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The inverse problem of reconstructing a current distribution from measurements of its magnetic field is considered. The focus is on vacuum arcs in high voltage, vacuum circuit breakers. The feasibility and applicability of the standard method of least squares approximation by regularized singular value decomposition is studied, with reference to typical geometries and conditions that are met in industrial applications. Several factors that affect the quality of the solution to the identification problem are studied, including the number of sensors, the effect of experimental noise and of regularization methods and their parameters. The functional dependence of the minimal error attainable with a given regularization method from the signal-to-noise ratio has been deduced. A theoretical upper bound is provided on the error on current reconstruction, and confirmed by the numerical tests.