Longitudinal and transverse polarizability of the conducting double sphere

Abstract

Exact expressions are derived for the longitudinal and transverse polarizability of two overlapping conducting spheres of arbitrary radii and with arbitrary angle of intersection. The transverse polarizability is expressed as a single integral, which can be performed if the angle of intersection is a rational fraction of π, i.e., the angle of intersection can be expressed as mπ/n, with m and n integers. The longitudinal polarizability can be expressed as a single integral if the two spheres are equal. For unequal spheres it involves two integrals, as well as the capacity, which itself was expressed as a single integral earlier. For equal spheres the second integral vanishes by symmetry, and the capacity is not needed. Both integrals can be performed if the angle of intersection is a rational fraction of π. In earlier work by the authors the longitudinal and transverse polarizability were found only for discrete angles of intersection π/n with n integer. Our result for the longitudinal polarizability of two equal overlapping conducting spheres shows that an earlier result of Radchik et al. [J. Appl. Phys. 76, 4827 (1994)] for overlapping dielectric spheres is incorrect.