to X/47T, and on further replacing ub and Uf by the values found for them, the total current becomes

ipX /_ 4tt0 \ CX ,,.AA.

In place of equation (569), the equation of propagation is

\ 'PV+N?'''))

As in § 600, the solution is

x=AeiPte±®+ir)x (601),

(7+,v, = -^(1 + i^)+_ilS- (602).

Non-conducting media.

610. For a non-conducting medium r' = oo , so that the last term in equation (602) vanishes, and the right-hand member becomes wholly real. For certain values of 6, this right-hand member is negative, so that q = 0, shewing that light is transmitted without diminution; the medium is perfectly transparent.

609, 610] Electron Theory 553

For transparent media we may take fi = 1, and the velocity of propagation V is given by

1 _r2_l_f 4tt<9

F2 _p» C3V l-«0/

If v is the refractive index of the medium, as compared with that of a vacuum, V= CJv, so that

Arr-0

*° = 1 + i^ (603)-

whence $£-"-***=? (604)'

in wmcn a = 1, cs = — --. — , so that a and c. are constants.

Clearly (cf. § 609) the value of a can be calculated if we make assumptions as to the arrangement of the molecules in the medium. On assuming that the molecules are regularly arranged in cubical piling, k is found to have the value §tt, so that a becomes equal to 2.

Formula (604) in which a is neglected altogether becomes exactly identical with the well-known Sellmeyer or Ketteler-Helmholtz formula for the dispersion of light, of which the accuracy is known to be very considerable. If a is put equal to 2, the formula becomes identical with dispersion formulae which have been suggested by Larmor and Lorentz.

It has been shewn by Maclaurin* that formula (604) will give results in almost perfect agreement with experiment, at least for certain solids, if a is treated as an adjustable constant. The agreement of the formula is so very good that little doubt can be felt that it is founded on a true basis. Mac- laurin finds for a values widely different from 2 (for rocksalt a = 5"51, for fiuorite a=l"04), the differences between these numbers and 2 pointing perhaps to the crystalline arrangement of the molecules. For liquids and gases we should expect to find a equal to 2.

Since M is proportional to p, the density of the substance, formula (604)

v'2 — 1 indicates that ought to vary directly as p when p varies. This law,

with a taken equal to 2, was announced by H. A. Lorentz f of Leyden and L. Lorenz^: of Copenhagen in 1880. Its truth has been verified by various observers, and, in particular, by Magri§ for a large range of densities of air.

From equation (604) it also follows that the values of for a mixture

i,2

-1

of liquids or gases ought to be equal to the sum of the values of — for its

• Proc. Roy. Soc. A, 81, p. 367 (1908). j Wied. Ann. 9, p. 641 (1880).

% Wied. Ann. 11, p. 70 (1880). § Phys. Zeitschrift, 6, p. 629 (1905).

554 The Electromagnetic Theory of Light [oh. xviii

ingredients, a law which is also found to agree closely with observation on taking a = 2.

611. For certain other values of 6, the right hand of equation (602) (in which t' is taken infinite) is found to be real and positive. We now have r = 0 and the solution (601) becomes

x = AeiPte±9x (605),

shewing that there is no wave-motion proper, but simply extinction of the light. Thus there are certain ranges of values of p (namely those which make (q + ir)2 positive in equation (601)) for which light cannot be transmitted at all ; these must represent absorption bands in the spectrum of the sub- stance.

Clearly (q + ir)2 becomes positive when 6 is large and negative. It will be noticed that 6, as given by equation (598), becomes infinite when p has any of the values n2, n2, ..., changing from — oo to + oo as p passes through these values. Thus the absorption bands will occur close to the frequencies of the natural vibrations of the molecule. But just in these regions we have to consider certain new physical agencies which cannot legitimately be neglected when p has values near to nlt n2, ..., although probably negligible in other regions of the spectrum.

612. Equation (593) is not strictly true with the value we have assigned to <l>s. For, in the first place the vibrations represented by the changes in <ps are subject to dissipation on account of the radiation of light, and of this no account has been taken. In the second place there must be sudden forces acting in liquids and gases occasioned by molecular impacts and requiring the addition of terms to <E>S throughout the short periods of these impacts. There must be analogous changes to be considered in the case of a solid, although our ignorance of the processes of molecular motion in a solid makes it im- possible to specify them with any precision.

The effect of these agencies must be to throw the $/s of the different molecules out of phase with one another and also out of phase with X and Y. The analysis of § 609 has made the ratios of X : T : <j>s wholly real (cf. equa- tions (596) and (597)), indicating that X, V and </>s are exactly in the same phase. The considerations just brought forward shew that these ratios ought also to contain small imaginary parts.

The process of separating real and imaginary parts in equation (602) now becomes much more complicated, but it will be obvious that for all values of p, both q and r will have some value different from zero. Thus there is always some extinction of light and some transmission, for all values of p, and there is no longer the sudden change from total extinction to perfect trans- mission. The edges of the absorption band become gradual and not sharp.

