116 Induction of Electric Currents in Infinite Plates. [Jan. 29, 



When the shell is infinitely thin, the action of the induced currents 

 is expressed by the following statement, which, it will be seen, is an 

 extension of Maxwell's result for a plane sheet. Divide the time into 

 an infinite number of equal intervals, and at the commencement of 

 any of these let a positive image of the external system be formed at 

 the place occupied by its electric image at the surface. Let the parts 

 of this image move towards the centre in straight lines, so that the 

 logarithmic decrement of their distances from the centre is constant 

 R 



and equal to (R being the resistance of the shell and a its 



radius), and let the intensity of the image increase at each point with 



R 



a constant logarithmic rate At the end of the interval let an 



exactly equal but negative image be formed in the place of the former 

 and move toward the centre in the same manner, and let these opera- 

 tions be repeated at the beginning and end of every interval : the 

 action of the sheet on external points will be that due to the above 

 train of images. 



Induction in a Rotating Conductor. 



When the conductor rotates uniformly about an axis of symmetry 

 the currents will become, after a time, steady. In the case of an infi- 

 nite plate of finite thickness, the vector potential and currents will have 

 the same general forms as before, but the electric potential will not be 

 zero. The solution is now expressed by taking 



dp d(p 4>7T 



ffx = w (P + P ), fv 2 P=^(P + P ), 



4.7T d(p 



(w being the angular velocity) . 



When the inducing magnetism lies all on the positive side of the 

 plate, we may express P within it in the form 



2 A cos m(£>J m (Kp)e KZ , 



the origin being on the positive face. The value of P at any external 

 point on the positive side may be expressed by 



P = 20 cosm0J m (^)e-^, 



, C k 3 — w 3 e fxb_ e -^b 27rmu . 



where — = — . — — — — — , h*=k*-— — . i. 



A 2k K{ef J - b -\-e~i J - b )—p:(e^—e- b ) a 



