A new quasi-static model of space-charge limiting

The classical quasi-static model of space-charge limiting estimates time averages of the potential distribution and the current flow. But numerical studies by Birdsall and Bridges have indicated important limitations in the classical model. In particular, it makes no estimate of the fluctuation in the electric field. This paper proposes an alternative model that does estimate the fluctuation. The model leads to a pair of coupled second-order equations that closely resemble the classical equations. First-order solutions are obtained by an iterative process. The location of the points of zero field are only in fair agreement with the values obtained by discrete-charge methods, but the predicted fluauation in the peak electric field is considerably better. In particular, the model predicts a much larger fluctuation at the anode than at the cathode (for currents near the threshed of instability) in agreement with Birdsall and Bridges´ numerical results.