Two-dimensional electromagnetic Child-Langmuir law of a short-pulse electron flow

Two-dimensional electromagnetic particle-in-cell simulations were performed to study the effect of the displacement current and the self-magnetic field on the space charge limited current density or the Child-Langmuir law of a short-pulse electron flow with a propagation distance L of and an emitting width of W from the classical regime to the relativistic regime. Numerical scaling of the two-dimensional electromagnetic Child-Langmuir law was constructed and it scales with (L/W) and (L/W)² at the classical and relativistic regimes, respectively. Our findings reveal that the displacement current can considerably enhance the space charge limited current density as compared to the well-known two-dimensional electrostatic Child-Langmuir law even at the classical regime.