Simple approximate treatment of unipolar space-charge-dominated coronas: The Warburg law and the saturation current

Unipolar space-charge distribution ρ(t) drifting with constant ion mobility μ in gas under time and space dependent electric fields will evolve according to the unipolar charge drift formula [1/ρ(t)]−[1/ρ(t0)] = (μ/ε0)(t−t0), neglecting diffusion effects. This general formula is especially useful as a starting point for approximate treatment of space-charge effects, both in ion drift and diffusion experiments, and in electrical coronas. A first simple derivation of an equivalent of the Warburg cos5ϑ current-density distribution for point-to-plane coronas is given, together with a formula for the maximum possible unipolar ion current (the unipolar saturation current) Is ≊2με0 V2/d, for a given corona voltage V and point-to-plane distance d. It is demonstrated that corona currents in excess of this limit always involve free electron or bipolar conduction phenomena, like streamers.