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Imagine we have a tree during thunderstorm. Due to electric field in the atmosphere, a corona discharge appears on top of it. My question is: is there a formula describing this effect?

I've already tried to figure something by myself, and here's what I got: there is a parameter, which I would call $U_h$, - the potential difference per meter from Earth's surface (something around 100 V/m). So the higher the tree, the easier it would be to start corona discharge. The other important parameter is $r_0$ - radius of curvature of emitting tip, the smaller the radius, the better. There is also A - electron work function, which is constant to every material; the greater it is, the worse. So, in general formula would be something like this:

$$E = \frac{A}{{U_h} * h * r_0}$$

Where E - electric field strength of the storm needed to start corona discharge, h - height of the tree.

Obviously the formula is incomplete, at least because dimensions don't match. What is missing, and was any similar formula described ever before?

In case you think that I ask silly questions out of boredom, this topic is very important to me, and this question bothers me for quite some time, so if you have a clue, you answer would be very welcome.

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This is called Townsend Discharge. Wikipedia has a pretty complete article on it.

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  • $\begingroup$ Thank you very much for your answer. Perhaps my question was not perfectly correct, because Townsend Discharge was not exactly what I was looking for. But it gave me a push in a right direction: it is field emission, especially cold field emission. The formulas in question most likely are: $ r = \frac \alpha A * U^\frac 3 2$ and $ E_0 = U*(\frac {1-\alpha}{\frac r 4 * \ln{\frac {4R} r}}+\frac \alpha r)$, where A - work function, r - tip radius, $\alpha$ - tip shape-based constant (roughly 0,1 in this case), U - voltage needed for discharge, $E_0$ - field at the tip, R - anode-cathode distance. $\endgroup$
    – realGuybrush
    Commented Jan 19, 2023 at 23:06

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