Stability of an Electrically Charged Droplet

The stability of an electrically charged droplet with respect to mechanical deformations is studied under the assumptions that the liquid is perfectly conducting, the medium devoid of external fields of force, and the sum of the electrical and mechanical energies in the system conserved. Unlike Rayleigh's case, which dealt with small perturbations of spherical drops, the deformations considered in the present case are allowed to be large in size but confined in shape to ellipsoids of revolution. The energy of the deformed droplet is expressed as a function of a geometrical parameter and the ratio α between the electrical and surface energies of the corresponding spherical shape. Likewise, the dependence of the extremal points on α is investigated. Conforming with Rayleigh, the spherical droplet is shown to be unstable for α > 4 and stable for α < 4. However, for a certain range of α in the latter case, it is found to be only in a metastable state. In addition, both one prolate and one oblate ellipsoid of minimal energy are shown to exist for every α > 4.