A minimum principle for the space charge region around a conductor within the depletion approximation

The space charge region Omega(0) around a conductor at potential V-0, as calculated in the depletion approximation, has a minimum property, which can be expressed in terms of the functional F = W - V(0)Q, where W is the system energy and Q the charge on the conductor. It is shown that F attains its minimum value for the actual region Omega(0), relative to other regions with the same uniform charge density but different shape and extent. The identification of Omega(0) through the minimum principle is illustrated in two cases, where the spherical symmetry allows an analytical treatment.