558 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 



Let the forces in the field be those due to the circuits A and B, then 

 the electromagnetic momentum of A is 



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



(34). 



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



__ 

 dt dx 



-,, S* (35), 



dt dy 



dt dz 



j 



where / 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 $ can always, however, 

 be determined in any particular case when we know the actual conditions of 

 the question. The physical interpretation of \l> 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 , and let its ex- 



at dt at 



ds 

 tremities slide along two parallel conductors with a velocity -r- . Let us find 



the alteration of the electromagnetic momentum of the circuit of which this 

 arrangement forms a part. 



(i,oc dt] dz 

 In unit of time the moving conductor has travelled distances -j- , - , -5- 



along the directions of the three axes, and at the same time the lengths of 



ds 

 the parallel conductors included in the circuit have each been increased by -7- . 



