Subtleties on energy calculations in the image method

TL;DR

This paper highlights a common mistake in calculating electrostatic energy with the image method, showing that naive formulas are off by a factor of 1/2, and provides corrected general results for various conductor shapes.

Contribution

It identifies a subtle error in electrostatic energy calculations using the image method and offers corrected formulas applicable to arbitrary-shaped conductors.

Findings

Naive image method formulas are incorrect by a factor of 1/2

Corrected energy expressions are derived for arbitrary conductors

Examples include point charges near conducting walls and spheres

Abstract

In this pedagogical work we point out a subtle mistake that can be done by undergraduate or graduate students in the computation of the electrostatic energy of a system containing charges and perfect conductors if they naively use the image method. Specifically, we show that the naive expressions for the electrostatic energy for these systems obtained directly from the image method are wrong by a factor 1/2. We start our discussion with well known examples, namely, point charge-perfectly conducting wall and point charge-perfectly conducting sphere and then proceed to the demonstration of general results, valid for conductors of arbitrary shapes.