  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 increases with the
injected current even after the formation of virtual cathode in both the
non-relativistic and relativistic cases. The increase is found to be a power
law. In other words, the time averaged fraction $f$ of electrons reaching the
anode varies with the input current as, $f\sim J_{\rm IN}^{-\beta}$ where
$\beta < 1$. In contrast, for an infinite parallel plate diode, $f \sim J_{\rm
IN}^{-1}$. The possibility of asymptotic saturation is also discussed.
