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



Conversely, if we know the distribution induced on a conductor PQ... at 

 potential zero by a unit charge at a point 0, then by inversion about we 

 obtain the distribution on the inverse conductor P'Q'... when raised to 



potential ^ . As before, the ratio of the densities is given by equation (132). 



Examples of Inversion. 



230. Sphere. The simplest electrical problem of which we know the 

 solution is that of a sphere raised to a given potential. Let us examine 

 what this solution becomes on inversion. 



If we invert with respect to a point P outside the sphere, we obtain the 

 distribution on another sphere when put to earth under the influence of a 

 point charge P. This distribution has already been obtained in 214 by 

 the method of images. The result there obtained, that the surface-density 

 varies inversely as the cube of the distance from P, can now be seen at once 

 from equation (132). 



So also, if P is inside the sphere, we obtain the distribution on an 

 uninsulated sphere produced by a point charge inside it, a result which can 

 again be obtained by the method of images. 



When P is on the sphere, we obtain the distribution on an uninsulated 

 plane, already obtained in 208. 



231. Intersecting Planes. As a more complicated example of inversion, 

 let us invert the results obtained in 212. We there shewed how to find 



FIG. 71. 



the distribution on two planes cutting at an angle , when put to earth 



under the influence of a point charge anywhere in the acute angle between 

 them. If we invert the solution we obtain the distribution on two spheres, 

 cutting at an angle ir/n, raised to a given potential. By a suitable choice 



