  In this paper we study the time-dependent Ginzburg-Landau equations on a
smooth, bounded domain $\Omega \subset \Rn{2}$, subject to an electrical
current applied on the boundary. The dynamics with an applied current are
non-dissipative, but via the identification of a special structure in an
interaction energy, we are able to derive a precise upper bound for the energy
growth. We then turn to the study of the dynamics of the vortices of the
solutions in the limit $\ep \to 0$. We first consider the original time scale,
in which the vortices do not move and the solutions undergo a "phase
relaxation." Then we study an accelerated time scale in which the vortices move
according to a derived dynamical law. In the dynamical law, we identify a novel
Lorentz force term induced by the applied boundary current.
