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′ + iϵ″. 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′ < −1, and a correction is proposed for this case, involving image multipoles of infinite magnitude which regularize the divergent line integral. The number of these multipoles depends on the position of ϵ relative to the resonant values ϵ = −1 − 1/n for positive integer n. The internal potential and dipole sources are also considered.An image system for a point charge outside a dielectric sphere is presented for all complex values of relative permittivity ϵ = ϵ′ + iϵ″. 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′ < −1, and a correction is proposed for this case, involving image multipoles of infinite magnitude which regularize the divergent line integral. The number of these multipoles depends on the position of ϵ relative to the resonant values ϵ = −1 − 1/n for positive integer n. The internal potential and dipole sources are also considered.