186 Methods for the Solution of Special Problems [CH. vm 



If R, Q are any two inverse points in the sphere, and P any point on the 

 surface, we have 



RP:PQ = OC :OQ, 



so that 



00 

 Thus the image of a charge e at Q is a charge ^TTTT at R, or the 



image of any point at a distance / from the centre of a sphere of radius a 

 is a charge -. at the inverse point, i.e. at a point on the same radius 



a 2 

 distant -? from the centre. 



Let us take polar coordinates, having the centre of the sphere for origin 

 and the line OQ as = 0. Our result is that at any point 8 outside the 

 sphere, the potential due to a charge e at Q and the charge induced on the 

 surface of the sphere, supposed put to earth, is 



ea 



QS RS 



ea 



Vr 2 +/ 2 -2/rcos0 , / 9 a* . a 2 , 



r ~ r cos 



2 



where r, are the coordinates of 



214. We can now find the surface-density of the induced charge. For 

 at any point on the sphere 



_R__ J^8F 



~47r~ 4-7T dr ' 



in which we have to put r- = a after differentiation. Clearly 



_ T7 - x /. m \r 2? cos 

 dV e(r-fcos0) y / 



a 



-/. O 



/ ( r 2 + - 2 , r cos 



a 4 



- , 



Putting r = a we obtain 



afcosO a 2 / 2 -a 3 / cos 



y 2 _ 2 /a cos ^)* (a 2 / 2 + a 4 - 2a 3 /cos 



a -/ 2 /a 



e (/ 2 -a 2 ) 

 47T 



