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Energetics of charge distributions

We consider isotropic static distributions of a given electric charge in a sphere and show that the minimal energy occurs for the uniform surface distribution, as in a charged spherical conductor in electrostatic equilibrium. Moreover, we show that the energy of such uniform distribution is minimal and stationary with respect to perturbations of the charge density, in particular dipole fluctuations. We argue that the use of the minimum energy principle to explain static distributions of electric charge provides interesting physical insights.