Deriving the method of images for conductors

TL;DR

This paper develops a systematic approach to derive the method of images for potentials outside conductors by relating it to the known inside conductor potentials, extending the technique to multiple dimensions.

Contribution

It introduces a new framework linking inside and outside conductor potentials, enabling derivation of image charges for outside potentials from inside solutions.

Findings

Derived a relation between inside and outside conductor potentials.

Extended the image method to n-dimensional spaces.

Provided a systematic way to determine image charge configurations.

Abstract

We start from the simple fact that the method of images can always be used to obtain {\Phi}_{\sigma} (r^\to_in), which is the potential inside conductor produced by induced surface charges. We use this fact to construct image method for outside potential {\Phi}_{\sigma} (r^\to_out). We show that if we can find a relation between {\Phi}_{\sigma} (r^\to_out) and {\Phi}_{\sigma} (r^\to_in), then the image method for {\Phi}_{\sigma} (r^\to_in) can be used to derive the image method for {\Phi}_{\sigma} (r^\toout). The number, the position and the amount of charge of the images can be directly derived. The discussion can be extended to the general n dimensions.