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 of $ζ$ 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 $(ζ/W)$ and $(ζ/W)2$ 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.