Approximate shape factors for soft magnetic composites

A general theory of the shape tensor is outlined, providing estimates of the shape or demagnetisation factors for either single or collections of magnetic particles. In the general case of multiple interacting magnetic particles, shape factors are insufficient to completely characterise the magnetic field. This paper provides approximate expressions for the shape factors for isolated magnetic cylinders, isolated blocks, and collections of spheres. The key assumption is that the stray field is approximately spatially constant over each magnetic particle, leading to a system of equations for the shape factors. These equations are linear for magnetic particles of very high susceptibility, and imply shape factors have non-local properties. Non-random particle alignment can lead to very small shape factors, and corresponding high susceptibility, while random particle alignments tend to produce shape factors approximating their isolated values. The special case, when the matrix system for the shape factors is singular, corresponds to the threshold when the composite susceptibility changes from depending on, to being independent of, the particle susceptibilities. An appendix summarises the magnetic fields from uniformly magnetised point dipoles, spheres, cylinders, ellipsoids and blocks.