What it does
When the voltage across a gas gap is high enough, a few free electrons gain enough energy between collisions to ionise atoms, each new electron does the same, and an avalanche turns into a spark or glow. Paschen’s law says that for flat electrodes this breakdown voltage depends only on the product of pressure and gap, p × d:
V = B p d / (ln(A p d) − ln(ln(1 + 1/γ)))
where A and B describe how the gas is ionised and γ is how many electrons each ion knocks out of the cathode. The curve has a minimum: below it, no DC voltage can break the gas down at any gap. To the left of the minimum, the gas is so thin that electrons rarely hit anything, so a wider gap breaks down more easily, which is why vacuum gaps and neon signs behave so differently from air gaps.
How to use it
- Choose the gas. Its rounded A and B fill in; type your own from a source you trust if you have them.
- Set γ (0.01 is a common default; it depends on the cathode metal).
- Give the pressure and the gap, in any of the units offered.
- Optionally type a voltage to check: the page says which gaps it can break down at that pressure.
- Tick Show every gas to compare curves.
Worked example
The page opens on dry air at 1 atm across a 1 mm gap with γ = 0.01. p × d is 76 Torr·cm, and the breakdown voltage comes out at about 5035 V, a field of 50.3 kV/cm. The Paschen minimum for air with these constants is 305 V at 0.836 Torr·cm, a gap of 11 µm at this pressure. A 1000 V supply could break down gaps from 4.60 µm to 123 µm at 1 atm, so the 1 mm gap holds it. At 1 cm the voltage is 35.5 kV, the familiar “about 30 kV per cm” for air (the simple law overestimates long gaps slightly).
Tips
- Lower the pressure and watch the operating point slide down the curve towards the minimum: that is how low-pressure lamps light with a few hundred volts.
- Paschen’s law is for uniform fields. Sharp points and edges break down at much lower average fields through corona.
- At very small gaps in air (below about 10 µm) real breakdown comes from field emission, well below the curve.
FAQ
Why do my measured values differ? A and B are fits over limited E/p ranges, γ depends on the electrode surface, and humidity, temperature and electrode shape all matter. Treat the result as an estimate.
Does temperature matter? Paschen’s law really depends on gas density. For a hot gas, use the pressure scaled to room temperature: p × 293 K / T.
Can I use this to design insulation? No: it is for study and planning, not a substitute for professional judgement or the clearance rules in electrical safety standards.
Where the numbers come from
- Paschen, F. (1889); Townsend breakdown theory as given in Raizer, Y. P., Gas Discharge Physics (Springer, 1991), chapter 7, and Lieberman and Lichtenberg, Principles of Plasma Discharges.
- A and B for each gas are rounded textbook values of the Townsend fit (orders of magnitude widely quoted in teaching texts), each valid only over a limited E/p range; they are editable on the page.
- Checked against an independent SciPy script (bounded minimisation for the minimum, Brent's method for the gap range) on the same formula.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.