# Paschen's Law

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#### 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 7.5mum 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 mum distance, but would arc at a 7.5 mum 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
• γse = secondary electron emission coefficient at the cathode

## References

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