Paschen's law
[4] Paschen studied the breakdown voltage of various gases between parallel metal plates as the gas pressure and gap distance were varied: For a given gas, the voltage is a function only of the product of the pressure and gap length.
He found an equation that fit these curves, which is now called Paschen's law.
[3] 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.
An arc would sometimes take place in a long irregular path rather than at the minimal distance between the electrodes.
For example, in air, at a pressure of one atmosphere, the distance for minimal breakdown voltage is about 7.5 μm.
is the secondary-electron-emission coefficient (the number of secondary electrons produced per incident positive ion),
They are determined experimentally and found to be roughly constant over a restricted range of
The composition of the gas determines both the minimal arc voltage and the distance at which it occurs.
For air at standard conditions for temperature and pressure (STP), the voltage needed to arc a 1-metre gap is about 3.4 MV.
[8] It takes into account non-uniformity in the electric field and formation of streamers due to the build up of charge within the gap that can occur over long distances.
Breakdown voltage can also differ from the Paschen curve prediction for very small electrode gaps, when field emission from the cathode surface becomes important.
Since electrons are much smaller, their average distance between colliding with molecules is about 5.6 times longer, or about 0.5 μm.
This is a substantial fraction of the 7.5 μm spacing between the electrodes for minimal arc voltage.
The first ionization energy needed to dislodge an electron from nitrogen molecule is about 15.6 eV.
A chain reaction then leads to avalanche breakdown, and an arc takes place from the cascade of released electrons.
[10] More collisions will take place in the electron path between the electrodes in a higher-pressure gas.
is high, an electron will collide with many different gas molecules as it travels from the cathode to the anode.
Collisions reduce the electron's energy and make it more difficult for it to ionize a molecule.
Energy losses from a greater number of collisions require larger voltages for the electrons to accumulate sufficient energy to ionize many gas molecules, which is required to produce an avalanche breakdown.
The electron mean free path can become long compared to the gap between the electrodes.
In this case, the electrons might gain large amounts of energy, but have fewer ionizing collisions.
A greater voltage is therefore required to assure ionization of enough gas molecules to start an avalanche.
To calculate the breakthrough voltage, a homogeneous electrical field is assumed.
(For very large applied voltages also field electron emission can occur.)
The number of ionization depends upon the probability that an electron hits a gas molecule.
It can be calculated using the equation of state of the ideal gas The adjoining sketch illustrates that
that a charged particle can get between a collision depends on the electric field strength
Noble gases like helium and argon are monatomic, which makes them harder to ionize and tend to have smaller effective diameters.
Ionization potentials differ between molecules, as well as the speed that they recapture electrons after they have been knocked out of orbit.
All three effects change the number of collisions needed to cause an exponential growth in free electrons.
