Formula for Electrical Arc Length I was playing with some High-Voltage the other day, when a question popped into my head. Can you calculate length of an electrical arc? It probably would be proportional to :-
1. Voltage of the source
2. Spark gap
3. Relative Humidity (in air only)
4. Resistivity of Medium
5. Shape of the Electrodes
Can anyone come up with a dimensional formula from this...?
 A: Paschen already did.  See Paschen's Law
The underlying physical intuition here has to do with the mean free path of electrons, electron-impact ionization and coulomb collisions.  
Electric fields will separate electrons from atoms in a gas, for example, and cause them to accelerate.  Those electrons will (1) lose energy (coulomb collisions) on other electrons as they accelerate and (2) may knock out other electrons (impact ionization) if they are sufficiently energetic.  These are two competing terms.  
If for example, the mean free path for electron-electron collisions is small compared to the gap size, and the electric fields are relatively low, then (1) will dominate and an arc is not sustainable.
On the other hand, if the mean free path is on the order or larger than the gap size, and or the electric fields are very large, then (2) will dominate and an arc is sustainable.
If you want a characteristic length, the mean free path is a good rule of thumb.  It should be obvious based on the above discussion how the voltage of the source  and the gap come into play (voltage / gap = electric field).  The resistivity of the medium in the traditional sense (the cold matter resistivity) doesn't matter.  Once you have the arc, what matters is the plasma resistivity which depends largely on the mean free path for coulomb collisions.  The shape of the electrodes can affect the electric fields, especially near the electrodes.  
