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Paschen's Law dates back to the 19th century, and is often referenced when in voltage breakdown calculations. For this effort you will find Friedrich Paschen in the Microwave Hall of Fame!
The empirical equation that Paschen came up describes the relationship between voltage breakdown, distance and gas pressure:
Vb=(apd)/(ln(pd)+b)
The constants a and b take on different values for different gases, For dry air, a=43.6*10E6 and b=12.8. Vb is the breakdown voltage in Volts, p is the pressure, d is the gap distance. Given these constants, the units of p (pressure) are atmospheres and d (distance) are meters. We have been alerted there is a problem with the figure below (or maybe our constants), we're working to correct it. Meanwhile, here is another site that presents the math.
An approximate number for breakdown electric field is 3 million volts per meter, or 30,000 volts per centimeter, for a gap of 7 mm at 1 atmosphere. Quoting an exact value for breakdown in air is not possible unless you consider gap size, pressure, and humidity.... even the geometry of the electrodes matters. Paschen's law applies to spherical electrodes.
Paschen videos!
Below we present two videos we found on Youtube. One highlights some experimenters, and one highlights the theory. We hope you enjoy one or both of them!
Here's a video of produced by Mohamed Diab conducting experiments in breakdown of gases at high voltage. Don't try this at home! The presentation is a little slow, but it is certainly better than watching Dancing With the Stars. If you pay attention you will see that the shape of the electrodes determines the critical electric field. Repeat this: pointy is bad... flat is good. Unless you are designing or buying a lightning rod, then you want a sharp point to maximize field strength and pull the lightning away from people and buildings....
Breakdown in gasses: high voltage testing
For some theory using Paschen's law, here's a really nice video high-schooler David Shlivko did back in 2015 that plots Paschen's law for air, and uses the resulting curve to talk about the creation of plasma. It was his entry to the Breakthrough Junior Challenge, a competition that encourages students ages 13 to 18 to create and submit original videos that bring to life a concept or theory in the life sciences, physics or mathematics.
Plasma and Paschen's Curve
Bravo to both video producers!