The General Theory of Electromagnetic Induction in a Conducting Half-Space

Abstract

Summary The general theory of electromagnetic induction in a conducting halfspace by an external magnetic source is developed in a new way which simplifies and consolidates the classic treatment of A. T. Price. The novel features of the theory are a systematic application of integral transforms and the use of electric and magnetic Hertz vectors aligned normal to the surface of the conductor. It is shown that the solutions associated with the electric Hertz vector correspond only to a free decay of currents within the conductor so that the entire theory of the induction problem is developed in terms of the one scalar component of the magnetic Hertz vector. The general solution of the magnetic Hertz potential corresponding to induction by an arbitrary time-dependent source is obtained in the form of a closed integral involving just one unknown function which is a Fourier transform of the magnetic Hertz potential of the source evaluated at the surface of the conductor. Results corresponding to the special cases of aperiodic and periodic fields are developed and explicit expressions for the electric and magnetic field vectors are also derived. The general theory is illustrated by considering three specific sources: (i) an aperiodic magnetic dipole normal to the surface of the conductor, (ii) a periodic magnetic dipole parallel to the surface of the conductor, and (iii) a periodic line current flowing parallel to the surface of the conductor.