Magnetostatics of the uniformly polarized torus

Author:

Beleggia Marco1,De Graef Marc2,Millev Yonko T.3

Affiliation:

1. Institute for Material Research, University of Leeds, Leeds LS2 9JT, UK

2. Department of Materials Science and Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213-3890, USA

3. American Physical Society, 1 Research Road, Ridge, NY 11961, USA

Abstract

We provide an exhaustive description of the magnetostatics of the uniformly polarized torus and its derivative self-intersecting (spindle) shapes. In the process, two complementary approaches have been implemented, position-space analysis of the Laplace equation with inhomogeneous boundary conditions and a Fourier-space analysis, starting from the determination of the shape amplitude of this topologically non-trivial body. The stray field and the demagnetization tensor have been determined as rapidly converging series of toroidal functions. The single independent demagnetization-tensor eigenvalue has been determined as a function of the unique aspect ratio α of the torus. Throughout the range of values of the ratio, corresponding to a multiply connected torus proper, the axial demagnetization factor N z remains close to one half. There is no breach of smoothness of N z ( α ) at the topological crossover to a simply connected tight torus ( α =1). However, N z is a non-monotonic function of the aspect ratio, such that substantially different pairs of corresponding tori would still have the same demagnetization factor. This property does not occur in a simply connected body of the same continuous axial symmetry. Several self-suggesting practical applications are outlined, deriving from the acquired quantitative insight.

Publisher

The Royal Society

Subject

General Physics and Astronomy,General Engineering,General Mathematics

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