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

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Electromotive Force in a Circuit.

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


where 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 and are the currents in A and B, and


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


where is a function of , and , 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 can always, however, be determined in any particular case when we know the actual conditions of the question. The physical interpretation of 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 , move with a velocity whose components are , and let its extremities slide along two parallel conductors with a velocity . 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 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 .

Hence the quantity