The Potential 



r^\r - 



this 



The potential V at any point in the nnedium must satisfy Laplace's Equation V V = 0. Expanded, 



is f \y + *: Y + f y = in Cartesian coordinates. The equivalent expression in spherical 



coordinates can'TDe found from the transformation equations between the two coordinate systems: 



r = [x^+y2+z^]'^ e = tan-'?5!±yi ^ <f -_ tan"' | . 



The result gives 





(1) 



We have already seen that V does not depend on (f . The last term, therefore, will be zero. The 

 resulting equation is a partial differential equation for V in terms of r and 9. 



The general solution of this equation is well known (MacRobert 1948) and is given in terms of the 

 sum of an infinite series: 



V= C,[Po + rP, + r'P2+r'P3+ ] 



(2) 



Ci and C2 are arbitrary constants that must be chosen so as to satisfy boundary conditions. The 



functions Pg, Pj, P21 ^^re functions of the angle 6 known as Legendre Polynonnials. They are 



defined as follows (Jahnke and Emde 1945): 



When u = cos6, the first few polynomials are 



Pq = I , p, = COS e , P2 = -^ ( 3 COS e + I ) 



P, = -5 (5 COS 30+ 3cose ), p."" Jz (35cos4e + 20cos2e+9) 



(3) 



8 



64 



I 



P5 = T^ (63cos5e + 35cos3e + 30cose ) 



(4) 



17 



