Spacing between conductor without sparking formula
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Answer:
There are two ways to get a high voltage. One is to make a high voltage intentionally.
The other is that you can get a high voltage unintentionally by breaking a large current in an inductive circuit. As all conductors are inductive to some extent, a sufficiently high current going through an opening switch will create a spark as the contacts open and try to stop the current. Supplying a headlight bulb from a 12v battery via jump leads, and then pulling one away will usually make a spark as the connection opens.
Before the spark, there is no current at all, only a voltage (potential difference) between two points.
Arc discharge happens when the voltage is high enough to overcome the gap, and then continues when conductors are drawn apart until the plasma dissipates. This depends on how large the gap is; you can easily draw visible sparks from a 12V power supply by rubbing some conductors together. Tiny arcs form across the few microns of gap between surfaces that aren't perfectly flat.
Once an arc is struck it's a fairly good conductor, so the voltage across it will drop and current increase until it's limited by the rest of the system.
Van der Graff generators and similar "static electricity" systems are effectively capacitors charged to huge voltages that produce a fairly high current for an extremely short duration. This enables them to produce long, brief sparks.
Conversely arc welders operate with comparatively low voltages, maybe as low as 20V, but extremely high currents (hundreds or thousands of amps). This requires a very short distance - you have to touch the material being welded with the electrode.