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# Paschen's Law

EmilyB.Paschen's Law

#### The Math / Science

**Paschen's Law** is an equation that gives the breakdown voltage, the voltage necessary to start a discharge or electric arc, between two electrodes in a gas as a function of pressure and gap length. At higher pressures and gap lengths, the breakdown voltage is approximately *proportional* to the product of pressure and gap length, and the term Paschen's Law is sometimes used to refer to this simpler relation. However this is only roughly true, over a limited range of the curve.

Early vacuum experimenters found a rather surprising behavior. An arc sometimes takes place in a long irregular path rather than at the minimum distance between electrodes. For example, in air, at a pressure of 10^{-3} atmospheres, the distance for minimum breakdown voltage is about 7.5`mu`m gap, the required voltage is 533 V, nearly twice as much. If 500 V were applied, it would not be sufficient to arc at the 2.85 `mu`m distance, but would arc at a 7.5 `mu`m distance. It was found that breakdown voltage was described by the equation:

`V_B=(B·p·d)/(ln(A·p·d)-ln[ln(1+1/γ_(se))]`,

where:

- A = saturation ionization in the gas at a particular electric field (112.501(kPa*cm)
^{-1}) - B = related to the excitation and ionization energies (2737.50 V/(kPa*cm))
- p = pressure
- d = gap distance
- γ
_{se}= secondary electron emission coefficient at the cathode

## References

Wikipedia (https://en.wikipedia.org/wiki/Paschen%27s_law)