Kelvin transformation and inverse multipoles in electrostatics

TL;DR

This paper explores the Kelvin transformation in electrostatics, demonstrating how it relates different problems and allows solutions of complex configurations by mapping from simpler symmetric systems, with a novel perspective on multipole expansion.

Contribution

It introduces a new viewpoint on multipole expansion using Kelvin transformation, enabling solutions of complex electrostatic problems through mappings from symmetric systems.

Findings

Exact solutions for nontrivial electrostatic problems obtained via Kelvin transformation.

Revised understanding of multipole expansion with sources far from the origin.

Demonstrated duality between different electrostatic configurations.

Abstract

The inversion in the sphere or Kelvin transformation, which exchanges the radial coordinate for its inverse, is used as a guide to relate distinct electrostatic problems with dual features. The exact solution of some nontrivial problems are obtained through the mapping from simple highly symmetric systems. In particular, the concept of multipole expansion is revisited from a point of view opposed to the usual one: the sources are distributed in a region far from the origin while the electrostatic potential is described at points close to it.