The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 34 of 39

624. Consider afresh the problem of which a preliminary discussion has already been given in § 572, of a single electron moving with a velocity u parallel to Ox. Since the field necessarily moves with the electron, the rate of change of any quantity % as we follow it in its motion must be nil. Thus we must have

(

d

dt

• ute)*-°

so that -r may be replaced by — u =- throughout our equations.

Ctt COG

Equation (617) becomes

dx" Xdx* dy" dz*

or, since a2 = G2JK/j,,

{1~~u^)d^ + df + dF-° (618)-

Also equations (613), (614) assume the forms

^O-B-g-S • <->'

5E-S-S <->■

622-627] Force on a moving Electron 561

625. In most problems, the velocity of motion u is small compared with the velocity of light, so that u/G may be treated as a small quantity. Equation (619) shews that the magnetic field set up by a moving charge may be regarded as small if u/C is small. The same is of course true of the field set up by any number of moving charges provided all their velocities are small compared with that of light.

When u/G is small, equation (620) shews that

d_Z_dY dy dz

will be a small quantity of the second order. Let us suppose, until the con- trary is stated, that u/G is so small for each moving charge that u2/C2 may legitimately be neglected. Then

dy dz

so that the forces X, Y, Z are derivable from a potential O. When u2/C2 is neglected equation (618) reduces to V2^ = 0. This equation is satisfied by X, Y, Z separately, and therefore also by O. Since X, Y, Z also satisfy equation (615), or

dX dY ZZ=^

dx dy dz "'

it is clear that the values of X, Y, Z are exactly the same as if the moving charge were instantaneously at rest.

626. This is exactly the assumption we made in § 572 in calculating the magnetic force from a moving charge. The forces there calculated, namely

a = 0, P = -Cri> y=G^ (621)'

are now seen to be accurate provided u2/C2 may be neglected, but not otherwise.

The Force acting on a Moving Electron.

627. The assumption we have made that u/G is small is the same as assuming to a first approximation that G is so great that the medium may be supposed to adjust itself instantaneously to changes occurring in it, just as an incompressible fluid would do. The time taken for action to pass from one point to another may be neglected. We may accordingly assume that at any instant the mechanical actions of any two parts of the field upon one another are such that action and reaction are equal and opposite.

J. 36

562

The Motion of Electrons

[CH. XIX

From equations (621), it appears that an electron moving with velocity u, 0, 0 at the origin will exert a force of components

0, -

ue mz

ue my G^

upon a magnetic pole of strength m at x, y, z. It follows that a magnetic pole of strength m at x, y, z will exert a force of components

0,

ue viz

ue my

.(622)

upon the moving electron at the origin.

628. If we have a number of magnetic poles, the resultant force upon the moving electron has components

0,

ue ^ mz ue ^ my

.(623)

G ~" r3 ' G r

and the components of magnetic force at the origin are given by (cf. § 408)

~mx

<x = — Z—r, etc.

r3

Thus the force on the moving electron may be put in the form

ue 0, -^7,

.(624).

Plainly the force on the electron will be given by formulae (624), whether the magnetic field arises from poles of permanent magnetism or not. It is clearly a force at right angles both to the direction of motion of the electron, and to the magnetic force a, /3, 7 at the point. If H is the resultant magnetic force, and 6 the angle between the directions of H and the axis of x, -then the resultant of the mechanical force is ueH sin 6/G.

629. If the electron has components of velocity u, V, w, the component of the mechanical force on it will be

jjiyv-pw), ^(aw-ryu), ^(@u-av) (625).

Since the mechanical force is always perpendicular to the direction of motion, it does no work on the moving particle ; and, in particular, if a charged particle moves freely in a magnetic field, its velocity remains con- stant.

The existence of this force explains the mechanism by which an induced current is set up in a wire moved across magnetic lines of force. The force (625) has its direction along the wire and so sets each electron into motion, producing a current proportional jointly to the velocity and strength of the field — i.e. to dN\dt.

627-632] Force, on a moving Electron 563

The "Hall Effect"

630. Very direct evidence of the existence of this force is provided by the "Hall Effect." Hall* found that when a metallic conductor conveying a current is placed in a magnetic field, the lines of flow rearrange themselves as they would under a superposed electromotive force at right angles both to the direction of the current and of the magnetic field. The same effect has also been detected in electrolytes and in gases.

