Exact solutions for the shape of a two-dimensional conducting liquid drop in a non-uniform electric field

The problem of possible equilibrium configurations of the free surface of a conducting liquid deformed by a non-uniform external electric field is considered. The liquid rests on a flat electrode; the opposite electrode is a straight thin conducting filament placed parallel to it. It is assumed that the problem has plane symmetry: the surface is invariant under shift along the filament. Using the method of conformal mapping, a family of exact solutions of the problem is obtained; equilibrium is achieved as a result of the balance between capillary and electrostatic forces. According to the solutions, the two-dimensional drop considerably deforms with an increase in the strength of the external electric field (this increase can be related to changes in the linear charge density of the filament or in the electrode spacing). It turns out that the drop aspect ratio varies from unity (the charge density equals zero) to 11/2 (the filament recedes to infinity).