610-615]

Electron Theory

555

But the molecular model now in use probably does not represent the details of molecular action with sufficient truthfulness to make it worth trying to represent the conditions now under discussion in exact analysis.

Conducting media.

613. For  a  conducting  medium  we  retain  t  in  equation  (602),  and  on 
     

equating imaginary parts we obtain, in place of equation (572) of § 600,

qr = -

./a , mT

.(606),

T2 +

N2e4

P

where tp is given by equation (586). Thus equation (573) of § 600 becomes replaced by

q = 2-^ (607).

Tp

For visible light this gives a very much smaller value of q than that discussed in § 600, and the value of q will obviously be still further modified by the considerations mentioned in § 612.

614. On  comparing  the  total  current,  as  given  by  formula  (600),  with  the 
     

value --- -n assigned to it in the analysis of §§ 594 — 598, we see that all

this earlier analysis will apply to the present problem if we suppose K to be a complex quantity given by

47rC2

K=v2 +

, m .

.(608),

where v is given by formula (603). If, as in § 603, we put

we find

W2 = ^?=(a-M73)2,

H>2

a--/32 =

flo

m 4ttC2

v —

Ne2

TTr

.(609),

a 27rG2 ap =

PTpftz

so that the reflecting power B, of a metal may be calculated from equation (577) in terms of rp.

615. On comparing formulae (609) with experiment, the general result appears to emerge, that, in order to account for the optical properties of conductors in this way, the number of free electrons in conductors must be comparable with the number of atoms. According to a paper by Schuster,

556 The Electromagnetic Theory of Light [ch. xviii

published in 1904*, the ratio of the number of free electrons to atoms ought to range from 1 to 3 in various substances; Nicholson f, as the result of a more elaborate investigation, obtains values for this ratio ranging from 2 to 7.

This result discloses a difficulty from which the electron theory, in the form in which we have so far considered it, has shewn little power of extri- cating itself.

Specific Heats and Electrical Conductivity.

616. According to the well-known law of Dulong and Petit the atomic heats of a large number of elements have values which are approximately all equal. Nernst and Lindemann have recently determined the specific heats of a large number of elements, and have found that, for all the elements they have examined, the atomic heats measured for constant volume {i.e. after correction for expansion arising out of change of temperature) have all the same value 595. Now the atomic heat represents the increase per unit rise of temperature in the energy of the solid measured per atom of its structure. This energy can be regarded as the sum of two contributions, namely the energy of the atoms and the energy of the free electrons. The energy of the atoms can be calculated by the well-known methods of the Kinetic Theory of matter, and it is found that this energy will provide a contribution to the atomic heat equal exactly to the total amount of the atomic heat, namely 595; in other words the contribution from the energy of the free electrons is as small as the experimental error. But the contribution from a given number of free electrons also admits of theoretical calculation if we make the assump- tion that their motion conforms to the ordinary dynamical laws. If there were as many free electrons as one-tenth of the number of atoms, the contri- bution to the atomic heat would be '30, so that the total atomic heat would be 6'25, a number much too large to be reconciled with the experiments of Nernst and Lindemann.

617. The foregoing figures refer only to matter at comparatively high temperatures. The specific heats of the elements have however been deter- mined by Nernst and Lindemann through a very wide range of temperatures, namely from normal temperatures down to the lowest temperatures now available in the laboratory. And it has recently been shewn by Debye that the atomic heats found by these experiments are, at all temperatures, almost exactly equal to those to be expected on theoretical grounds on the supposition that the free electrons contribute nothing to the specific heat. The observed atomic heats agree so well with those calculated from theory, for all substances examined and at all temperatures available, that the conclusion seems to be inevitable that the number of free electrons is very small compared with the number of atoms.

• Phil. Mag. February 1904. t Phil. Mag. Aug. 1911.

615-619] Electron Theory 557

618. Thus we are led to the conclusion that although the present electron theory may shew a certain power of explaining the optical properties of metals, qualitatively at least, yet this explanation demands the presence of a far greater number of free electrons than can be reconciled with the values of the specific heats.

If the present electron theory were in other respects satisfactory, the difficulty just revealed might be thought to constitute a serious defect in the electromagnetic theory of light. But the present electron theory is far from satisfactory in other respects; indeed a difficulty very similar to that just disclosed has been found to arise in connection with a much simpler pheno- menon, namely the conductivity of metals. We have seen (§ 345 a) that the electron theory requires that in' a good conductor the number of free electrons should be large; approximately how large it must be is a matter which can also be determined by further analysis. The requisite analysis has been given by Drude.