The Hall Effect is of interest as exhibiting a definite point of divergence between Maxwell's original theory and the modern electron theory. Accord- ing to Maxwell's theory, a magnetic field could act only on the material conductor conveying a current, and not on the current itself, so that if the conductor was held at rest the lines of flow ought to remain unaltered f. The electron theory, confirmed by the experimental evidence of the Hall Effect, shews that this is not so, and that the lines of flow must be altered in the presence of a transverse magnetic field.

Motion of a charged 'particle in a uniform magnetic field.

631. Let a particle of charge e move freely in a uniform magnetic field of intensity H. Let its velocity be resolved into a component A parallel to the lines of force, and a component B in the plane perpendicular to them. By what has just been said (§ 629) both A and B must remain constant throughout the motion, and there will be a force eHBjG acting on the particle in a direction perpendicular to that of B, and in the plane perpendicular to the lines of force. Thus if m is the mass of the particle, its acceleration must be eHB/mC in this same direction.

Considering only the motion in a plane perpendicular to the lines of force, we have a velocity B and an acceleration eHB/mC perpendicular to it. This latter must be equal to B2Jp, where p is the curvature of the path. Thus

p = — jj- , a constant, shewing that the motion in question is circular.

Combining this circular motion with the motion parallel to the lines of force we find that the complete orbit is a circular helix, of radius BmGjeH, described about one of the lines of magnetic force as axis.

632. By measuring the curvature of an orbit described in this manner, it is found possible to determine ejm experimentally for electrons and other charged particles (cf. § 665 below). Incidentally the fact that curvature is observed at all provides experimental confirmation of the existence of the force acting on a moving electron.

• Phil. Mag. 9 (1880), p. 225.

t Maxwell, Electricity and Magnetism, § 501.

36—2

564 The Motion of Electrons [ch. xix

The Zeeman Effect.

633. When a source of light emitting a line-spectrum is placed in a strong magnetic field, the lines of the spectrum are observed to undergo certain striking modifications. The simplest form assumed by the pheno- menon is as follows.

If the light is examined in a direction parallel to the lines of magnetic force, each of the spectral lines appears split into two lines, on opposite sides of, and equidistant from, the position of the original line, and the light of these two lines is found to be circularly polarised, the direction of polarisation being different for the two.

If the light is examined across the lines of force, these same two lines appear, accompanied now by a line at the original position of the line, so that the original line now appears split into three. The side lines are observed to be plane polarised in a plane through the line of sight and the lines of force, while the middle line is plane polarised in a plane perpendicular to the lines of force.

634. These various phenomena were observed by Zeeman in 1896, and an explanation in terms of the electron theory was at once suggested by Lorentz.

Let us first examine a simple artificial case in which the spectrum contains one line only, assumed to be produced by the oscillations of a single electron about a position of equilibrium.

If the frequency of this oscillation is p/Ztt, the equations of motion of the

electron must be of the form

d?x m -tt2 = - mp2x, etc (626),

in which x, y, z are the coordinates of the electron referred to its position of equilibrium.

Next suppose the electron to move in a field of force of intensity H parallel to the axis of x. In addition to the force of restitution of components

• mp2x, - mp2y, — mp-z, the electron will be acted on by a force (cf. formulae

(625)) of components

n eH dz eH dy

' TTdV ~~Cdf In place of the former equations, the equations of motion are now

&

\r

mdt2= ~mP2x

d?y , eH dz

mw-=-mpy+-cTt

d2z eH dy

.(627).

633-635] The Zeeman Effect

The solutions of these equations are x = A cos ( pt — e),

y = A1 cos {qxt - ex) + A2 cos (q2t - ea), z = Az sin (qxt - e,) + ^2 sin (g2£ — ea), in which A, Alt A2> e, e^ ea are constants, and qlt q2 are the roots of

b§o

— mq* = — mp* +

eH

c-1

.(628).

For even the strongest fields which are available in the laboratory, the value of the last term in this equation is small compared with that of the other terms, so that the solution of equation (628) may be taken to be

eH

The original vibrations of the electron, all of frequency p, may now be replaced by the three following vibrations :

I. x = A cos(pt— e), 2/ = 0, z=0.

II. x = 0, y = Ai cos III. x = 0, y — A2 cos

p +

eH

2mC.

eH\

t

(p-hrc)1-'*

z= J.1sin

z = — J..2sin

p +

P

eH

2mC

eH

2mC.

t

t-e.

