An extension of Thomson's theorem and its application for determining induced charge densities

Thomson’s theorem of electrostatics states that the equilibrium distribution of charge on a system of conductors is that of minimum energy relative to other charge distributions with the same amount of charge on each conductor. By an appropriate modification of the energy functional, the theorem is extended to the case where the potential of some conductors, rather than their charge, is fixed. This extension is of interest in the analysis of electrostatic induction. The basic theorem and its extension are applied to a spherical conductor in the presence of a point charge or placed in a uniform external field. The known expressions for the induced surface charge density in different conditions are obtained from the direct minimization of the new functional on the basis of a simple argument.