• In uniform fields, the Townsend's criterion for breakdown in electropositive gases is given by the following equation,

or                                                                 αd = ln (1/ + 1)

where the coefficients α and γ are functions of E/p and are given as follows:

and                                                                              

where E₀ is the applied electric field, and p the gas pressure. In a uniform field electrode system of gap distance d,

where U_b is the breakdown voltage and E_b the corresponding field intensity. E_b is equal to the electric strength of the dielectric under given conditions. When the applied field intensity E₀ = E_b , the Townsend's criterion for breakdown in electropositive gases in uniform field can be represented in terms of the product of the gas pressure and the electrode gap distance 'pd' as,

or                                                                                                                                                                   (10.2)

This is known as Paschen's law. The scientist, Paschen, established it experimentally in 1889 from the measurement of breakdown voltage in air, carbon dioxide and hydrogen. It can be shown that this law is also applicable to electronegative gases. The values of α depend upon the particular gas and of   upon the electrode material. The breakdown voltage of a gas in uniform field is, therefore, a unique function of the product of gas pressure, 'p' and the gap distance between electrodes 'd'   for a given electrode material and its condition.

• At low value of Elp, that is at high pressures, where a steady state can be achieved, experiments have been performed on the spatial growth of ionization in a large number of both, electropositive and electronegative gases. In these experiments the conduction gap currents were maintained below 0.1µA, so that the field distortions due to space charge remained at a minimum. The results have shown that this generalised Townsend's theory of breakdown is applicable over a wide range of physical conditions. The values of the breakdown voltages U_b estimated theoretically have been found in good agreement with those observed experimentally.

• A schematic of the variation of U_b with respect to 'pd' is shown in Fig. 10.1. Equation 10.1 does not imply that the breakdown voltage varies linearly with 'pd', although in practice it is found to be nearly linear over certain regions. The breakdown voltage attains a minimum value U_{b min} around a particular value of the product (pd)_min.