The capacity coefficients of spherical conductors

Abstract

Maxwell pointed out that the self-capacity coefficient of a conductor is numerically equal to its charge when its potential is unity and all neighbour­ing conductors are at zero potential. He considered that the “proper definition of the capacity of a conductor” is to define it as being equal to the self-capacity coefficient. Adopting this definition, we may consider that the self-capacity coefficient is the capacity of the condenser formed by the conductor, on the one side, and, on the other side, all neighbouring conductors connected with the earth. This gives a simple physical meaning to the self­ capacity coefficient, and in one or two simple cases it enables us to compute its value. In the case of a spherical conductor, however, we can give an equally simple way of regarding it, which leads to easier methods of com­puting its value. As a knowledge of the self-capacity coefficients of spheres is essential in certain practical problems, for instance, when computing the electric stress at which a spark will occur between unequal spherical electrodes when the dielectric between them is at a given temperature and pressure, simplified methods of finding their values are useful. It is proved below that the self-capacity coefficient of a spherical conductor equals its radius together with the capacity of the condenser formed between the surface of the sphere and the images in the sphere of all external conductors, including the earth connected in parallel. Many of the formulæ given by the author in his papers are connected by very simple relations. For brevity, we shall refer to these papers as X and Y respectively. The approximate formulæ given in Y for spherical con­densers can be usefully employed for computing the capacity coefficients for external spheres, and, conversely, we can use the tables given by Kelvin and in X, p. 529, for computing the values of the capacities of spherical condensers.