Multiscaled asymptotic expansions for the electric potential: Surface charge densities and electric fields at rounded corners

We are interested in computing the charge density and the electric field at the rounded tip of an electrode of small curvature. As a model, we focus on solving the electrostatic problem for the electric potential. For this problem, Peek's empirical formulas describe the relation between the electric field at the surface of the electrode and its curvature radius. However, it can be used only for electrodes with either a purely cylindrical, or a purely spherical, geometrical shape. Our aim is to justify rigorously these formulas, and to extend it to more general, two-dimensional, or three-dimensional axisymmetric, geometries. With the help of multiscaled asymptotic expansions, we establish an explicit formula for the electric potential in geometries that coincide with a cone infinity. We also prove a formula for the surface charge density, which is very simple to compute with the Finite Element Method. In particular, the meshsize can be chosen independently of the curvature radius. We illustrate our mathematical results by numerical experiments.