367THE ELECTROMAGNETIC FIELD DUE TO THE ELECTRONS.

( )( ) ( )0,yx z

vv vt x y z

ρρ ρρ ∂∂ ∂∂+ + + =

∂ ∂ ∂ ∂

, 0,y yx xz zd hd hd h

x y z x y zρ

∂ ∂∂ ∂∂ ∂+ + = + + =

∂ ∂ ∂ ∂ ∂ ∂

Proceedings of the London Mathematical Society, Vol. 1, 1904, p. 367-372.(Retyped for readability, with same page breaks).

ON AN EXPRESSION OF THE ELECTROMAGNETIC FIELDDUE TO ELECTRONS BY MEANS OF TWO SCALAR

POTENTIAL FUNCTIONS

By E. T. Whittaker.

[Received 11 November, 1903, -- Read 12th November, 1903]

1. Object of Paper

The object of the present paper is to show that when any number of electronsare moving in any manner, the functions which define the resulting electrodynamicsfield, namely, the three components of dielectric displacement in the aether and thethree components of the magnetic force at every point of the field, can be expressedin terms of the derivates of two scalar potential functions. (Previous writers haveexpressed them in terms of a scalar potential function and a vector potential function,which are equivalent to four scalar potential functions.) These two scalar potentialfunctions are explicitly evaluated in terms of the charges and co-ordinates of theelectrons. It is then shown that from these results the general functional form of anelectrodynamic disturbance due to electrons can be derived.

2. Explanation of Notation, and Summary of previously known Results.

The work of previous writers, so far as it concerns the present investigationand explains the notation used, may be briefly summarized as follows: --

Let D be the volume density of electricity at any place and time, and let vx, vy,vz be the components of its velocity, and c the velocity of light in the aether. Let dx,dy, dz be the three components of the dielectric displacement in the aether, and hx, hy,hz the three components of the magnetic force. Then the fundamental equations ofelectrodynamics may be written in Lorentz’s form (the units being suitably chosen):

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368 Mr. E. T. Whittaker [Nov. 12, 1903]

, ,y yx xz zx

h dd hh dc v c

y z t y z tρ

∂ ∂ ∂ ∂∂ ∂− = + − = − ∂ ∂ ∂ ∂ ∂ ∂

, ,y yx xz zy

d hh dh dc v c

z x t z x tρ

∂ ∂∂ ∂∂ ∂ − = + − = − ∂ ∂ ∂ ∂ ∂ ∂

, .y yx xz zz

h dh dd hc v c

x y t x y tρ

∂ ∂ ∂ ∂∂ ∂− = + − = − ∂ ∂ ∂ ∂ ∂ ∂

1, ,yx z

z x

aa ad h

c t x y zφ ∂∂ ∂∂

= − − = −∂ ∂ ∂ ∂

222 2 2 2 2

2 2, ,x

x x

ac c c a c v

t tφ

φ ρ ρ∂∂

∇ − = − ∇ − = −∂ ∂

2 22 2 2 2

2 2, ,y z

y y z z

a ac a c v c a c v

t tρ ρ

∂ ∂∇ − = − ∇ − = −

∂ ∂1

0.yx zaa a

x y z c tφ∂∂ ∂ ∂

+ + + =∂ ∂ ∂ ∂

1( , , , ) , ( , , , ) ,

4 cos( , ) 4 cos( , )x

x

vec ex y z t a x y z t

cr r v v r cr r v v rφ

π π= =

+ +∑ ∑

In place of dx, dy, dz, hx, hy, hz, we can define the field by a scalar potentialfunction N and three functions ax, ay, az, which are usually regarded as the threecomponents of a vector potential. The quantities dx, dy, dz, hx, hy, hz are given interms of N, ax, ay, az by the equations

and four similar, equations dy, dz, hy, hz.

The scalar potential and the three components of the vector potential satisfythe system of equations

For the fundamental case, namely, that in which the field is due to any numberof electrons moving in any way, the scalar potential and the three components of thevector potential are given by the equations

and two similar equations for ay and az, where e is the charge of a typical electron, r isits distance from the point (x, y, z), v is its velocity, (vx, vy, vz) the components of v,(v, r) the angle between the direction of v and r, and the bars over the letters meanthat the position of the electron considered is that which it occupied at a time

and the summation is taken over all the electrons. We shall assume;t r c−throughout this paper that the velocities of all the electrons are less than the velocityof radiation.

