On equilibrium shapes of charged flat drops

TL;DR

This paper provides an explicit solution to the equilibrium shape problem of charged flat drops, showing that balls are the unique minimizers under certain conditions related to charge and area.

Contribution

It offers a complete explicit solution to the variational problem governing equilibrium shapes of charged drops, identifying conditions for minimizers and their uniqueness.

Findings

Balls are the unique minimizers at fixed charge and area below a critical charge.

No minimizers exist for charges exceeding the critical value.

Results apply to drops with fixed potential or charge.

Abstract

Equilibrium shapes of two-dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while for larger charge minimizers do not exist. Analogous results hold for drops whose potential, rather than charge, is fixed.