The Electrostatic Field about a Slender Dielectric Body

The problem of determining the electrostatic potential field around and inside a slender dielectric body of revolution is studied. The part of the potential outside the body that is due to the presence of the body is represented as the superposition of potentials due to point singularities distributed along a portion of the axis of the body and lying entirely inside the body. The potential inside the body is represented as a Taylor series in the radial distance from the axis of the body, with unknown coefficients. The boundary conditions on the body surface lead to a pair of coupled integral equations for the singularity density and the Taylor series coefficients. The leading terms in the uniform asymptotic expansion of the solution to these equations is obtained in terms of the slenderness ratio ε of the body. For the case of an axially symmetric external field, the solution is found to depend in a nonuniform manner on the ratio δ of the dielectric constants and three separate cases are considered, which involve different relative magnitudes of δ and ε. The cases of uniform external fields parallel and perpendicular to the body axis are considered in detail. The far field behavior of the potential is determined and the polarization charge density on the body surface is computed and discussed.