Understanding the Magnetic Polarizability Tensor

The aim of this paper is to provide new insights into the properties of the rank 2 polarizability tensor M proposed by Ledger and Lionheart for describing the perturbation in the magnetic field caused by the presence of a conducting object in the eddy-current regime. In particular, we explore its connection with the magnetic polarizability tensor and the Polya-Szego tensor and how, by introducing new splittings of M, they form a family of rank 2 tensors for describing the response from different categories of conducting (permeable) objects. We include new bounds on the invariants of the Polya-Szego tensor and expressions for the low-frequency and high-conductivity limiting coefficients of M. We show, for the high-conductivity case (and for frequencies at the limit of the quasi-static approximation), that it is important to consider whether the object is simply or multiply connected but, for the low-frequency case, the coefficients are independent of the connectedness of the object. Furthermore, we explore the frequency response of the coefficients of M for a range of simply and multiply connected objects.