28. The equation expressing the relation between the current and the voltage is very complicated even in the case of a uniform rate of production of ions between the plates. An approximate 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 = αN2,
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
NeuV
i = -----
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