Vibration I of frequency p is a linear motion of the electron parallel to Ox, the direction of the lines of magnetic force. The magnetic force in the emitted radiation is accordingly always parallel to the plane of yz and vanishes immediately behind and in front of the electron (cf. § 618). Thus there is no radiation emitted in the direction of the axis of x, and the radiation emitted in the plane of yz will be polarised (§ 592) in this plane.

Vibrations II and III represent circular motions in the plane of yz of

eH

frequencies p ± = — ^ . Clearly the radiation emitted along the axis of x will

be circularly polarised, while that emitted in the plane of yz will be plane polarised in a plane through the line Ox and the line of sight (the motion along the line of sight sending no radiation in this direction). Thus the observed appearances are accounted for.

635. The analysis just given explains the observed facts of the normal Zeeman Effect, but only in terms of a model which is known not to be in accordance with the actual facts of atomic structure. As was pointed out by Larmor, the explanation just given can be easily generalised so that the atomic model shall at least accord better with the facts of nature than that we have just had under discussion.

566

The Motion of Electrons

[ch. XIX

If an electron is moving in a field of magnetic force of intensity H parallel to the axis of x, its equations of motion will be

d-

X

= Fm

•(629),

mdf-~^

dry -r, eH dz

mcd=F»+u it

d2z _ p eH dy mdP~ z~ G dt)

where Fx, Fy, Fz are the components of the force which acts on the electron apart from the superimposed magnetic field H. These equations of course contain equations (627) as a special case.

If x, y, z were coordinates measured with reference to a system of axes rotating with uniform angular velocity co about the axis of x in the direction from Oy to Oz, the component of the velocity of the point x, y, z in space would be given by

.(630),

dx

1

u-

= dt

V

_dy

' dt

(OZ >

w

dz

= di +

coy

J J

and the accelerations in space by

Dv Dt

dv dt

dv dx

dv

■

By

cv

dz

= —, + u - + v - + w - = -^ - 2 co ^ - tfy

cPy

dt2

dz dt'

•(631),

and similar equations.

When the angular velocity is so small that co2 may be neglected, the system of accelerations, as given by equations such as (631), reduce to

Du Dt

x

dt2

Dv _ d2y dz

Di~~~~dt2 W dt

Dw _ d2z dy

DidP+ tt)

.(632).

Thus if co is defined by the equation

co =

eH 2mC

.(633),

635, 636] The Zeeman Effect 567

the equations of motion (629) of the electron in the superimposed magnetic field become

mlTt=F*'

■UU TP

mT^ = Fy,

mnt=F"

which would be precisely the equations of motion of the electron referred to axes at rest with the magnetic field non-existent. Thus the superposition of the magnetic field parallel to the axis of x is seen to have had precisely the same effect on electronic motion as the setting of the axes in rotation with an angular velocity <o defined by equation (633).

Before the magnetic field is superposed, let the electron describe a path such that when its coordinates are resolved into simple-harmonic terms by Fourier's theorem, one of the constituent simple-harmonic vibrations is of the form

x = 0, y = A1 cos (pt — ej), z = Ax sin (pt — e^).

Assume that one of the lines in the spectrum of the atom when in its natural state corresponds to a frequency pj2ir. The superposition of a magnetic field has the same effect on the coordinates x, y, z as the setting of the axes in rotation with an angular velocity co, so that when this field is superposed the coordinates of the electron may be taken to be

x = 0, y = A1cos[(p + (o)t — d], z = A^in^p + ^t — e2J.

It is at once seen that the vibration is identical in general type with the vibration II that we found in § 634, so that the discussion of the polarisation and change of frequency there given will apply to the present case also.

636. The discussion of the last section is applicable to any electron describing an orbit such that its motion can be resolved into oscillations of definite frequencies. It shews that each spectral line ought in general to be resolved into a triplet of three equidistant lines, a line initially at p giving place to lines at p ± 8p where

bp=^tch <634>-

This represents what is normally observed, and a formation of triplets of this type is commonly spoken of as the normal Zeeman Effect. Certain lines separate out in a more complex way in the presence of a magnetic field, these lines having generally appeared as multiple lines (doublets, triplets, etc.) even before the magnetic field was turned on. This is known as the complex or abnormal Zeeman Effect, and is not covered by our simple theory.

