Unique shape detection in transient eddy current problems

Transient excitation currents generate electromagnetic fields which in turn induce electric currents in proximal conductors. For slowly varying fields, this can be described by the eddy current equations, which are obtained by neglecting the dielectric displacement currents in Maxwell?s equations. The eddy current equations are of parabolic?elliptic type. In insulating regions, the field instantaneously adapts to the excitation (elliptic behavior), while in conducting regions, this adaptation takes some time due to the induced eddy currents (parabolic behavior). The subject of this work is to locate the conductor(s) surrounded by a non-conducting medium from electromagnetic measurements, i.e., from knowledge of the excitation currents and measurements of the corresponding electromagnetic fields. We show that the conductors are uniquely determined by the measurements, and give an explicit criterion to decide whether a given point is inside or outside the conducting domain. This criterion serves as a base for rigorously justified non-iterative numerical reconstruction strategies.