On the sharpness of a needle

Abstract

In a recent paper [Oct. 2003], Van Bladel discussed in detail the behavior of fields and surface charges near singularities of metallic structures, such as a hollow circular cylinder, a circular cone, a thin prolate spheroid, and a uniformly charged segment. At times, the reasoning in [Van Bladel, Oct. 2003] leads to conclusions not as definitive as one might wish because of the various approximations introduced, such as the electrostatic limit for the field or idealized singularities for the conductors (e.g., the tip of a cone). In this work, an example is provided that may assist in shedding some light on the questions raised in [Van Bladel, Oct. 2003], because it consists of an exact, closed-form, simple solution to a boundary-value problem that is valid at all frequencies. The problem consists of a plane wave propagating in free space and axially incident on the convex side of a paraboloid of revolution. This problem was solved by Schensted [1955] for a perfectly conducting (PEC) paraboloid, and by Roy and Uslenghi [Oct. 1997] for an isorefractive paraboloid, which comprises the PEC paraboloid as a particular case. A remarkable fact is that these exact solutions are also the geometrical optics solutions.