Paschen's Law, named after Friedrich Paschen, was first stated in 1889. He studied the breakdown voltage of gas between parallel plates as a function of pressure and gap distance. The voltage necessary to arc across the gap decreased up to a point as the pressure was reduced. It then increased, gradually exceeding its original value. He also found that for normal pressure the voltage needed to cause an arc reduced with the gap size, but only up to a point. As the gap was reduced further, the required voltage began to rise and again, exceeded its original value.
Paschen curve
Early vacuum experimenters found a rather surprising behavior. An arc would sometimes take place in a long irregular path rather than at the minimum distance between the electrodes. For example, at a pressure of 10-3 atmospheres, the distance for minimum breakdown voltage is about 7.5 mm. The voltage required to arc that distance is 327 V and is smaller for gaps that are either wider or narrower. For a 3.75 mm 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 mm distance, but would arc at a 7.5 mm distance. Wikipedia, Paschen's Law
See Also
Law of Vibrating Atomolic Substances
Plasma
Principle of Regeneration
Spark