= Jared P. Nelson
{c}
<Jared P. Nelson> coauthored the generalized gas-discharge study; its contribution statement credits validation, formal analysis and investigation. The published affiliation places this work in Embry-Riddle's Space and Atmospheric Instrumentation Lab.<A Generalized Townsend's Theory for Paschen Curves in Planar, Cylindrical, and Spherical Geometries in Planetary Atmospheres by Jeremy A. Riousset and others>[Byline, affiliation and Author contributions]{ref}
A simple illustration of the geometric effect uses coaxial conducting cylinders. With inner radius $a$, outer radius $b$ and applied <electric potential difference> $V$, <Gauss's law> gives the charge-free field magnitude
$$
E(r)=\frac{|V|}{r\log(b/a)}.
$$
For $a=1\,\mathrm{mm}$ and $b=10\,\mathrm{mm}$, the field at the inner surface is ten times that at the outer surface. Equal gap width and voltage therefore do not imply a uniform ionization environment. This independently derived example illustrates why the field near a curved electrode can differ strongly from the average field across the gap.
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