Page:A Dynamical Theory of the Electromagnetic Field.pdf/25

 Rh Electromotive Force in a Circuit.

(63) Let $$\xi$$ be the electromotive force acting round the circuit A, then

where $$ds$$ is the element of length, and the integration is performed round the circuit.

Let the forces in the field be those due to the circuits A and B, then the electromagnetic momentum of A is

where $$u$$ and $$v$$ are the currents in A and B, and

Hence, if there is no motion of the circuit A,

{{MathForm2|(35)|$$\left.\begin{array}{l} P=-\frac{dF}{dt}-\frac{d\Psi}{dx},\\ \\Q=-\frac{dG}{dt}-\frac{d\Psi}{dy},\\ \\R=-\frac{dH}{dt}-\frac{d\Psi}{dz}.\end{array}\right\} $$}}

where $$\Psi$$ is a function of $$x, y, z$$, and $$t$$, which is indeterminate as far as regards the solution of the above equations, because the terms depending on it will disappear on integrating round the circuit. The quantity $$\Psi$$ can always, however, be determined in any particular case when we know the actual conditions of the question. The physical interpretation of $$\Psi$$ is, that it represents the electric potential at each point of space.

Electromotive Force on a Moving Conductor.

(64) Let a short straight conductor of length a, parallel to the axis of $$x$$, move with a velocity whose components are $$\tfrac{dx}{dt},\tfrac{dy}{dt},\tfrac{dz}{dt}$$, and let its extremities slide along two parallel conductors with a velocity $$\tfrac{ds}{dt}$$. Let us find the alteration of the electromagnetic momentum of the circuit of which this arrangement forms a part.

In unit of time the moving conductor has travelled distances $$\tfrac{dx}{dt},\tfrac{dy}{dt},\tfrac{dz}{dt}$$ along the directions of the three axes, and at the same time the lengths of the parallel conductors included in the circuit have each been increased by $$\tfrac{ds}{dt}$$.

Hence the quantity

$\int\left(F\frac{dx}{ds}+G\frac{dy}{ds}+H\frac{dz}{ds}\right)ds$