In this paper we analyse a mixed finite-element method and boundary-element method coupling for a time-dependent eddy current problem posed in the whole space and formulated in terms of the electric field Ee. The coupled problem is obtained by first proposing a mixed formulation of the interior problem in order to handle efficiently the divergence-free constraint satisfied by E in a dielectric material. Next we incorporate the far-field effect in the latter formulation through boundary integral equations defined on the coupling interface. We show that the resulting degenerate parabolic problem (with saddle point structure) is well-posed and use Nédélec edge elements and standard nodal finite elements to define a semidiscrete Galerkin scheme. Furthermore, we introduce the corresponding backward Euler fully discrete formulation and analyse the asymptotic behaviour of the error in terms of the discretization parameters for both schemes.