3. Introduction and Evaluation of the two Scalar Potentials.

Now let denote the position of the electron e at time t; and( ), ( ), ( )x t y t z t′ ′ ′let be used to denote so that are known (.../369)x′ ( ),x t r c′ − , ,x y z′ ′ ′4-2

369THE ELECTROMAGNETIC FIELD DUE TO THE ELECTRONS.

SER. 2. VOL. 1. NO. 843.

{ }1( ) ( ) ( ) ( )x y z

r rr x x x x v y y v z z v

x c x∂ ∂′ ′ ′ ′= − − − − + − + −∂ ∂

( ).

cos( , )

r c x xx cr r v v r

′∂ −=

∂ +

1 2 31

( )( ).

cos( , )x y zx x f v f v f vf

fx cr r v v r

′ − + +∂= − +

∂ +

1 2 3( ).

cos( , )x y zc f v f v f vf

t c v v r

+ +∂=

∂ +

{ }12

1

2 2( , , , ) sinh ,

4 ( ) ( )

e z zF x y z t

x x y yπ− ′ −

=′ ′− + −

∑

1( , , , ) tan ,4

e y yG x y z t

x xπ− ′ −

=′ −∑

{ }12

2 2( , , , ) log ( ) ( ) ,4

ex y z t x x y yψ

π′ ′= − + −∑

,4 cos( , )

,4 cos( , )

1,

4 cos( , )

1 1.

4 cos( , )

x

y

z

vG ex y cr r v v r

vG ey x cr r v v r

vF ez c t cr r v v r

F ecc t z cr r v v r

ψπ

ψπ

ψπ

ψπ

∂ ∂− =

∂ ∂ +

∂ ∂+ =∂ ∂ +

∂ ∂+ =

∂ ∂ +

∂ ∂+ =

∂ ∂ +

∑

∑

∑

∑

functions of when the motions of the electrons are known; we have, , , ,x y z t and therefore2 2 2 2( ) ( ) ( ) ,r x x y y z z′ ′ ′= − + − + −

or

More generally, if f be any function of the there quantitiesand if f1, f2, f3 denote its derivates with respect to these three, , ,x x y y z z′ ′ ′− − −

arguments respectively, we easily find that

Similar equations hold for and while is given by thef y∂ ∂ ,f z∂ ∂ f t∂ ∂equation

Now define functions F, G, R by the equations

where the summation is taken over all the electrons. Using the formulae just obtainedfor the derivates of a function of the kind f, we find that

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370 Mr. E. T. Whittaker [Nov. 12, 1903]

1, , ,x y z

G G Fa a a

x y y x z c tψ ψ ψ∂ ∂ ∂ ∂ ∂ ∂

− = + = + =∂ ∂ ∂ ∂ ∂ ∂

1.

Fc t z

ψϕ

∂ ∂+ = −

∂ ∂

1, ,yx z

x x

aa ad h

c t x y zϕ ∂∂ ∂∂

= − − = −∂ ∂ ∂ ∂

2 2 2 2 2 2

2 2 2

1 1 1, , ;x y z

F G F G F Fd d d

x z c y t y z c x t z c t∂ ∂ ∂ ∂ ∂ ∂

= + = − = −∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

2 2 2 2 2 2

2 2

1 1, , .x y z

F G F G G Gh h h

c y t x z c x t y z x y∂ ∂ ∂ ∂ ∂ ∂

= − = − − = +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

{ }12

1

2 2( , , , ) sinh ,

4 ( ) ( )

e z zF x y z t

x x y yπ− ′ −

=′ ′− + −

∑

1( , , , ) tan ,4

e y yG x y z t

x xπ− ′ −

=′ −∑

1, 0,

rF f t G

r c = − =

Combining these results with the expressions already found for N, ax, ay, az, we have

Substituting these results for N, ax, ay, az in the equations of the type

which give the components of the dielectric displacement and the magnetic force, wefind that R disappears automatically, and we obtain

These equations show that the six components of the dielectric displacementand the magnetic force can be expressed in terms of the derivates of two scalarpotentials F and G, defined by the equations

where the summation is taken over all the electrons in the field.

