Stability of a Conducting Droplet under the Influence of Surface Tension and Electrostatic Forces

Abstract

The Lagrange Equations of Motion are written in generalized coordinates which describe small departures from the spherical equilibrium configuration of a conducting liquiddroplet. It is initially assumed that the actual shape differs only very slightly from the equilibrium sphere. The equation representing the surface is, then, written as a series of surface zonal harmonics in which the coefficients are shown to be the normal coordinates of the droplet. The frequency of oscillation of the normal coordinates is shown to depend on the total charge on the droplet in such a manner that for all values of charge below a certain limit, the frequency is real. For all values of charge above a certain limit, the frequency is imaginary; and, thus, the droplet is unstable. This paper presents a detailed derivation of a result communicated by Rayleigh in 1882. The results of this communication have been widely quoted but, until now, this particular derivation has not appeared in the literature.