  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.
