Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/234

 This discovery seems to have been reserved for Sir W. Thomson, who has developed it into a method of great power for the solution of electrical problems, and at the same time capable of being presented in an elementary geometrical form.

His original investigations, which are contained in the Cambridge and Dublin Mathematical Journal, 1848, are expressed in terms of the ordinary theory of attraction at a distance, and make no use of the method of potentials and of the general theorems of Chapter IV, though they were probably discovered by these methods. Instead, however, of following the method of the author, I shall make free use of the idea of the potential and of equipotential surfaces, when ever the investigation can be rendered more intelligible by such means.

Theory of Electric Images.

156.] Let $$A$$ and $$B$$, Figure 7, represent two points in a uniform dielectric medium of infinite extent. Let the charges of $$A$$ and $$B$$ be $$e_1$$ and $$e_2$$ respectively. Let $$P$$ be any point in space whose distances from $$A$$ and $$B$$ are $$r_1$$ and $$r_2$$ respectively. Then the value of the potential at $$P$$ will be

$V=\frac{e_{1}}{r_{1}}+\frac{e_{2}}{r_{2}}.$|undefined

The equipotential surfaces due to this distribution of electricity are represented in Fig. I (at the end of this volume) when $$e_1$$ and $$e_2$$ are of the same sign, and in Fig. II when they are of opposite signs. We have now to consider that surface for which $$V = 0$$, which is the only spherical surface in the system. When $$e_1$$ and $$e_2$$ are of the same sign, this surface is entirely at an infinite distance, but when they are of opposite signs there is a plane or spherical surface at a finite distance for which the potential is zero.

The equation of this surface is

Its centre is at a point C in AB produced, such that

$AC:BC::e_{1}^{2}:e_{2}^{2},$

and the radius of the sphere is

$AB\frac{e_{1}e_{2}}{e_{1}^{2}-e_{2}^{2}}.$|undefined

The two points A and B are inverse points with respect to this