An Electrostatics Problem on the Sphere Arising from a Nearby Point Charge

TL;DR

This paper studies how a nearby point charge influences the equilibrium charge distribution on a sphere under Riesz potential, revealing conditions where negative charges appear on spherical caps and exploring related polynomial zero properties.

Contribution

It introduces a detailed analysis of charge distributions on spheres affected by nearby charges under Riesz potentials, linking to polynomial zero properties.

Findings

Identification of critical distances causing negative charge regions

Characterization of equilibrium distributions under Riesz potentials

Connection to special polynomials with unique zero distributions

Abstract

For a positively charged insulated d-dimensional sphere we investigate how the distribution of this charge is affected by proximity to a nearby positive or negative point charge when the system is governed by a Riesz s-potential 1/r^s, s>0, where r denotes Euclidean distance between point charges. Of particular interest are those distances from the point charge to the sphere for which the equilibrium charge distribution is no longer supported on the whole of the sphere (i.e. spherical caps of negative charge appear). Arising from this problem attributed to A. A. Gonchar are sequences of polynomials of a complex variable that have some fascinating properties regarding their zeros.