Justification of the law of Peek into electrostatic

We consider the computation of the electrostatic charge density at the tip of a rounded corner. The relation between the curvature radius and the electrostatic field is given by Peek's empirical law which is valid only for thin, cylindrical or spherical, geometries. In this Note, we justify mathematically this law and extend it to other geometries. With the help of multiscaled asymptotic expansions, we derive an expression for the charge density for geometries which coincide at infinity with a cone. A numerical illustration is provided.