568

The Motion of Electrons

[CH. XIX

In the normal Zeeman Effect, the frequency difference, predicted by- equation (634), is constant for all the lines of the spectrum. Observationally this is found to be the case, and equation (634) makes it possible to determine a value of ejm from the observed separation of spectral lines in a magnetic field of known strength. The value so obtained proves to be in good agree- ment with values for ejm measured by other and more direct methods.

The General Equations of Moving Electrons.

637. We now return to the general equations of §621, namely

4W df\ dry d/3 ,nnw

.(636),

dy

G dt~dy dz'

and discuss the field set up by the motion of electric charges when there is no restriction as to the smallness of their velocities.

On multiplying both sides of equation (635) by /x and differentiating with respect to the time, we obtain

G dt\pU+ dt)~ dt\dy~dz)' Using relations (636) we readily find that the right-hand member

_dy\dx dy J dz\dz dxj^

= C

v*x

_a/SI 3F dZ\ dx \ dx dy dz)

Putting 47r/"= KX and -^ + ^— + -^ = 4nrp, this becomes

V2X

dx Kud'X

dz

47T/i d

C* dt2 C*

, . 4-7T dp

638. This is the differential equation satisfied by X. Similar equations are of course satisfied by Fand Z. If we divide both sides of equation (636) by /j, and differentiate with respect to the time, we readily find that a satisfies the differential equation

Kfid'a 4tt

V2a-

C2 dt*

G

!<P*)-!<pr)].

We shall shortly obtain these differential equations in another way.

636-640] General Equations 569

Introduction of the Potentials.

639. With equations (636) we may combine the relation

da db dc _ /ri>7\

dx dy dz ' '

(equation (616)), and it follows, as in §443, that we can find a vector-potential of components F, G, H connected with a, 6, c by the relations

-?-£.«* <«»>

and with X, Y„ Z by the relations (cf. § 530)

x4f=-^-etc <639>-

in which ^ is a function, at present undetermined in the general case, which becomes identical with the electrostatic potential when there is no motion.

640. We have seen (§ 442) that equations (638) are not adequate to determine F, 0, H completely, and hence \P also (cf. equation (639)) is not fully determined.

Let F0, G0, H0y yVQ be any special set of values satisfying equations (638) and (639). Then the most general values of F, G, H are given by (cf. § 442)

F=F0 + d^, etc (640),

where % is any arbitrary single- valued function.

To find the most general value of W, we have from equation (639)

dx + G\dt dxdtj dx Gdxdt '

so that, on integration,

13v

^ = ^0--,^ + a constant (641).

From (640) and (641) we obtain

dx + dy dz+ G dt dx dy dz + G dt X G3 dt*

(642).

The function % is entirely at our disposal, so that

x C2 dt*

may have any value we please to assign to it. Let us agree to give to x such a value, for every instant of time and all values of x, y, z, as shall make the right-hand member of equation (642) vanish.

570 The Motion of Electrons [oh. xix

The value of ^ is now fixed, except for a set of values of ^ such that

x c* dt*

at every instant and point, these values of ^ representing of course contribu- tions that might arise from a set of disturbances propagated through the medium from outside.

Except for such additional values of %, the values of F, G, H, ty are now uniquely determined by equations (640) and (641). The vector-potential will in future mean the special vector of which these values of F, G, H are the components, while the corresponding special value of ^ will be called the " Electric Potential."

From equation (642) it follows that the vector-potential and the electric potential are connected by the relation

dF dG dH^^dV

dx + dy+dz~ a dt (b4d>

Differential Equations satisfied by the Potentials.

641. If we differentiate equations (639) with respect to x, y, z and add, we obtain

/ax by d_z\ id /a? do ,atf\ VnF

\dx + dy^dz) + Gdt\dx^ 8*/ dzj~ '

which, on substituting from equations (643) and (639), becomes

V * & dt' K (b*^'

the differential equation satisfied by M*. We notice that for a steady field it becomes identical with Poisson's equation, while in regions in which there are no charges it becomes identical with the equation of wave-propagation.

