When pulsed voltage is applied between two electrodes, streamers will
form once one electrode breaks down. The streamers (narrow channels
of plasma) grow towards the other electrode and will short out the gap
once they bridge the gap. If one electrode is highly field enhanced,
then streamers forms at the beginning of the pulse and the streamer
velocity is the gap length divided by the time until the gap is shorted
and voltage collapses.
Sources: Private communication with Ian Smith (for water when V>1MV
[although he writes it as Ft^0.5 = 0.09 where F is the electric field])
and a report by H.G.Herbert left for me by Ron
Boller and labelled "SSWA/HGH/6610/104" that describes the emperical fit to
data from breakdown experiments with a sewing needle and a 2" ball (for
Pulse generators were a 1.0 MV transformer with 500 ns rise time and 1.3 MV Blumlein with 45 ns pulse width.
Input parameter "d" is not required to calculate velocity for oil and
polyethylene (just for water) although it is required to calculate closure
time for all three cases.
These equations can be used to estimate the
closure time of a switch where one electrode is highly enhanced or to
determine the minimum possible gap closure time.