theory, which is of utility in interpreting the experimental results, can however be simply deduced if the disturbance of the potential gradient is disregarded, and the ionization assumed uniform between the plates.
Suppose that the ions are produced at a constant rate q per cubic centimetre per second in the gas between parallel plates distant l cms. from each other. When no electric field is applied, the number N present per c.c., when there is equilibrium between the rates of production and recombination, is given by q = αN^2, where α is a constant.
If a small potential difference V is applied, which gives only a small fraction of the maximum current, and consequently has not much effect on the value of N, the current i per sq. cm. of the plate, is given by
i = NeuV/l,
where u is the sum of the velocity of the ions for unit potential gradient, and e is the charge carried by an ion. uV/l is the velocity of the ions in the electric field of strength V/l.
The number of ions produced per second in a prism of length l and unit area of cross-section is ql. The maximum or saturation current I per sq. cm. of the plate is obtained when all of these ions are removed to the electrodes before any recombination has occurred.
Thus I = q . l . e,
and i/I = NuV/(ql^2) = uV/(l^2[sqrt](qα)).
This equation expresses the fact previously noted that, for small voltages, the current i is proportional to V.
Let i/I = ρ,
then V = ρ . l^2[sqrt](qα)/u.
