### A new theory for metal detectors?

Following our MIRAN Land Mine clearance meeting in Manchester in 2012 there has been a collective effort to understand how to characterize conducting objects in low frequency magnetic fields. This is the metal detector length scale so without the object the problem is magneto-quasi static and the object satisfies the Eddy Current Approximation.

The search for the right theory began with Tony Peyton and my work on airport metal detectors that can locate and characterize metallic objects. This was in collaboration with Rapiscan and the project was called EMBody. It resulted in a successful prototype walk through metal detector which was trailed in Manchester Airport (who were also project partners).

So when Sir Bobby Charlton approached us to help with land mine detection, resulting in the founding of the charity Find A Better Way we though first to improve metal detectors so they could locate multiple objects with multiple coils and discriminate between objects based on tehir shape and electrical properties (conductivity and permeability).

This resulted in a long campaign by many mathematicians to understand this case. Habib Ammari at the Ecole Normale Supérieure Paris was at an advantage. With A. Buffa and J.-C. Nédélec he had written a paper A justification of the eddy currents model for Maxwell's equations.,

which laid the theoretical foundation for the eddy current model. With Kang he had also litterally written the book on Polarization Tensors, a way of representing the response of an object in a field using a small number of coefficients. The problem was to a large extent cracked in Ammari et al's paper Target identification using dictionary matching of generalized polarization tensors. Interestingly the metal detector, especially Unexploded Ordnance (UXO) community had long argued heuristically that the response to a conductive object should be given by a rank 2 symmetric tensor (six numbers) . Ammari's paper uses a rank 4 tensor, which could be 81 numbers, but with symmetries cuts down to 27. My own work with Paul Ledger from Swansea now shows that this is in fact a rank two tensor in disguise and the UXO literature was right all along. In a way everyone was right! Now we can calculate this tensor from the shape and electromagnetic properties of a material and work towards better metal detectors for land mine clearance.

The search for the right theory began with Tony Peyton and my work on airport metal detectors that can locate and characterize metallic objects. This was in collaboration with Rapiscan and the project was called EMBody. It resulted in a successful prototype walk through metal detector which was trailed in Manchester Airport (who were also project partners).

So when Sir Bobby Charlton approached us to help with land mine detection, resulting in the founding of the charity Find A Better Way we though first to improve metal detectors so they could locate multiple objects with multiple coils and discriminate between objects based on tehir shape and electrical properties (conductivity and permeability).

This resulted in a long campaign by many mathematicians to understand this case. Habib Ammari at the Ecole Normale Supérieure Paris was at an advantage. With A. Buffa and J.-C. Nédélec he had written a paper A justification of the eddy currents model for Maxwell's equations.,

which laid the theoretical foundation for the eddy current model. With Kang he had also litterally written the book on Polarization Tensors, a way of representing the response of an object in a field using a small number of coefficients. The problem was to a large extent cracked in Ammari et al's paper Target identification using dictionary matching of generalized polarization tensors. Interestingly the metal detector, especially Unexploded Ordnance (UXO) community had long argued heuristically that the response to a conductive object should be given by a rank 2 symmetric tensor (six numbers) . Ammari's paper uses a rank 4 tensor, which could be 81 numbers, but with symmetries cuts down to 27. My own work with Paul Ledger from Swansea now shows that this is in fact a rank two tensor in disguise and the UXO literature was right all along. In a way everyone was right! Now we can calculate this tensor from the shape and electromagnetic properties of a material and work towards better metal detectors for land mine clearance.

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