# What is the equation for the electrical arc length of an arc that has been struck?

I have been experimenting with voltages at 2,320 volts AC. I am curious and haven’t found any equation to calculate the electrical arc distance that is drawn after striking an arc by touching the two contacts and pulling the two contacts away.

For example: My output power is 1,380 Watts, 2,320 Volts and 0.60 Amps. The contacts get as close as 0.25 of an inch and then an arc is attained. As I pull the two contacts away the arc is sustained up to 4 inches. I want to find the equation for that. So if I was given a set voltage and amperage I would be able to figure out the arc length.

So my question is this, what is the equation for the struck arc length distance that an arc with a certain power can sustain?

I would greatly appreciate any input as this is a very important variable in my next project. I call my project the 10,000 project because it will output a minimum of 10,000 volts and a maximum of 25,000 volts. It will also use 5,520 watts of power which is enough power to dim the lights on separate circuits.

I should add that I am very well aware of the possible death hazards as the primary will output 2.4 amps on the secondary. Which makes this very deadly. I forgot to also add that I am well aware of Paschen's law and I have found that this law isn’t the right law for what I’m looking for.

                          **UPDATE:**


I am using solid what looks to be around 18 gauge wire high voltage specified wire. Ive hooked the wire directly up to the transformer. I’m arcing the electricity from those two wires.

Thanks to all of your contributions.

For a DC arc what you are looking for is called Paschen's law. Wikipedia knows about as much as I do, but in brief Paschen's law is a phenomenological model which gives the breakdown voltage for an electrode of given gap size filled with a gas at a given pressure. The breakdown is thought to be an avalanche process in which free electrons are accelerated by field in the diode and then slammed into gas molecules to break free more electrons. This only works best if the electrons have lots of gas molecules to hit, but no so many that they collide before having gained enough kinetic energy to break apart the gas. The number of gas molecules in-between the electrodes is proportional to $$p d$$ where $$p$$ is the pressure (i.e. density) and $$d$$ is the distance. This paschen's law gives a formula for $$V(pd)$$ which is the voltage at which breakdown will occur. There is a particular $$pd$$ for which breakdown is most likely which is referred to as the Paschen minimum.

Importantly we can expect many properties of a discharge to scale (roughly) with $$V/pd$$.

Now the situation can be more complicated with AC current. But if the breakdown occurs fast enough then the AC current can be thought of as DC. The electron mobility (pdf link to a Computed electron drift velocity in moist air by Wyatt 1967) in air depends on the ratio $$V/pd$$, but assuming that you are at breakdown and that you have a large $$pd$$ (i.e. not near the Paschen minimum) then you have a $$pd$$ on the order of 40 V/cm/mm Hg and a velocity on the order of 50km/sec. Assuming you have AC at 60hz then you are near the top of the sin wave for at least 2 ms and so electrons can drift through 50 m. Now that doesn't mean that you will be able to arc of 50 m, you still need enough collisions to cause a breakdown, but I think 1m would be a good scale distance (and require 3.4 MV--more than you could reasonably achieve).

Of course there can be much more to the puzzle than that. For example, you observed that you could stretch your arc. This is likely because after first establishing breakdown you created a plasma which helps conduct the current. But for an extended arc there is also the trouble that every half cycle the applied field goes to zero and the gas starts to recombine. There is also a question of geometery since Paschen's law applies to parallel plates. Moreover at higher AC frequencies (like RF waves) then you get new phenomena. For example if you were at RF and added magnetic fields and build an inductively coupled plasma.

As I look over my post I will add another reference which seems quite useful for phenomenological describing arcs. And lastly be safe! As you must know by now, the voltage/currents you are using are no joke.

Edit Since you describe you describe arcing between 'contacts' I might suppose that you are using some sort of 'tip' or pointy contact to arc.

If we use two spheres then we can calculate the voltage (i.e. image charge method). For two infinite planes we have $$E=v/d$$, but for two spheres (far apart from each other) then we have $$E=V*a/d^2$$ where $$a$$ is the radius of the spheres. If the spheres are put close--$$d \approx a$$--then there is significant field enhancement (in the form of extra image charges) which will help lead to arcing.

In practice the result will greatly depend on the geometry of your problem!

Some googling shows a result for cylindical diodes--"Minimum Breakdown Voltage in Cylindrical Diode"

• I know Paschen's law and that has to do with the voltage breakdown of a certain gap. And that law doesn’t give me the right values I need. – Scientist Smith YT Dec 1 '18 at 20:46
• That second section to your post describes how it works. But I need some sort of formula besides Paschen's law because that doesn’t give me the right values. – Scientist Smith YT Dec 1 '18 at 20:50
• A note to take is that I live in an area where the air is very dry. – Scientist Smith YT Dec 1 '18 at 20:52
• Yes I’m using two wires to arc across from. – Scientist Smith YT Dec 4 '18 at 20:37
• I suspect that it's difficult to get reproducible results for arc length as a function of voltage and/or current. For one thing, long arcs are unstable due to convection set up by the hot arc carrying the middle of the arc upwards. – Philip Wood Sep 10 at 22:08