We consider the scenario in which the high-energy electrons responsible for hard X-ray production in the impulsive phase of solar flares are accelerated by a large-scale direct electric field. We point out that both Ampere's and Faraday's laws require that the current pattern associated with the accelerated electrons be highly filamented, with the degree of filamentation dependent on the assumed structure of the preflare current pattern. Recognizing that cospatial return currents are not permitted in such models, we consider the closure of the current pattern using a cross-field drift of protons at the chromospheric footpoints of the elementary magnetic flux tubes. We demonstrate that there is a sufficient rate of ionization (both collisional and radiative) and recombination to create and absorb, respectively, the necessary electron fluxes, and we also demonstrate that pressure gradients in the chromosphere are adequate to drive the opposite flows of protons and hydrogen atoms against the frictional forces there. We further argue that the current closes within a vertical layer of order the thickness of the current channel, i.e., a few meters, and that this layer is most likely situated at the base of the transition region, where the conductivity tenser first becomes roughly isotropic. A possible diagnostic of this model is the polarization of the H alpha line produced by an unequal population of the n = 3 hydrogen atom sublevels in the presence of the anisotropic proton distribution function commensurate with the above current structure.