Electromagnetic Induction in a Stratified Conducting Half-Space by an Arbitrary Periodic Source

Abstract

A general theory of induction in a horizontally stratified plane conductor by an external, time-periodic, magnetic source is presented. The analysis is a generalization to the case of an N-layered conductor of a previously published theory for induction in a uniform conducting half-space, in which the electromagnetic field was expressed in terms of electric and magnetic Hertz vectors oriented normally to the surface of the conductor. With the aid of this representation the entire theory is developed in terms of the one scalar component of the magnetic Hertz vector. Solutions for the electric and magnetic fields above and within the conductor are obtained in the form of double integrals whose integrands are related through a recursion formula to the Fourier transform of the magnetic Hertz potential of the source evaluated at the surface of the conductor. Special formulas applicable to 1- and 2-layer conductors are derived and the form of solution for some elementary sources is also discussed. As an illustration of the theory, numerical calculations are given for an infinite line current above a 10-layer conductor whose conductivity increases (i) linearly and (ii) exponentially with depth.