Electrostatic fields in inhomogeneous dielectrics

Abstract

Conditions under which simple, closed-form solutions to the electrostatic field equations in linear inhomogeneous dielectrics can be obtained are investigated. It is shown that such solutions can be found in general orthogonal coordinates (q1,q2,q3), in which equipotential surfaces are given by constant values of the axial q1 coordinate, whenever the permittivity can be expressed as ε(q1,q2,q3)=ε0(h1/h2h3)f(q1) g(q2,q3). In this formula, h1,h2, and h3 are arc length factors, while f(q1) and g(q2,q3) are essentially arbitrary functions of the coordinates. These solutions have the property that the E and D fields have components only in the axial q1 direction. Explicit formulas for field, displacement, polarization, potential, capacitance, etc. are obtained in terms of the generalized coordinates and the functions f and g. These expressions encompass virtually all the ‘‘workable’’ capacitor problems posed in standard texts. In the case of Cartesian coordinates where x is the axial coordinate, the conditions for closed-form solutions of this type reduce to ε=ε0 f(x)g(y,z), where f(x) and g(y,z) are arbitrary functions, and it is found that Ex depends only on x, while Dx depends only upon y and z. Effects arising from the presence of volume distributions of free charge are investigated, and it is found that their presence severely restricts the conditions under which simple solutions may exist. The methods used, and the results obtained, are applicable to structures such as capacitors, superlattices, optical fibers, inhomogeneous thin films, semiconductor heterojunctions, etc.