In this paper, we analyze a mixed form of a time-dependent eddy current problem formulated in terms of the electric field<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper E"><mml:semantics><mml:mi mathvariant="bold-italic">E</mml:mi><mml:annotation encoding="application/x-tex">\boldsymbol {E}</mml:annotation></mml:semantics></mml:math></inline-formula>. We show that this formulation admits a well-posed saddle point structure when the constraints satisfied by the primary unknown in the dielectric material are handled by means of a Lagrange multiplier. We use Nédélec edge elements and standard nodal finite elements to define a semi-discrete Galerkin scheme. Furthermore, we introduce the corresponding backward-Euler fully-discrete formulation and prove error estimates.