In this paper, the eddy current problem in a two-dimensional conductor containing a crack is studied. The decomposition of the electric field into a piecewise regular part and a singular part deriving from scalar potentials localized at the crack tip and at the crack mouth is proved. At the crack mouth, the electric field is shown to have standard singularities inside the conductor, but presents a singularity outside the conductor that does not belong to the classical L 2 -space. Well-posedness of the E-based model and the A � ψ-formulation of combined potentials are proved and an un-gauged discretization of the latter formulation is discussed. The present paper is concerned with the analysis of eddy currents in the presence of cracks. Eddy current simulations have become an important research area due to numerous applications in electrical engineering. Eddy current testing, for example, as a particular technique of non destructive testing, remains one of the most popular tools in the quality control of conducting test pieces, and an important number of papers in the electric engineering community, mostly based on integral equation techniques, is devoted to numerical methods for crack detection (see e.g. (11, 12, 14, 15, 27)). In a typically configuration of eddy current testing, a coil carrying an alternative current is placed in proximity to the conducting test piece. The coil's magnetic field induces the eddy currents in the conductor which generate measurable variations of the impedance of the coil. In the presence of a crack, the eddy currents are deviated and the electromagnetic response allows for the detection of the defect. Due to the limited penetration depth of the currents, eddy current testing is mostly used to detect cracks that are situated near the surface. In the mathematical community, eddy current models have gathered much interest in relation with the topological properties of the device. Several formulations for the eddy current problem have been suggested (see