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



can imagine it to be replaced by a conductor and the charges on it will be 

 in equilibrium. These charges now become charges induced on a conductor 

 .at potential zero by charges outside this conductor. 



From the analogy with optical images in a mirror, the system of point 

 -charges which have to be combined with the original charges to produce zero 

 potential over a conductor are spoken of as the " electrical images " of the 

 original charges. For instance, in the example already discussed, the field is 

 produced partly by the charge at A, partly by the charge induced on the 

 infinite plane : the method of images enables us to replace the whole charge 

 induced on the plane by a single point charge at A'. So also, if A were a 

 candle placed in front of an infinite plane mirror, the illumination in front of 

 the mirror would be produced partly by the candle at A, partly by the light 

 reflected from the infinite mirror ; the method of optical images enables us to 

 replace the whole of this reflected light by the light from a single source at A'. 



210. In an electrostatic field produced by any number of point charges, 

 we can, as we have seen, select any equipotential and replace it by a con- 

 ductor. The charges on either side of this equipotential are then the 

 " images " of those on the other side. 



Thus if we can write the equation of any surface in the form 



.(125), 



e e e A 



- 4- -. + -T> + . . . = , 

 r r r 



where r is the distance from a point outside the surface, and r, r", . . . are the 

 distances from points inside the surface, then we may say that charges 

 e' ', e", ... at these latter points are the images of a charge e at the former 

 point. 



Charges induced on Intersecting planes. 



211. It will be found that charges 

 e at a, 1 , y, 0, 



e at a;, y, 0, 



e at #, y, 0, 

 e at x, y, 



give zero potential over the planes #=0, y = 0. 

 The potential of these charges is therefore the 

 same, in the quadrant in which x, y are both 

 positive, as if the boundary of this quadrant 

 were a conductor put to earth under the in- 

 fluence of a charge e at the point x, y, 0. 



It will be found that a conductor consisting 

 of three planes intersecting at right angles can 

 be treated in the same way. 



FIG. 59. 



