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

Two-dimensional electromagnetic (EM) 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 (CL) 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 (SCL) current density as compared to the well-known two-dimensional electrostatic (ES) Child-Langmuir law even at the classical regime.