Potential and field of a homogeneous magnetic spheroid of arbitrary direction in a homogeneous magnetic field in Cartesian coordinates

Abstract

Purpose

The purpose of this paper is to present new expressions in Cartesian coordinates for the potential and magnetic field of prolate and oblate spheroids with arbitrary direction of the symmetry axis in a homogeneous field.

Design/methodology/approach

The potentials found in prolate or oblate spheroidal coordinates are transformed to Cartesian coordinates. These results are represented in such a form that they depend only on expressions, which are invariant under rotations around the symmetry axis. Thus, it is easy to change to arbitrary directions of both the symmetry axis and of that of the primary field. The gradients of the potentials are calculated and transformed exactly to the simplest form possible.

Findings

The paper presents simple expressions for the magnetic perturbations due to homogeneous prolate or oblate spheroids in a homogeneous magnetic field.

Research limitations/implications

Results are exact for single non‐ferromagnetic spheroids in a homogeneous field.

Practical implications

Superposition of these perturbations presupposes small values of the magnetic susceptibilities of both the spheroids and their environment as in biological tissues.

Originality/value

The paper presents novel formulas for fields of homogeneous spheroids in a homogeneous magnetic field which are very useful for modelling biological tissues in studies of magnetic resonance imaging and magnetic resonance spectroscopy.

Citation

Kraiger, M. and Schnizer, B. (2013), "Potential and field of a homogeneous magnetic spheroid of arbitrary direction in a homogeneous magnetic field in Cartesian coordinates", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 32 No. 3, pp. 936-960. https://doi.org/10.1108/03321641311305845