E/N and electron energy, $\varepsilon$

Equations 2.2 and 2.3 can be combined to show that the electron mobility is inversely proportional to the gas number density:

$$\mu_e \propto \frac{1}{N}$$ (2.4)

For low to moderate fields where $v_e\!\ll\!\bar{v}$, the proportionality relationship in Equation 2.4 can be combined with Equation 2.1 to produce the following relationship:

$$v_e \propto \frac{E}{N} \qquad \mbox{if} \quad v_e \ll \bar{v}$$ (2.5)

The above proportionality is not strictly true for larger $v_e$ since $\bar{v}$ and hence $\nu_m$ will increase (Equation 2.3) leading to a decrease in $\mu_e$ (Equation 2.2). This decrease in $\mu_e$ will affect the relationship between $v_e$ and $E$ (Equation 2.1). However, it is generally true that $v_e = f(\frac{E}{N})$, since the increase in $\bar{v}$ is a function of $E/N$. It follows that the electron energy, $\varepsilon\!=\!\frac{1}{2}m{v}^2$, also follows the same similarity law:

$$\varepsilon = f({\textstyle \frac{E}{N}})$$ (2.6)

This is a particularly important insight since the electron energy controls the breakdown physics, as will be reviewed in the next section. This fundamental property of breakdown physics has been known since Townsend's pioneering work in the early 1900's.