Yesterday we dumped a bucket of electrons on the Statue of Liberty and watched what happened. The most important thing we saw is that all the charge immediately distributes itself over the outside surface of the statue in such a way so that the electric field within the statue is zero. We also noted that the field was high on sharp points like the spikes in the crown, but we left that without further explanation. Today we fix that.
Fields can be funny things. In the case of electric fields and gravitational fields and numerous others, the fields are vector fields. This means that a given point will have an electric field with a strength and a direction for it to point. These vector fields can be a pain in the neck to deal with and thus physics often uses a concept called potential to simplify the situation. A potential tells you how fields are set up without reference to vectors. To use gravity as a quick example: if you’re sitting on a piece of cardboard on a hill, the field is the direction gravity makes you slide along with the strength of the pull. The potential is the height of the hill at every given point. If the potential is changing rapidly the hill is steep and the field is high along the slope. And you’re pulled along rapidly by gravity. In short, a field is just the rate of change of potential with distance.
In that spirit let’s talk about the charges on the Statue in terms of potentials instead of fields. There being no field in the statue means the potential is constant – if potential isn’t changing with distance, there’s no field. That’s true within the statue. It’s also going to be true along the surface. If there were a potential gradient, there would be a field. If there were a field, the charges would be moving, which they’re not or they wouldn’t be at their equilibrium positions. From that we have to conclude that the potential along the surface is constant. The only thing left is to deduce what charge configuration produces a constant potential.
To answer that question, let’s break out an analogy we first used with black holes: the physics choir. Let’s give them some music: I suggest “Battle Hymn of the Republic” on account of the reference to lightning. Now each member of the choir represents an electron. The volume of their voices represents the potential the charged electrons produce. It’s high in magnitude as you’re close to the choir, and low in magnitude as you stand far away.
A statue is a complicated shape, so let’s simplify the problem by replacing it with the simplest charge configuration we can: a piece of straight wire. So take the choir out to a field, lay a long line of tape down to represent the wire, and have the choir distribute themselves evenly along it. Is the potential (the volume of the singing) the same everywhere along the wire? Heck no. If you stand at the center of the wire and listen, no choir member is more than half the length of the wire away from you. If you stand at the end of the wire, exactly half of the choir is farther from you than the farthest member was when you were at the center. As a result the choir is going to sound a lot quieter to you when you’re at the end of the wire. Which means the wire isn’t at a constant potential, which means that it doesn’t represent the way electrons actually distribute themselves.
So how will they distribute themselves? You might think that if the choir arranges itself so that it’s more spaced out near the center and more closely packed near the ends, the volume might be constant everywhere in the wire. And you’d be right. The fact that the volume has to be constant necessarily concentrates more singers near the ends. In two and three dimensions, electrons do the same thing. To produce a constant potential, they have to densely pack in to the places where geometry doesn’t give them a lot of room to produce the same potential by the expedient of low density over a large area.
If we wanted to argue this in a mathematically specific way we’d have a pretty hard time of it. Even very simple geometries quickly become to difficult to handle exactly. Nonetheless it’s possible to show that the sharper the point the higher the charge density. This is responsible for everything from the way sparks fly off a Tesla coil to the shape of lightning rods.
And that’s why electric charge tends to collect at sharp points.