A note on fractional calculus and the image method for dielectric spheres

Abstract

In this note, we present an alternative mathematical way of expressing electrostatic image charges for dielectric spheres using the concept of fractional calculus. It is shown that the distributions of image charges for the dielectric sphere (Lindell's solution) and the image charges for the perfectly conducting ungrounded spheres (Lord Kelvin's solution) can be mathematically related via fractional differintegration, and that the order of such fractional operators depend of the dielectric property of the sphere, i.e., its dielectric constant.