208, 209] Images 183 



potential inside this region which satisfies these conditions (cf. 186), so that 

 this value must be that given by equation (124). 



To the right of 8 the potential is the same, whether we have the 

 charge e at A' or the charge on the conducting plane 8. To the left of $ 

 in the latter case there is no electric field. Hence the lines of force, when 

 the plane 8 is a conductor, are entirely to the right of 8, and are the same 

 as in the original field in which the two point-charges were present. The 

 lines end on the plane 8, terminating of course on the charge induced on 8. 



We can find the amount of this induced charge at any part of the plane 

 by Coulomb's Law. Taking the plane to be the plane of yz, and the point A 

 to be the point (a, 0, 0) on the axis of x, we have 



j, 9F 



47TCT = R = 



a 



3 IV(# of + if + z* *J(x + a) 2 

 where the last line has to be calculated at the point on the plane 8 at which 

 we require the density. We must therefore put x after differentiation, 

 and so obtain for the density at the point 0, y, z on the plane 8, 



4}7T(T = 



or, if a 2 4- y 2 + z* = r 2 , so that r is the distance of the point on the plane 8 

 from the point A, 



ae 



~2wT 8 ' 



Thus the surface density falls off inversely as the cube of the distance 

 from the point A. The distribution of electricity on the 

 plane is represented graphically in fig. 58, in which the 

 thickness of the shaded part is proportional to the surface 

 density of electricity. The negative electricity is, so to 

 speak, heaped up near the point A under the influence 

 of the attraction of the charge at A. The field produced 

 by this distribution of electricity on the plane S at any 

 point to the right of 8 is, as we know, exactly the same as 

 would be produced by the point charge e at A'. 



209. This problem affords the simplest illustration of a .021 

 general method for the solution of electrostatic problems, 

 which is known as the " method of images." The principle 

 underlying this method is that of finding a system of electric FlG 58 



charges such that a certain surface, ultimately to be made 

 into a conductor, is caused to coincide with the equipotential F = 0. We 

 then replace the charges inside this equipotential by the Green's equivalent 

 stratum on its surface (cf. 204). As this surface is an equipotential, we 



