Branching Electrical Discharges

Most recent answer: 10/22/2007

Why does an electrical discharge, such as lightning, have a branching pattern? Why is not the pattern in a plasma ball spherically symmetric because of the spherically symmetric electric field inside?
- Fenasi Kerim (age 18)
Istanbul, Turkey
Hi Fenasi,

Thanks for the nice pair of related questions. The main answer is that the simple principle of "electrical currents follow the path of least resistance" is guiding the path of the electrical discharge, but that the path of least resistance depends on where the electricity has already flowed, giving an interesting an nonlinear feedback situation. We’ll explain:

A lightning discharge releases energy by neutralizing opposite charges which have built up between clouds or between a cloud and the earth during a thunderstorm. These charges build up over large volumes of air and ground. As the voltage difference increases between the clouds and the ground, it eventually rises to the point where electrons can be torn away from the negatively charged ions and fly through the air. The built-up electric field between the cloud and the ground provides the energy. As an electron travels through the field, it is accelerated. If the electron collides hard enough with a gas molecule, it can knock some of the molecule’s electrons loose, and the process repeats in an avalanche-like manner. The end result is a tube of plasma, a gas consisting of positive and negative ions (in addition to remaining unionized gas molecules). The electrical resistivity in the plasma is much less than that of ordinary air because of the presence of charged particles which are free to move. Once this tube has been established, electrical currents will flow very easily along it.

A lightning strike may consist of many separate discharges, all down the same tube of ionized air. But some areas of charge buildup may not lie directly on the ends of the tube, but instead are off to the sides. The path of least resistance is for the current to stay in the main bolt as long as possible, but then branch off to the region where the opposite charge buildup is, taking a short side path. It’s a little like planning a trip in a car. You follow small branch roads to find the nearest interstate freeway (which offers the fastest travel with the least resistance). You follow that as long as you can, and then branch off again on small roads to go to your eventual destination. The nonlinear feedback here is that people will drive their cars the most where people have driven their cars before, which caused the interstate freeway to be built in the first place.

The same sorts of rules apply to many instances of self-organizing distribution networks. These include: blood vessels in animals, tree trunks and branches and roots, and rivers and streams. These all follow crazy, jagged paths, which are needed to distribute a mobile substance either from or to a volume. This happens when the distribution channels have a lower dimension than that in which they are distributing their cargo. Lightning bolts and blood vessels are effectively one-dimensional, but they must distribute electrical charge or blood to a three-dimensional volume, and therefore must take jagged, branching paths to fill up the space in the most efficient manner. The same principles apply to crystal growth. Snowflakes develop by attaching water molecules on the ends of frozen crystals, and the easiest place to make new ice is at the free ends. The raw material, water in the gaseous phase, is distributed in three dimensions around the snowflake and each molecule attaches to the closest part of the currently existing ice crystal. Snowflakes and other crystals therefore often have branches and dendrites on their ends.

The discharges in a plasma ball obey essentially the same rules -- charge travels the easiest down already established conducting plasma discharge paths, and then fork out to the places where the charge needs to go. You raise the very interesting question of "how does the discharge decide where to make the plasma path in the first place, if the electric field is spherically symmetrical?" The answer is a very interesting topic in its own right, with the name "spontaneous symmetry breaking." A symmetrical equation can have a non-symmetrical solution. The electrical discharge in a plasma ball just picks a direction or directions at random when it is turned on, if it is truly symmetrical. The same sort of thing happens if you push down on on a ruler that is held vertically in contact with a table. The system is left-right symmetric, but if you push hard enough, the ruler will bow to the left or right.

If there are asymmetries in the plasma ball, such as someone’s hand touching the glass, then the discharge paths are affected by the asymmetry.

Here is an explanation of .

Tom, Mike, and Nigel

(published on 10/22/2007)