Fields and charges near the apex of a hyperbolic cone

The sharp point of a cone is known to accumulate charges, thus representing a singularity for the electric field. However, a real pointed object has a finite radius of curvature at the apex. Thus, we investigate in this paper how the 'roundness' affects the behaviour of fields and charges at the tip of a cone. Two cases are considered: a double cone being the asymptote of a two-sided hyperboloid of two sheets and a one-sided cone being the asymptote of a similar one-sided hyperboloid. Our analysis employs the prolate spheroidal system of coordinates and uses Legendre functions of the first kind of non-integer degree. The behaviour of the surface charge density is illustrated graphically in terms of the distance to the apex. Simple approximate formulas for the charge density valid near the apex are given