Image theory for a sphere with negative permittivity

An image system for a point charge outside a dielectric sphere is presented for all complex values of relative permittivity $ϵ={ϵ}^{\mathrm{\prime }}+i{ϵ}^{\mathrm{\prime }\mathrm{\prime }}$. 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 ${ϵ}^{\mathrm{\prime }}<-1$, 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 $ϵ$ relative to the resonant values $ϵ=-1-1\mathrm{/}n$ for positive integer $n$. The internal potential and dipole sources are also considered.