Two-interval sturm-liouville theory for the electrostatic potential of a point charge near a dielectric cone

摘要

This paper provides the electrostatic potential of a point charge near a dielectric cone of circular cross section. Our theory applies to any pair of dielectrics on either side of the conical interface and any cone angle. Separation of variables yields two associated Legendre differential equations which are coupled through boundary conditions on the surface of the cone. We solve this boundary value problem by consideration of a two-interval Sturm-Liouville operator which acts on the direct sum of two L2 spaces, as suggested by Wang, Sun, and Zettl [J. Math. Anal. Appl., 328 (2007), pp. 390-399]. We proceed further to determine the inverse-integral-operator, we prove that it is compact and symmetric, we determine its eigenvectors, which form an orthogonal basis, and we use that basis, along with a classical Fourier series, to formulate the solution of the aforesaid electrostatic problem. Our analytical solution confirms van Blade?s prediction for the field at the tip of the cone, an outcome of his numerical investigation [IEEE Trans. Antennas and Propagation, 33 (1985), pp. 893-895]. As for the far field, we show that it is as if generated by the very same point charge in an unbounded homogenous medium with an effective dielectric constant wherein the media occupying the exterior and interior of the cone contribute according to the cone angle. Finally, we present briefly the formal derivation of our solution via the classical Mellin transform method and discuss its disadvantages over our proposed method.