619. We may suppose, as a rough approximation to the truth, that in a conductor each free electron moves freely for a certain length of time t between two consecutive collisions with molecules. In the notation already used in § 345 a, the momentum gained in this time will be Xet. If we suppose this momentum to be entirely checked at each collision (cf. §§ 355, 373), the average forward momentum of all the electrons at any instant will be hXet, and since this is equal to mu in the notation of § 345 a, we have

u = \ — t (610),

2 m

and hence (by equation (6), § 345 a)

i = Neu = ± — tX (611).

2 m

Thus the quantity 7 of § 345 a is, as regards order of magnitude at least,

equal to — , and the specific resistance r of a substance will be given by t

1 1 /V>2

• = ~ — t (612),

t 2 m

where N is the number of free electrons per cubic centimetre. Now for silver or copper t = 1'6x 10-6 ohms = l'8 x 10-18 in electrostatic units. The value of

1 e2 -it in electrostatic units is 1*26 x 108, and hence to give to t the value

2 m

appropriate for silver or copper we must have Nt = 5 x 109 approximately. In silver or copper the number of atoms per cubic centimetre is of the order of 1023, so that if the observed values of the specific heats do not allow of N being more than one-hundredth part of this we must at most suppose that N is of the order of 1021, and this requires t to be comparable with 5 x 10~12 at least.

558 The Electromagnetic Theory of Light [ch. xviii

Since the average velocity of the free electrons is believed to be about 107 cms. per second (§ 345 a), this would require each electron to travel an average distance of 5xl0-s cms. between consecutive violent collisions. This appears to be too large to be reconciled with present beliefs as to the structure of matter.

The difficulty becomes much worse when we consider the phenomenon at low temperatures. Kamerlingh Onnes has found for silver at a temperature of 13'88° abs. a resistance only equal to 0"7 per cent, of that at 0° C. Thus in silver at this low temperature we must have Nt of the order of 1012, so that if we take N = 1021 as above, t = 10-9. This velocity of free electrons at this low temperature is of the order of 2 x 106, so that the average distance travelled would be about -^ cm.

620. We have now found that contradictions exist in connection with the Electromagnetic Theory of Light, the theory of Specific Heats of metals, and the theory of Electric Conductivity, so long as we treat these questions in terms of ordinary dynamical laws and Maxwell's electromagnetic equations. A large accumulation of evidence, of which our discussion has touched only on a small fringe, suggests that a new system of dynamics and a new electron theory is needed. So far as can be seen the special feature of this new theory must be that the interaction between electrons and radiation is of an entirely different nature from that imagined by the classical laws. The new theory is in existence and is generally known as the Quantum-theory. A brief intro- duction to it will be found in the last chapter of the present book.

CHAPTER XIX

THE MOTION OF ELECTRONS

General Equations.

621. The motion of an electron or other electric charge gives rise to a system of displacement currents, which in turn produce a magnetic field. The changes in this magnetic field give rise to new electric forces, and so on. Thus the motion of electrons or other charges is accompanied by magnetic and electric fields, mutually interacting. To examine the nature and effects of these fields is the object of the present chapter.

The necessary equations have already been obtained in § 574, but the current u, v, w will now be regarded as produced by the motion of charged bodies. If at any point x, y, z there is a volume density p of electricity moving with a velocity of components u, v, w, then the current at x, y, z has components pu, pv, pw in electrostatic units. Since u, v, w in equations (529) are measured in electromagnetic units, they must be replaced by pu/C, pv/C, p w/C, and the equations become

4tt/ df\ dy 3/3 /iMmx

Equations (528), namely

Ida dZ dY

-CTt=dy--di>etc <614)'

remain unaltered, and the two sets of equations (613) and (614) provide the material for our present discussion.

When we had these same equations under review in § 574, G was regarded merely as the ratio of the units. We may now regard C as being the velocity of light, this being also the velocity of any other electromagnetic disturbance in free space.

622. If we differentiate equations (613) with respect to x, y, z and add, we obtain

9/v3/%,3/v d fdf da dh^

dx vy dz r dt\dx dy dz,

We have also, as an equation of continuity, expressing that the increase in p in any small element is accounted for by the flow of electricity across the faces by which the element is bounded,

560 The Motion of Electrons [ch. xix

By comparison with the equation just obtained, we have

d(df + dg + dh\dp dt\dx dy dz) dt'

of which the integral is our former equation (63), namely

di + dy + dz-p (615)-

Similarly, on differentiating equations (614) with respect to x, y, z and adding, we obtain

d (da db dc\

dt \dx dy dz) '.

of which the integral is our former equation (362), namely

da db dc _ /«,«x

s+£+s-° •. (616)-

623. At a point at which there is no electric charge (p = 0), equations (613) and (614) become identical with the systems of equations (A) and (B) of § 577, and the quantities X, Y, Z, a, /3, 7 must all satisfy the differential equation (534), namely

^ = «2V2% (617).

Force of a Moving Electron.