Forced Vibrations of Electromagnetic Systems. 401 



In the case m=l, the waste of energy per second is 



(189) 



(/i)W* 



3p v 



due to the uniform alternating field of impressed force of 



intensity . „, 



J (/i) cos nt 



within the sphere. 



In reality, the impressed force must have been an infinitely 

 long time in operation to make the above solutions true to an 

 infinite distance, and have therefore already wasted an infinite 

 amount of energy. If the impressed force has been in operation 

 any finite time t, however great, the disturbance has only 

 reached the distance r = vt + a. Of course the solutions are 

 true, provided we do not go further than r = vt — a. We see, 

 therefore, that the real function of the never-ceasing waste of 

 energy is to set up the sinusoidal state of E and H in the 

 boundless regions of space to which the disturbances have not 

 yet reached. The above outward waves are the same as in 

 Rowland's solutions*. Here, however, they are explicitly 

 expressed in terms of the impressed forces causing them. 



w ffl =0 makes the external field vanish when m is odd ; and 

 w a =0 when m is even ; that is, when the sinusoidal state has 

 been assumed. It takes only the time 2a/v to do this, as regards 

 the sphere r=a ; the initial external disturbance goes out to 

 infinity and is lost. This vanishing of the external field 

 happens whatever may be the nature of the external medium 

 away from the sphere, except that the initial external disturb- 

 ance will behave differently, being variously reflected or 

 absorbed according to circumstances. 



28. Conducting Medium. m = l. — Now consider the same 

 problem in an infinitely extended conductor of conductivity k. 

 We may remark at once that, unless the conductivity is low, 

 the solution is but little different from what it would be were 

 the conductor not greatly larger than the spherical portion 

 within it on whose surface lie the vortex lines of the impressed 

 force, owing to the great attenuation suffered by the dis- 

 turbances as they progress from the surface. In a similar 

 manner, if the sphere be large, or the frequency of alternations 

 great, or both, we may remove the greater part of the interior 

 of the sphere without much altering matters. 



We have now 



q — (4tt/x %>) i = (1 + i)a, 

 if 



x = (2irfjL knf (190) 



* In paper referred to in § 18. 



