Exact solutions for equilibrium surface configurations of a conducting liquid in the electric field of a charged straight filament

The problem of possible equilibrium configurations of the free surface of a conducting liquid in the electric field of a charged horizontal straight filament is considered. Using the method of conformal mapping, a one-parameter family of exact solutions of the problem is obtained, which take into account the effect of capillary forces. Conditions are found for which the region occupied by the liquid ceases to be simply connected and a charged drop is formed on the liquid surface