The magnetic field of a plane current sheet with piecewise uniform electrical conductivity, with results for three insertions

Abstract

A system of N coupled linear boundary Fredholm integral equations of the second kind is derived to describe the electric current system and the magnetic field distribution in space for an infinite plane electrical conducting sheet with N non-overlapping insertions, permeated by a uniform parallel electric field. The cases of one or two insertions obtained earlier are recovered. The system for three insertions is derived and solved numerically to provide solutions for new problems involving elliptic or square insertions. The lines of current are plotted and the results for each case are discussed to assess the efficiency of the numerical method. The solved problems provide a detailed study of the complex behaviour of harmonic functions in space and in the plane, represented, respectively, by the magnetic scalar potential and the current function, at lines of discontinuity of the electrical conductivity. It is noted that the case of holes can be treated equally well. The method and the obtained results allow to evaluate the magnetic field on the Earth’s surface on and around a multitude of islands. They may be useful in non-destructive testing of current sheets through magnetic field measurement. The tackled problems display different types of discontinuities of the magnetic field components. The numerical results indicate that the case of three or more insertions may differ qualitatively from that of one or two insertions, namely the possibility of existence of current loops.