The lightning-rod fallacy

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Abstract
It is accepted generally that the electric field strength on the surface of an isolated charged conductor is greatest where the surface curvature is greatest. We show here that there is, in fact, no relationship between these two maxima and that, in general, they are located at different points on the surface. Two classes of analytic examples are offered: one using conformal mapping techniques and the other involving small perturbations of a conducting spherical surface.

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