Relationships among charges, fields, and potential on spherical surfaces boundary value problems

Abstract

This paper suggests a new point of view for the Poisson equation and its solution for the potential and field on the d dimensional sphere, Sd, on which point charges are distributed. The available solutions for the potential on multidimensional spheres in the literature are purely mathematical, while the solution suggested here is motivated by physical intuition and requires minimal background; namely, basic laws of electrostatics and dimensional analysis. In this study, the modified Coulomb’s law is presented by means of “dimensional reduction” and the use of equivalence between a point charge on Sd and a charged ray in Rd+1.Besides formal detailed solutions and theorems, this paper presents concrete physical examples (unstudied or studied partly), such as distribution of charges/sources on a two-sphere; Dirichlet problem for currents on a truncated sphere; and fields and potentials created by “infinite” cones. Well-known statements about special charge distributions in Euclidean space must be reformulated and amended when dealing with the case of charges embedded in a spherical manifold.