= Paschen law
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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 $\gamma(e^{\alpha d}-1)=1$ gives
$$
V_B=\frac{Bpd}{\log(Apd)-\log\!\bigl(\log(1+1/\gamma)\bigr)}.
$$
Here $E=V_B/d$, $A$ 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.<A Generalized Townsend's Theory for Paschen Curves in Planar, Cylindrical, and Spherical Geometries in Planetary Atmospheres by Jeremy A. Riousset and others>[Equation (6), empirical Townsend coefficient; planar self-sustaining criterion]{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.
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