A THEORETICAL EXPLANATION OF THE LAWS OF WARBURG AND SIGMOND

Charge drift methods have recently been developed for use in field pro blems as an alternative to satisfying the charge continuity equation. Here they are applied to hyperboloidal point-plane geometry in order t o obtain analytic forms for the DC charge and planar current distribut ions beneath a coronating point. The charge distribution is linked to the ionic trajectories and the modelling of these loci is given carefu l consideration. The current distribution is then found, and this allo ws a comparison to be made with the widely quoted empirical current la w of Warburg and also with the more recent work of Sigmond. The new th eory clearly shows why Warburg's choice of a cosine power law in terms of theta, the semi-vertical cone angle of the discharge, is so succes sful. It initially gives a pure Laplacian prediction for the per-unit current distribution in the form cos(s) theta f(theta) where the index s=3 and f(theta)=2/(2+sin(2) theta), which is similar to Sigmond's ex pression. This form is then modified to take into account the gaseous space-charge dependencies and other point geometries. For discharges i n air it has been shown that there are further cosine factors and it f ollows that taking s=4 (or slightly less) with the above form of f(the ta) is appropriate. This Poissonian result is equivalent to taking s s imilar or equal to 5 and f(theta)=2/root(4-3 sin(4) theta-sin(6) theta ), where this new form for f(theta) slowly increases from 1 with theta . Hence, putting f(theta)=1 and treating s as a parameter with values a little smaller than 5 should empirically represent all planar corona currents. This is Warburg's result.