Variational formulation for the equilibrium condition of a conducting fluid in an electric field

Abstract

In this paper the variational formalism is used to derive a set of equations describing the equilibrium configuration of a conducting fluid in an applied electric field. The validity of the variational equations is confirmed by application to the well-defined problem of concentric spherical electrodes. It is further shown that a cone, including the so-called Taylor cone used to model the equilibrium configuration of liquid metal ion sources, is inconsistent with the general equations. An analysis of the Taylor derivation suggests that reasons for the disagreement are the omission of the pressure difference term in the Laplace formula and use of only an approximate solution to the electrostatic cone problem. Finally a quasi-empirical method is suggested for the self-consistent solution of the variational equations.