Abstract
The magnetic polarizability tensor (MPT) has attracted considerable interest due to the possibility it offers for characterizing conducting objects and assisting with the identification and location of hidden targets in metal detection. An explicit formula for its calculation for arbitrary-shaped objects is missing in the electrical engineering literature. Furthermore, the circumstances for the validity of the magnetic dipole approximation of the perturbed field, induced by the presence of the object, are not fully understood. On the other hand, in the applied mathematics community, an asymptotic expansion of the perturbed magnetic field has been derived for small objects and a rigorous formula for the calculation of the MPT has been obtained. The purpose of this paper is to relate the results of the two communities, to provide a rigorous justification for the MPT, and to explain the situations in which the approximation is valid.
Highlights
T HE characterization of a highly conducting object from the measurements of the low-frequency time harmonic magnetic field, which is perturbed by its presence, has important applications in metal detection, where the goal is to locate and identify a concealed conductive permeable inclusion in an otherwise low conducting background
We have shown that the dipole expansion (4) agrees with the asymptotic expansion (11) for (Hα − H0)(x) as α → 0 for both uniform and nonuniform background fields provided that x is away from Bα, ν = O(1), H0 is analytic in Bα, and we choose H0|Bα = H0(z)
We have emphasized the advantages provided by the asymptotic formula in (11) over the multipole expansion (4) for (Hα − H0)(x) with x away from Bα, namely, that it provides a measure accuracy of the approximation and an explicit formula for the magnetic polarizability tensor (MPT) in the form of M, which holds for general objects
Summary
T HE characterization of a highly conducting object from the measurements of the low-frequency time harmonic magnetic field, which is perturbed by its presence, has important applications in metal detection, where the goal is to locate and identify a concealed conductive permeable inclusion in an otherwise low conducting background. In the applied mathematics community, an asymptotic formula for the perturbed magnetic field for the eddy-current problem of a conducting (permeable) object in a low-frequency time harmonic background field has been rigorously derived for the case where the object size tends to zero [4], [5]. For this case, an explicit formula for the computation of the MPT coefficients has been derived [21], [22] and our previous analysis clearly states what the requirements are on the material parameters, frequency, and object size in order for it to apply. We close with some concluding remarks, which justify the validity of the multipole expansion and its connection with the aforementioned asymptotic expansion
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