Forced Vibrations of Electromagnetic Systems. 397 



as the displacement through v due to unit vortex line at v v 

 Applying this result to a circular electric current, B=//, H 

 takes the place of D = (c/47r)E, as the flux concerned, whilst 

 if h be the strength of the shell of impressed magnetic force, 

 A/47T is the equivalent bounding electric current. The in- 

 duction through the circle v due to unit electric current in 

 the circle v x is therefore obtainable from (179) by turning c 

 to fi and multiplying by (47r) 2 . The result agrees with 

 Maxwell's formula for the coefficient of mutual induction of 

 two circles (vol. ii. art. 697). 



It must be noted that in the magnetic-shell application 

 there must be no conductivity, if the wave-formulse are to 

 apply. 



26. An Electromotive Impulse. m=l. — Returning to the 

 case of impressed electric force, let in a spherical portion of 

 an infinite dielectric a uniform field of impressed force act 

 momentarily. We know the result of the continued applica- 

 tion of the force. "We have, then, to imagine it cancelled by 

 an oppositely directed force, starting a little later. Let t x be 

 the time of application of the real force, and let it be a small 

 fraction of 2a/v, the time the travelling shell takes to traverse 

 any point. The result is evidently a shell of depth vt x at 

 r = vt + a, in which the electromagnetic field is the same as in 

 the case of continued application of the force, and a similar 

 shell situated at r=vt—a, in which H is negative. Within 

 this inner shell there is no E or H. But between the two 

 thin shells just mentioned there is a diffused disturbance, of 

 weak intensity, which is due to the sphericity of the waves, 

 and would be non-existent were they plane waves. In fact, 

 at time t — t x , when the initial disturbance H=/ 1 v/2/* u has 

 extended itself a small distance vt x on each side of the surface 

 of the sphere, there is a radial component F at the surface 

 itself, since, by (150), 



IW.cos^-g), . . . (180) 



so that the sudden removal of / x leaves two waves which do 

 not satisfy the condition E = fjo vR at their common surface of 

 contact. On separation, therefore, there must be a residual 

 disturbance between them. The discontinuity in B at the 

 moment of removing f x is abolished by instantaneous assump- 

 tion of the mean value, but it is impossible to destroy the 

 radial displacement which joins the two shells at the moment 

 they separate. Put on / x when t = 0, then — f x at time t x 

 later. The H at time t due to both is by (142), 



