8 Prof. J. Larmor on Electromagnetic Induction 



The equations which give the currents become 



a d® _ d <P + P n ) d^r_ 



r sin 0d(j> dt sin dd<f> rdd' 



rdd dt dd r sin 0d<f>' 



U ~ dr' 



where o- is the specific resistance of the medium. 

 These are satisfied, as before, by ijr = 0, 



But 



therefore 



r dt K ■ + r o> 



V 2 P=-^ ; 



7* 



VT=^|(P+P„), (3) 



which determines P to satisfy all the conditions of the pro- 

 blem. The boundary conditions are simply that the current 

 shall not become infinite at the surface, and 'therefore that the 

 distribution <D shall not have a finite surface-density,- this 



requires that — as well as P shall be continuous at the 



boundary. 



To obtain a solution, let us assume a varying external mag- 

 netic field given by © 



P„ = A fi<*tr n 'Y ' 



,i o — ■ a -o e ' x »> 



then 



P= <>«%,, 



wefind Q ' S a functi ° n ° {r; and sub stituting in equation (3), 



cPQ 2dQ »(n + l) 4tt ,. . s 



therefore 



<PQ 2dQ f n(n + l)\„ 



where 



47T/C 



cr 



