Identification of the free boundary of a space-charge region with the aid of a Laplacian potential

The possibility of identifying the free boundary of the space-charge region around a conductor, without going through the solution of the related Poisson's problem, is discussed in the depletion approximation. The use of Green's second theorem leads to an integral identity that involves this boundary and a Laplacian potential for the same configuration. The identity is applied to a spherical and a right-angle concave conductor. For the sphere, the method yields the exact solution; in the wedge case, with some additional conditions, an approximate boundary identification is obtained, which is in good agreement with published numerical results.