458 Prof. D. X. Mallik on Magnetic 



5. Ex. To find the solid angle subtended by a circular 

 wire at any point P, in zonal harmonies (spherical). 



Describe a sphere having the plane of the given circle 

 as one o£ its plane sections, and take its centre C as the 

 origin of coordinates. 



Let the axis of z be the axis of the zonal harmonics, and 

 a = angle subtended by the radius of the circle at the centre. 



Also, let v = angle between any radius (6, </>) and CP( = r) 

 and d$, the corresponding element of surface. 



Then 





dsl 



dc V + >- -2 re cos v)*' 



Ceos a f*2 



J i Jo 



, i£ r < c, 



But 



d/jid(pS(n-\-l)—. P n (cosv), 



where /jl= cos 0. 



m-n ( n m \ | 



P„(co S v)=P„0)P„0')+ 22 — — r j.Tr&.')T?(f)oosm(#- 



where T's are Tesseral harmonics. 



(*eoa a r n 



.-. a> = 27T 2(n 4-1)- • P„(/a)P„0')^ 



[1 r w r <iP -! cosa "i 



Similarly, when r > c. 



Ota. It is clear that this method will enable us to find 

 the potential due to a circular current in any system of 

 harmonics, provided we choose the equivalent shell appropriate 

 to these harmonics. 



6. To find the solid angle subtended at any point P by a 

 circular wire in spheroidal harmonics (zonal). 



Let the axis of the zonal harmonics be the axis of revolution 

 of a spheroid having the plane of the circle as one of its 

 plane-sections, and centre, the origin. 



Then, r , -, . 



But from the theory of confocals, since r = constant is a 



