Comparison of hyperbolic and hyperboloid conductor electrostatics

The potentials and fields of hyperbolic and hyperboloidal conductors are available analytically. A detailed comparison of the two-dimensional and three-dimensional problems shows strong similarities, but also interesting differences. The electric field near a hyperboloidal needle is stronger (ceteris paribus) than near a hyperbolic blade, and dies off faster. The field at the hyperbolic conductor varies as the 1/3 power of the local curvature. At the hyperboloid conductor the field varies as the 1/4 power of the local Gaussian curvature (which is the product of the two principal curvatures)