PaschenGesetz

Paschen's law, also known in German as the Paschen-Gesetz, describes the breakdown voltage between two electrodes in a gas as a function of the product p d, where p is the gas pressure and d is the electrode separation. The law expresses how the voltage required to initiate a discharge depends on gas density and the distance electrons must travel, with the specific gas entering through material constants.

In its common empirical form for a given gas, the breakdown voltage Vb is approximately Vb = (A p d) / [ln(p d) + B], where p is pressure (often in Torr) and d is the gap distance (in centimeters); A and B are constants that depend on the gas and electrode materials. Other equivalent forms exist depending on the chosen units and modeling details. The relation predicts a characteristic Paschen curve when Vb is plotted against p d.

A key feature is the Paschen minimum: there is a value of p d at which Vb is smallest. For many gases, including air at room temperature, the minimum occurs at a characteristic p d of around 0.5 Torr·cm, with a breakdown voltage on the order of a few hundred volts (roughly ~300–350 V for air, depending on conditions). For values of p d lower or higher than the minimum, the required breakdown voltage increases.

Physical basis and limitations: Paschen’s law emerges from Townsend discharge theory, relating ionization and secondary emission processes in the gap. It assumes a uniform field and simple parallel-plate geometry, and is most accurate for clean gases and moderate gaps. Real devices can deviate due to nonuniform fields, impurities, electrode effects, microgaps, and alternative discharge mechanisms.

Historically, the law is named after Friedrich Paschen, who formulated the relation in the late 19th century.