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Conventionally, space-charge (SC) limited current density is defined as themaximal cur- rent density allowed to traverse a diode under a DC voltage when atime-invariant flow is injected from the cathode. In this work, we study the SClimited current density under a time-varying injection for both classical andrelativistic regimes and determine the maxi- mal amount of limited currentdensity under certain conditions. Our simulations show that it is unlikely thata time-varying injection emitted from cathode exceeds the known SC limits, ineither classical or relativistic regime.

Space-charge effect of the time-varying electron injection in a diode: classical and relativistic regimes