# Image theory for a sphere with negative permittivity

## TL;DR

This paper develops a comprehensive image system for a point charge near a dielectric sphere with negative permittivity, addressing divergence issues and introducing multipole corrections for all complex permittivity values.

## Contribution

It introduces a corrected image method for negative permittivity values, including multipole solutions to regularize divergences in the classical approach.

## Key findings

- Standard image integral diverges for ε'<-1
- Proposed correction involves infinite multipoles
- Solution depends on ε relative to resonant values

## Abstract

An image system for a point charge outside a dielectric sphere is presented for all complex values of relative permittivity $\epsilon=\epsilon'+i\epsilon''$. The standard image integral solution of a point charge outside a dielectric sphere involving an image point charge plus a line source is shown to diverge for $\epsilon'<-1$, and a correction is proposed for this case, involving image multipoles of infinite magnitude that regularise the divergent line integral. The number of these multipoles depends on the position of $\epsilon$ relative to the resonant values $\epsilon=-1-1/n$ for positive integer $n$. The internal potential and dipole sources are also considered.