It can without difficulty be shown that, if any number of electrons whose totalcharge is zero are moving in any manner so as to retain always in the vicinity of agiven point (i.e., to be stationary motion), then the electromagnetic field thusgenerated is of the type given by

where r is the distance from the point and f is an arbitrary function; or, moregenerally, of a field of this type superposed on fields of the same type, but related tothe axes of y and x in the same way as this is related to the axis of z. This is perhapsof some interest in connection with the view advocated by some physicists that theatoms of the chemical elements consist of sets of electrons, whose total charge iszero, in stationary motion.4-4

371THE ELECTROMAGNETIC FIELD DUE TO THE ELECTRONS.

1 1, ,curl curl curl curl curl curl

c c= + = −&&d f g h f g

1grad curl

r =

a

4. Discussion of the Apparent Asymmetry of the preceding resultand its Vector Expression.

The formulae thus obtained are not asymmetrical with respect to x, y, and z.In order to discuss their relation to symmetrical formulae, we observe that they canbe written in the form of vector equations

where d and h are the electric and magnetic vectors, and f and g are vectors directedparallel to the axis of z, whose magnitude are F and G respectively. These vectorequations are quite symmetrical, and our result is that, if, instead of regarding theelectromagnetic field as defined by the vectors d and h, we regard it as defined byvectors f and g, connected with d and h by the above vector equations, then f andg are simple functions of the coordinates of the electrons, whereas d and h arecomplicated functions of their velocities and accelerations; and we have also obtainedthe result that without loss of generality we can take f and g to be everywhere, andat all times, parallel to some fixed direction in space (e.g., the axis of z), a fact whichmakes possible to specify them by two scalar quantities only.

It might be asked whether vectors f and g exist which satisfy the above vectorequations and which are perfectly symmetrical -- the answer to this is in the negative;in fact, although the equations are themselves invariantive, and can therefore beexpressed in the vector notation, yet they do not possess invariant solutions; just asthe vector equation

(where r is the scalar distance from the origin and a is a vector to be determined)possesses an infinite number of solutions a, which can readily be found, but each ofwhich is specially related to some line in space, so that no solution is symmetrical.

5. Deduction of the General Functional Form of an ElectrodynamicDisturbance in the Aether.

Having now shown that an electrodynamic field due to electrons iscompletely characterized by two scalar potential functions F and G, we canproceed to deduce its general functional form.

The functions F and G have singularities at those points which are actuallyoccupied by electrons; at all other points we find by direct (.../372)

4-5

372 Mr. E. T. Whittaker [Nov. 12, 1903]

2 2 2 2

2 2 2 2 2

10,

F F F Fx y z c t

∂ ∂ ∂ ∂+ + − =

∂ ∂ ∂ ∂2 2 2 2

2 2 2 2 2

10.

G G G Gx y z c t

∂ ∂ ∂ ∂+ + − =

∂ ∂ ∂ ∂

2 2 2 2 2 2

2 2 2

1 1 1, , ;x y z

F G F G F Fd d d

x z c y t y z c x t z c t∂ ∂ ∂ ∂ ∂ ∂

= + = − = −∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

2 2 2 2 2 2

2 2

1 1, , .x y z

F G F G G Gh h h

c y t x z c x t y z x y∂ ∂ ∂ ∂ ∂ ∂

= − = − − = +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

2

0 0( sin cos sin sin cos , , ) ,F f x u v y u v z u ct u v du dv

π π= + + +∫ ∫

2

0 0( sin cos sin sin cos , , ) ,G g x u v y u v z u ct u v du dv

π π= + + +∫ ∫

differentiation, or by substituting in the original electrodynamic equations thevalues of the components of dielectric displacement and magnetic force in terms ofF and G, that F and G satisfy the partial differential equations

Writing down the general solution of these latter equations,* we obtain theresult that the most general type of electrodynamic disturbances at a place in theaether not occupied by an electron is that for which the components of thedielectric displacement and magnetic force are represented by the equations

where

and f and g are arbitrary functions of their arguments.

*Cf. A paper by the author in Math. Ann., Vol. LVII., pp. 333-355, 1903.

4-6