50* Magnetic Induction in Spheroids 



4 ■ 



By Pkof. D. N. Mallik. 



The present paper deals with the problem of magnetic induc- 

 ition iu a magnetic substance iu the form of a prolate spheroid, 

 due to a current circulating in a wire wrapped round it along a 

 part of its length. 



1. The first step is to solve the equation A^V=0 in spheroi- 

 dal harmonics in the usual way. 

 For this. 



let X = h\/r^ — 1 sin 5 cos <p 



y=h\/r'^ — l sin 5 sin ^ 

 z = h r cos 6. 



Then, since — r h f- = 1, 



IfiK:,"^ -\y h\r^ -\) li'^r'*' ' 



vr = constant is a net of confocal prolate spheroids. 

 Now — = ^ig^-j— dr* + (r« - cos* e)d9»+{r*-l) sin' 6 df 



.-. transforniiug ^^V to r, B, <p, co-ordinates, we have 



_8 p^-cos^g) (r»- 1) _^i^d "1 ' SF 



8f\_ T^ — cosW J 8 



+ ^/l 1 r^9 iVTTi" 9 i3v i 



^^ L (r«-l) (r«-cos»6') J S5 



a r^-cos,^6 f«-co«»d -|' SF „ 



+ - I — :: — —- . T-^ — - ■ . .. . I — = 





g^L r»-l ' (r« - 1 ) sini* e J S^ 



8 . . ,. 8F 1 8 / . „ 8F\ r^-cos^e S^V 



^^" 8r 



/, ,xSF 18/. . SF\ . »*-co8'^ 8«F ^ 



If V is independent of <p, we have simply, 



8 , , , SF 1 8 / . « SF, ^ 