642. To obtain the differential equation satisfied by F, we transform equation (635) by the use of equations (638). We have

4nra f df\ dc db

dy\dx ay J dz\dz dxj

m IP* +*Jt + ?)-?

dx \dx dy dz J

whence, from equations (643) and (639),

w-*"-^-- <**

the differential equation satisfied by F. Similar equations are of course satisfied by G and H.

640-645] General Equations 571

Differential Equations satisfied by the Forces.

Kix d? 643. Operating on equation (639) with the operator V'2 — -~ -y-2 , we

have

KnffiX_ ldfr72Z, KpffiF\ d (^Jr Kf±cM>

C2 dt2

^x_K^d^__l^( Kd,d^_d_f

C2 dt2 Gdt\ C2 dt2) dx\

=^>>+l?l ^-

This is the differential equation satisfied by X, and similar equations are satisfied by Y and Z. These same equations were obtained by a more direct method in § 637.

644. For the differential equation satisfied by a, /3, 7 we have, from equations (638) and (645),

G2 dt2 /A G2 dt2)\dy dz)

— ~CVTy W\ (647)'

and similar equations for /3 and 7. These equations agree with those already obtained in § 638.

Solution of the Differential Equations.

645. It will be seen that all the differential equations are of the same general form, namely

Vs*-^=-4™ <648>-

where cr arises from electric charges, at rest or in motion.

Clearly the value of ^ may be regarded as the sum of contributions from the values of a in the different small elements of volume. The simplest solution for % is that arising from a distribution of cr at and close to the origin, cr being zero everywhere else.

For this special solution % is a function of r only, which must satisfy

x a2 dt2

everywhere except at the origin. Proceeding as in § 578, and rejecting the term which represents convergent waves, as having no physical importance, we obtain the solution (cf. equation (536))

' x = lf(r-at) (649),

where f is so far a perfectly arbitrary function.

572 The Motion of Electrons [ch. xix

Close to the origin, this reduces to

%-;/<-«*) (650),

and it now appears that in equation (648) the middle term becomes insig- nificant near the origin in comparison with the first term V2^. Thus close to the origin the equation becomes identical with Poisson's equation, and the integral is

<rdxdydz

//.

X = -^~ = ; (651),

where the integral is taken only through the element of volume at the origin in which cr exists, and t represents the integral of a taken through this element of volume.

On comparing solutions (650) and (651), both of which are true near the origin, we find that

/(-aO-r (652),

and this determines the function / completely. The general solution (649) is now fully known, and by summation of such solutions the general solution of equation (648) is obtained.

Let P, Q be any points distant r apart ; let t be any instant of time, and let tQ denote the instant of time r/a previous to it, so that t0 = t — r/a. Clearly t0 is the instant of departure from P of a disturbance reaching Q at t. We may speak of t0 as the "retarded time" at P corresponding to the time t at Q.

With this meaning assigned to t0, we have

f{r-at)=f-a{t--^=f(-at«) = r,

where r is evaluated at time t0 (cf. equation (652)). If we agree to denote by [<£] the value of </> estimated at the retarded time at the point at which <£ occurs, then this value of t will be expressed by [t], and solution (649) becomes

% = V (653).

The most general solution of equation (648), obtained by the summation of solutions such as (653), is

x rrrM^=2[i] (654)_

the last form applying when the distribution of a occurs only at points or in small regions so small that the variations of the retardation of time through each region are negligible.

The analogy of Poisson's equation and its solution in electrostatics (cf. §§ 49, 40, 41) is obvious.

645-647] General Equations 573

646. From equations (644) and (645) it follows that the potentials are given by

*4///[«^ (655),

-g///kg]<^y.e* (656).

These potentials are commonly spoken of as "Retarded Potentials." They differ from the ordinary potentials, in which the finite velocity of propagation is not taken into account, only in that the quantities in the numerators must be evaluated at the retarded times appropriate to the point.

The solution of equations (646) and (647) may be similarly written down, but it is usually easier to evaluate the forces by differentiation of the potentials.

If the moving electrons in formula (656) are conveying currents in linear circuits, the formula becomes (on taking p. = 1)

where the summation is over the different circuits and ix denotes the

doc ^-component of the current, which may also be expressed as i-r-. This

formula may be compared with (419), from which it differs only in that it takes account of the finite time required for the propagation of electro- magnetic action.

The Field set up by Moving Electrons.

