I've always been really interested in lightning. It's so cool, yet it's never really discussed in depth in typical physics courses.

How do you calculate some of the basic properties of lightning? For instance, things like the length of a typical bolt, the mean free distance travelled by the electrons, or the energy in a typical bolt.

(I have an answer I'm going to post, but I'll wait to accept so that others can take a crack at it too, or correct me. Constructive criticism welcome!)


Lightning occurs because of a charge build up on storm clouds (positive on top, negative on the bottom). If we know the height of the cloud above the ground (about 1e3 m e.g.) and that the voltage breakdown level in dry air is about 3e6 V/m then the voltage between the cloud and the ground must be around 3e9 V (V=Ed). It is usually less than this (100's of millions to billions of volts) because the air is wet during a thunderstorm and this lowers the breakdown level and there are also imperfections on the Earth (trees, hills, buildings) and on the cloud (bumps) which cause charge concentration and thus some higher voltage areas. However, this is a decent approximation of the voltage of a lightning bolt.

Once this breakdown level is reached, a 'step leader' will jump from the cloud to the ground. This step leader is 1-10 m in diameter and travels to the ground at about 100 mi/s so it makes it to the ground in about 5 ms. It carries with it about 0.5 C of charge and so the current is something like 100 A. The step leader serves the purpose of ionizing the air and creating a very good conductor (a kind of gaseous ion soup) for the rest of the lightning to follow.

The next step is the 'return stroke', where electrons between the cloud and the Earth are sucked down this ion channel to the Earth's magnetic core. During this period the current reaches 10,000-100,000 A. This, obviously, produces huge amounts of light and heat. As a result of this heat, when the return stroke finishes, a pressure wave shoots out, which we hear as thunder. During this stroke, about 5 C of charge are exchanged between the cloud and the Earth.

This process can then repeat 5-15 times as the necessary voltage for discharge is now significantly lowered because now, instead of air, there is the ion soup mentioned earlier, and a new step leader can follow that path to replenish the soup for the next return stroke. In total, these bolts can carry 25-50 C of charge off the cloud and then it needs to be recharged before it can thunder some more.

  • $\begingroup$ Hey, that's some interesting stuff. Do you have any references where I can read more in depth. Mostly because there wasn't many calculations. P.S. (Don't mean to be pushy, but you didn't upvote the question, yet still answered. Do you have reservations?) $\endgroup$ – Zach466920 Aug 10 '16 at 22:14
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    $\begingroup$ youtube.com/… This is a great lecture, the second half of which focuses on lightning. And sorry about not upvoting, I just forgot to. :P $\endgroup$ – Ulthran Aug 10 '16 at 22:16

If one looks at a typical bolt of lightning, it looks extremely branched and intricate. It definitely isn't a smooth curve or surface. Such objects are studied quite often under the umbrella term of Fractals. These objects are bizarre in that they can have infinite length, yet occupy a finite amount of space. They also have a "fractional" dimensionality associated with them.

To see this, consider the number of "boxes" of length $\epsilon$ it takes to cover a square of length $\delta$. The area of the large box is $\delta^2$ and the area of the small box is $\epsilon^2$. Thus, the number of boxes needed is $\left( \cfrac{\delta}{\epsilon} \right)^2$. For a general two-dimensional box with characteristic length $\delta$, the number of boxes, $N(\delta,\epsilon)$, it takes to cover the object is proportional to our previous quantity,

$$N(\delta,\epsilon)=\lambda \cdot \left( \cfrac{\delta}{\epsilon} \right)^2$$

We can also investigate a similar problem regarding the number of smaller lines it takes to cover one larger curve, we find,

$$N(\delta,\epsilon)=\lambda \cdot \left( \cfrac{\delta}{\epsilon} \right)^1$$

Note that the exponent in the first case is $2$, while for the second, the exponent is $1$. Thus, we can hypothesize that for a $d_f$ dimensional object, the following holds,

$$N(\delta,\epsilon)=\lambda \cdot \left( \cfrac{\delta}{\epsilon} \right)^{d_f}$$

If we estimate the length of the bolt contained in the box as proportional to the boxes linear existent, then we find that the length, $L$, of our whole bolt will be,

$$L=N \cdot (\mu \cdot \epsilon)=\beta \cdot \delta^{d_f} \cdot \epsilon^{1-d_f}$$

To make calculations tractable we need to make some assumptions; here they are.

1) For our case, we'll consider the lightning bolt to be the structure that connects from the cloud to the ground and lasts significantly longer than the other parts of the bolt.

2) We'll look at an average lightning bolt. This means that we'll look at the average path of the bolt. This means that the bolts diameter will be taken to be infinitely thin.

3) Finally, we'll assume that the visible part of lightning is made from electrons, travelling near the speed of light, interacting with particles in the air.

Using these three assumptions and standard values for the need quantities, we can calculate many things about lightning. Specifically, we can find the mean free path of the electron $\epsilon$, the time it takes lightning to travel from the clouds to the ground, and the energy in a typical lightning bolt.

$$\epsilon=70.79 \ nm$$ $$t_p \sim 0.09 \ s$$ $$E=1.11 \cdot 10^9 \ J$$

These should be compared to the real values,

$$\epsilon=69.17 \ nm$$ $$t_p \sim 0.1 \ s$$ $$E=1.11 \cdot 10^9 \ J$$

I actually made a video showing the full derivation of those values here. Don't worry, it doesn't use anything except some algebra.


http://www.wolframalpha.com/input/?i=mean+free+path+of+air https://en.wikipedia.org/wiki/Harvesting_lightning_energy https://en.wikipedia.org/wiki/Cloud_base


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