§ 6. Transformation of Maxwell’s equations for Pure Vacuum.
On the nature of the Electromotive Force caused by motion in a magnetic
field.
The Maxwell-Hertz equations for pure vacuum may hold for the stationary
system K, so that
$$ \frac {1}{c} \frac {\partial}{\partial t} [X, Y, Z] = \begin{vmatrix}
\frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac
{\partial}{\partial z} L & M & N \end{vmatrix} $$
and
$$ \frac {1}{c} \frac {\partial}{\partial t} [L, M, N] = \begin{vmatrix}
\frac {\partial}{\partial x} & \frac {\partial}{\partial y} & \frac
{\partial}{\partial z} X & Y & Z \end{vmatrix} $$ (1)
where [X, Y, Z] are the components of the electric force, L, M, N are
the components of the magnetic force.
If we apply the transformations in §3 to these equations, and if we
refer the electromagnetic processes to the co-ordinate system moving
with velocity _v_, we obtain,
$$ \frac {1}{c} \frac {\partial}{\partial \tau} [X, \beta (Y - \frac
{v}{c} N), \beta (Z + \frac {v}{c} M)] = \begin{vmatrix} \frac
{\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac
{\partial}{\partial \zeta} L & \beta(M + \frac {v}{c} Z) & \beta(N -
\frac {v}{c} Y) \end{vmatrix}
and
$$ \frac {1}{c} \frac {\partial}{\partial \tau} [L, \beta(M + \frac
{v}{c} Z), \beta(N - \frac {v}{c} Y)] = - \begin{vmatrix} \frac
{\partial}{\partial \xi} & \frac {\partial}{\partial \eta} & \frac
{\partial}{\partial \zeta} X & \beta(Y - \frac {v}{c} N) & \beta(Z +
\frac {v}{c} M) \end{vmatrix} $$ ... (2)
where
$$ \beta = \frac {1}{\sqrt {1 - \frac {v^2}{c^2}}} $$
The principle of Relativity requires that the Maxwell-Hertzian equations
for pure vacuum shall hold also for the system k, if they hold for the
system K, _i.e._, for the vectors of the electric and magnetic forces
acting upon electric and magnetic masses in the moving system k, which
are defined by their pondermotive reaction, the same equations hold, ...
_i.e._ ...
$$ \frac {1}{c} \frac {\partial}{\partial \tau} (X', Y', Z')
= \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac
{\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} L' & M' &
N' \end{vmatrix} $$ ,
$$ \frac {1}{c} \frac {\partial}{\partial \tau} (L', M', N') =
- \begin{vmatrix} \frac {\partial}{\partial \xi} & \frac
{\partial}{\partial \eta} & \frac {\partial}{\partial \zeta} X' & Y' &
Z' \end{vmatrix} $$ ... (3)
Clearly both the systems of equations (2) and (3) developed for the
system k shall express the same things, for both of these systems are
equivalent to the Maxwell-Hertzian equations for the system K. Since
both the systems of equations (2) and (3) agree up to the symbols
representing the vectors, it follows that the functions occurring at
corresponding places will agree up to a certain factor ψ(_v_), which
depends only on _v_, and is independent of (ξ, η, ζ, τ). Hence the
relations,
_v_ _v_
[X′, Y′, Z′] = ψ (_v_) [X, β(Y - ----- N), β(Z + ------ M)],
_c_ _c_
_v_ _v_
[L′, M′, N′] = ψ (_v_) [L, β(M - ----- Z), β(N + ----- Y)],
_c_ _c_
Then by reasoning similar to that followed in §(3), it can be shown that
ψ(_v_) = 1.
_v_ _v_
[X′, Y′, Z′] = [X, β(Y - ----- N), β(Z + ------ M)]
_c_ _c_
_v_ _v_
[L′, M′, N′] = [L, β(M - ------ Z), β(N + ----- Y)],
_c_ _c_
For the interpretation of these equations, we make the following
remarks. Let us have a point-mass of electricity which is of magnitude
unity in the stationary system K, _i.e._, it exerts a unit force upon a
similar quantity placed at a distance of 1 cm. If this quantity of
electricity be at rest in the stationary system, then the force acting
upon it is equivalent to the vector (X, Y, Z) of electric force. But if
the quantity of electricity be at rest relative to the moving system (at
least for the moment considered), then the force acting upon it, and
measured in the moving system is equivalent to the vector (X′, Y′, Z′).
The first three of equations (1), (2), (3), can be expressed in the
following way:—
1. If a point-mass of electric unit pole moves in an electro-magnetic
field, then besides the electric force, an electromotive force acts upon
it, which, neglecting the numbers involving the second and higher powers
of _v_/_c_, is equivalent to the vector-product of the velocity vector,
and the magnetic force divided by the velocity of light (Old mode of
expression).
2. If a point-mass of electric unit pole moves in an electro-magnetic
field, then the force acting upon it is equivalent to the electric force
existing at the position of the unit pole, which we obtain by the
transformation of the field to a co-ordinate system which is at rest
relative to the electric unit pole [New mode of expression].
Similar theorems hold with reference to the magnetic force. We see that
in the theory developed the electro-magnetic force plays the part of an
auxiliary concept, which owes its introduction in theory to the
circumstance that the electric and magnetic forces possess no existence
independent of the nature of motion of the co-ordinate system.
It is further clear that the asymmetry mentioned in the introduction
which occurs when we treat of the current excited by the relative motion
of a magnet and a conductor disappears. Also the question about the seat
of electromagnetic energy is seen to be without any meaning.
§ 7. Theory of Döppler’s Principle and Aberration.
In the system K, at a great distance from the origin of co-ordinates,
let there be a source of electrodynamic waves, which is represented with
sufficient approximation in a part of space not containing the origin,
by the equations:—
X = X₀ sin Φ
Y = Y₀ sin Φ
Z = Z₀ sin Φ
L = L₀ sin Φ
M = M₀ sin Φ
N = N₀ sin Φ
lx + my + nz
Φ = ω(t - ------------ )
c
Here (X₀, Y₀, Z₀) and (L₀, M₀, N₀) are the vectors which determine the
amplitudes of the train of waves, (_l_, _m_, _n_) are the
direction-cosines of the wave-normal.
Let us now ask ourselves about the composition of these waves, when they
are investigated by an observer at rest in a moving medium _k_:—By
applying the equations of transformation obtained in §6 for the electric
and magnetic forces, and the equations of transformation obtained in § 3
for the co-ordinates, and time, we obtain immediately:—