Paschen law

The Paschen law models gas-breakdown electric potential difference as a function of pressure times electrode spacing. For a uniform gap $d$, fixed gas conditions and effective secondary-emission coefficient $\gamma$, an empirical ionization law $\alpha =Ap\exp (-Bp/E)$ together with $\gamma (e^{\alpha d}-1)=1$ givesHere $E=V_B/d$, $A$ has units inverse to pressure times length, and $Bpd$ has units of potential. The denominator must be positive. These constants and the fitted range are part of the model.^[ref]

Writing $L=\log (1+1/\gamma )$ and $x=pd$ gives $V_B=Bx/\log (Ax/L)$. Differentiation places its minimum at $x=eL/A$, with $V_{B,\min }=BeL/A$. A pressure sweep at fixed gap can therefore first lower and then raise the threshold. Nonuniform fields, changing cathode surfaces, temperature changes or microscopic gaps can break this scaling; it is not a universal law for every electrical spark.