Exact Expressions for the Near Field of a Thin Uniform Circular Loop Current with Application to Loops Lying on a Half Space

Abstract

Exact analytic solutions for the electromagnetic field due to a thin, time varying uniform circular loop current are presented. The solutions are provided in the form of a power series with respect to wavenumber. The coefficients of the series are real functions of the spatial coordinates and loop radius and involve recursions of complete elliptic integrals or finite sums of elementary functions. Explicit expressions for the magnetic vector potential and electric and magnetic fields are provided for both cylindrical and spherical coordinate systems. The expressions are adapted for computing the electric field and axial magnetic field on the interface of two half spaces generated by a current loop lying on the half-space interface. Expressions for the self and mutual loop impedances are provided for both the free-space and interface case. Computed examples are given for specific frequency and half-space parameters and are compared to known solutions based on spherical Hankel functions or direct integration. The solutions are shown to be particularly efficient in the near field. Their derivation is motivated by recent developments of large sensors used in magnetic resonance sensing of minerals.