An Integro-Differential Equations System for a Quasi-Stationary Elec-tromagnetic Field in a Nonmagnetic Conductive Body Under a Dielectric Layer

Abstract

An initial-boundary value problem for a system of the equations of electrodynamics in a quasi-stationary approximation is considered for the case of a nonmagnetic conductive body, which is covered with a dielectric layer. It is assumed, that the conductor and the dielectric can be inhomogeneous in their conductive and dielectric properties, respectively. An electromagnetic field is induced by an external current, flowing in a limited area, located in a media, which external to the conductor and the dielectric layer; the external media has not any electrical and magnetic properties. At the boundaries of media, the usual conditions of conjugation must be satisfied: the tangential components of the tensions must be continuous; in addition, the normal component of the electrical induction must be continuous at the boundaries between non-conductive media. The initial-boundary value problem is considered in the classical formulation: the tensions of the electric and magnetic fields mast be smooth functions, that satisfy equations and boundary conditions in the usual (not generalized) sense. Under certain assumption about the connectivity of the region with conductor, dielectric and foreign current, as well as the smoothness of the boundaries of these regions, the uniqueness of solution for the considered initial-boundary value problem is proved. Also a system of integro-differential equations, which equivalent to the considered initial-boundary value problem, is derived; kernels of integral operators in this system have a weak singularity. The results are interest for the problems of eddy current flaw detection and thickness measurement.