In the framework of quantum electrodynamics (QED) in external potentials, we introduce a method to compute the time-dependence of the expectation value of the current density for time-dependent homogeneous external electric fields. We apply it to the so-called Sauter pulse. For late times, our results agree with the asymptotic value due to electron-positron pair production. For sub-critical peak field strengths, or results agree very well with the general expression derived by Serber for the linearization in the external field. In particular, the expectation value of the current density at intermediate times can be much greater than at asymptotic times. We comment on consequences of these findings for recent proposals to test the Schwinger effect with high intensity lasers using processes at intermediate times.