The Child–Langmuir law is investigated numerically using a fully electromagnetic particle-in-cell code for a closed axially symmetric diode. It is found that the average current transmitted to the anode (JTR) increases with the injected current (JIN) even after the formation of virtual cathode in both the nonrelativistic and relativistic cases. The increase is found to be a power law, JTR∼JIN1−β. In other words, the time averaged fraction f of electrons reaching the anode varies with the input current as, f∼JIN−β, where β<1. In contrast, for an infinite parallel plate diode, f∼JIN−1. The possibility of asymptotic saturation is also discussed.

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