Exact closed-form and asymptotic expressions for the electrostatic force between two conducting spheres

TL;DR

This paper derives exact and asymptotic formulas for the electrostatic force between two conducting spheres of arbitrary sizes, revealing conditions under which like-charged spheres attract or repel, and how to maximize repulsion.

Contribution

It provides the first complete closed-form and asymptotic expressions for the electrostatic force between two conducting spheres of any size ratio, including novel insights into force behavior at different separations.

Findings

Like-charged spheres attract at small separations unless voltages are equal at contact.

For large size asymmetries, repulsion increases with separation when charges are constant.

Optimal voltage adjustments can maximize repulsion at specific separations.

Abstract

We present exact closed-form expressions and complete asymptotic expansions for the electrostatic force between two charged conducting spheres of arbitrary sizes. Using asymptotic expansions of the force we confirm that even like-charged spheres attract each other at sufficiently small separation unless their voltages/charges are the same as they would be at contact. We show that for sufficiently large size asymmetries, the repulsion between two spheres $\textit{increases}$ when they separate from contact if their voltages or their charges are held constant. Additionally, we show that in the constant voltage case, this like-voltage repulsion can be further increased and maximised though an optimal $\textit{lowering}$ of the voltage on the larger sphere at an optimal sphere separation.