Understanding the Magnetic Polarizability Tensor

The aim of this paper is provide new insights into the properties of the rank 2 polarizability tensor $\check{\check{\mathcal M}}$ proposed in (P.D. Ledger and W.R.B. Lionheart Characterising the shape and material properties of hidden targets from magnetic induction data, IMA Journal of Applied Mathematics, doi: 10.1093/imamat/hxv015) 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 P\'olya-Szeg\"o tensor and how, by introducing new splittings of $\check{\check{\mathcal M}}$, 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 P\'olya-Szeg\"o tensor and expressions for the low frequency and high conductivity limiting coefficients of $\check{\check{\mathcal M}}$. 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 $\check{\check{\mathcal M}}$ for a range of simply and multiply connected objects.