The electrostatic problem of two conducting spheres

Abstract

In continuation of his previous paper on the spherical condenser † the author discusses the capacities, the mutual actions and the field in the neighbourhood of two electrified spherical conductors. After a brief introductory statement of this historical problem an attempt is made to solve it by elementary theory. The magnitudes of the errors introduced by this method are pointed out. Accurate formulæ are then obtained by the method described in the preceding paper. These formulæ enable the capacity coefficients to be computed in all cases to any required degree of accuracy without difficulty. A few simplified formulæ which will be useful in practice are also given. The discussion of the mutual actions between the spheres shows that when their potentials or charges are of like sign the force between them may be attractive or repulsive, depending on the ratio of the potentials or the charges. In this case there are always two ratios of the charges and of the potentials which make the mutual force zero. Methods of computing the forces in other cases are given. Simple formulæ are found for calculating the potential gradients at the points on the spheres where the line joining their centres cuts their surfaces. This tells us whether the charge on the spheres is all positive or all negative or partly positive and partly negative. In the Appendix a few more accurate but more elaborate formulæ are given.