647. An electron is a charge of total amount e spread through a very small volume. When we attempt to apply the equations just obtained to the motion of electrons, a complication arises. We must not integrate p or pv through the space occupied by the electron because the retarded time varies from one part of the electron to another. And this complication does not disappear even when we pass to the limit and suppose the electron to be of infinitesimal size.

Let the electron be moving with a velocity (not necessarily uniform) of which the components at any instant are u, v, w. Suppose we wish to evaluate the potentials at x ', y', z at time t.

Let x, y, z be the position of any element of the electron at the retarded time t0, defined by

t0=t-- where r2 = (as' - x)2 + (y' - y)2 + (z' - z)\

Qj

We may speak of x, y, z as the effective position of the element of the electron under consideration, since the element contributes to the potentials we are in search of, only when it is at x, y, z.

574

The Motion of Electrons

[CH. XIX

The retarded time t0 will be different for different parts of the electron. Let its value at the centre of the electron be 60. Let the position of the element under consideration at time 80 be x0) y0, z0. Then the element which is at x0, y0, zQ at time 60 has moved to x, y, z by time t0, so that

x = x0+ u(t0- #o) + § u(to-0o)2 + ... etc., where U, v, w refer to the velocity of the electron at time 60.

Remembering that t0 is a function of x, y, z, we obtain on differentiation with respect to x,

dr dt

and similarly,

lr -§{*+ &-•> + lHt0-e0y +...}.

Those elements of the electron which have their effective positions inside a small element of volume dxdydz occupy at the fixed time 60 an element of volume dx0dy0dz0. The ratio of these elements of volume is given by the usual Jacobian determinant

dx0dy0dz0 dxdydz

dx0

dx'

dx0 dy'

dx0

dz

ho

dx '

ho

dy'

ho

dz

dz0 'dx'

dz0 dy'

dz0 dz

On inserting the values of the differential coefficients as just calculated, and expanding the determinant, we readily obtain

dx0dy0dzt dxdydz

= 1- 2 d^{u+v(t0-d0) + hu(t0-6^+...},

z,y>z

all terms in u, v, w of degree higher than the first being found to disappear. If the electron is small in comparison with its distance from the point x , y', z', local variations of x, y, z throughout the electron may be neglected, so that t0 — 6q may be neglected, and in the above expression x, y, z, r may be supposed to refer to the centre of the electron. In this case we have dx0dy0dzQ y dt0 , . 1 / _dr

dxdydz x.y.zd®

or, if vr denote the radial velocity of the electron towards the point x', y', z

at the instant t = 6a,

dx0dyfidz0

dr dr dx dy dz

= 1

Vr

a '

dxdydz

Equation (655) may now be written in the form

*p dx„dy0dzQ

• K 
  

r 1-

it

647]

General Equations

575

where all quantities are evaluated at the time t = 0O, or since

fffpda}0dyodz0 = e, e

^ =

K

1-*

a

where square brackets signify that the quantity inside is to be evaluated at the retarded time as estimated at the electron. Similarly equation (656) becomes*

F=~C

u

v i r

a

r

Suppose it is required to calculate the field at a point 0 at time t. Let E be the position of one of the electrons in the p £

field at a time t0 such that

r

t0 = t — — , a

where r = EO. Then the quantities in square brackets must be calculated for this electron in the position E at the time t0.

Let the velocity of the electron at the time tQ 6 be V in a direction EF making an angle 6 with Fi8- 139-

EO, and let EF be the distance V (t — 10) which the electron would describe by the time t if its velocity remained constant.

If FG is the perpendicular from F on to EO, the intercept EG is given by

EG = EF cos 6 = V cosd(t - t0).

Now Y cos 6 is simply the component Yr of velocity along EO, while t - t0 = rfa. Thus EG = rVrja and

Vj

OG = r-EG = r 1

aj

The formulae for the potentials now become

1 e

V =

KOG

COG'

If squares of Y/C are neglected, the angle FOE in figure 139 is a small angle and OG is approximately equal to OF. Thus as far as terms of the first

• These formulae for & and F were first given by Lienard (L'Eclairage Electrique, 1898) and E. Wiechert (Arch. Neerland. 5, 1900, p. 549). Our proof has followed closely the method given by Lorentz (Theory of Electrons, p. 254) ; an alternative proof is given by Schott (Electromagnetic Radiation, 1912, p. 22).

