An upper bound to time-averaged space-charge limited diode currents

The Child-Langmuir law limits the steady-state current density across a one-dimensional planar diode. While it is known that the peak current density can surpass this limit when the boundary conditions vary in time, it remains an open question of whether the average current can violate the Child-Langmuir limit under time-dependent conditions. For the case where the applied voltage is constant but the electric field at the cathode is allowed to vary in time, one-dimensional particle-in-cell simulations suggest that such a violation is impossible. Although a formal proof is not given, an upper bound on the time-averaged current density is offered.