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

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.