[CH. XIX

576 The Motion of Electrons

order in VJC the potentials are

KOF' COF'

where F is the position of the electron at the instant at which the potentials are evaluated, except for a correction arising from accelerations or sudden changes in the motion of the electron.

648. In the case in which u, v, w are treated as small we can also write down the potentials directly from equations (655) and (656). For in this case dx0 dy0 dz0 becomes equal to dx dy dz and the equations assume the forms

0] F_/4^T]

■^ ~~ n ~ >

•\Jr _

Kr'

C

where r is the distance from the point x, y', z' at which the forces are measured to the effective position of the electron. Thus the magnetic forces are given by

a~a[dy' dz')-c{dy' r dz r J'etC {b°7)'

fj>\dy' dz' J G\dy Since [ew] is a function of t— r/a, we have

9 r , 1

orL J a

so that

dt

(ew)

= --[ew],

a

3 [ew]_y' — y d [ew] _ y'—y fl [ew] _ [ew]} dy' r r dr r r [a r r'1

and on substitution in equations (657) we obtain formulae for a, j3, y.

These formulae are seen to contain terms both in r_1 and r~\ At a great distance from the electron the former alone are of importance, and the com- ponents of force become

a = —

y -y

[ew]-Z—^-[ev], etc.

aC ( Similarly we find for the electric forces at a great distance

X = -£C^],etc

.(658).

O

.(659).

For a single electron moving along the axis of x with an acceleration u, in free space for which /i = K = 1, the components of force assume the simple forms

.(660),

these being accurate only at great distances from the electron.

647-650] Radiation of Energy 577

Radiation of Energy.

649. We saw in § 576 that the flow of energy across any closed surface is given by

jj(lUx + mUy + nUz)dS (661),

where

n,= ^.(F7-Z/3),etc.

In proving this the energy was assumed to be localised in the medium in the way imagined by Maxwell, but if we identify our closed surface with a sphere at infinity this assumption is no longer necessary. For independently of this assumption, the total energy in the whole of space is given by

T+W=jjj~ (X2 + F2 + Z*) + ^ (a2 + /32 + 74 dxdydz

and from this we can deduce formula (661) directly. On assigning to a, /3, y, X, Y, Z the value obtained in equations (660) for the forces from a single electron, we find

n*=o, uy=-^Xy, n,=^x/3,

IUX + mUy + nUz = {y'~y^rZf (^)a>

whence the flow of energy across a sphere of infinite radius is readily found to be

? ** (662)

This is Larmor's formula for the rate at which a single moving electron radiates energy. We notice that a steady velocity u contributes nothing to the radiation ; energy is radiated away from an electron which is undergoing acceleration but not from one in steady motion.

It must be added that the new dynamics referred to in § 620 seems to throw doubt on this formula for emission of radiation. Many physicists now question whether any emission of radiation is produced by the acceleration of an electron, except under certain special conditions. Bearing this caution in mind, we may proceed to examine some of the consequences of the formulae just obtained.

650. If each of a cluster of electrons is so near to the point x, y, z that differences of retardation of time may be neglected throughout the cluster, the radiation from the cluster is easily seen to be the same as that from a single electron of charge E moving with components of acceleration U, V, W, such that

EU=%eu, etc. J. 37

578 The Motion of Electrons [en. xix

The condition that there shall be no radiation from such a cluster is

2e u = Se v = 2e w = 0.

If this condition is not satisfied, the rate of emission of radiation is (cf. formula (662))

^s{(teuy+(Zevy+($ewy} (663).

651. Consider next the field produced by a particle of charge E oscillating along the axis of x with simple harmonic motion, its coordinate at any instant being x0 cos pt. We have

Eu=- Ep^xo cos pt; [Eu] = - Ep"x0 cos p[t — -),

and the field can be written down by substitution in formulae (660).

From formula (662) the average rate of emission of radiation is found to be

1 p'E'a:* _16ttE2x0C 3 C3~~~ 3V

where X is the wave-length of the emitted light.

A particle moving in this way is spoken of as a simple Hertzian vibrator. Its motion was taken by Hertz to represent the oscillating flow of current in an oscillatory discharge of a condenser. Such an oscillation formed the source of the waves in Hertz's original experiments (1888), and forms the source of the waves used in modern wireless telegraphy.