# Charge distribution on a conductor

1. Aug 24, 2012

### bcrowell

Staff Emeritus
Recently I've been trying to understand this subject in more detail. I thought I'd post some of what I've learned here in case either (a) it's useful to others, or (b) others can add to what I've found out. Surprisingly, this seems to be a subject on which people are still publishing papers, and there is even some controversy.

The general observation is that charge is more dense at places where the surface has more curvature.

First off, it's easy to show that this can't be a universal relationship. For example, you can have a Faraday cage. Virtually no charge will collect on the interior, no matter how big the local curvature.

There are two natural ways in which you can measure the curvature. One is the Gaussian curvature K, and the other is the mean curvature H. The Gaussian curvature is intrinsic, and the nature of the problem is extrinsic (forces go through the 3-d space in which the conductor is embedded), so you would expect that H would probably matter more. Actually it turns out that sometimes H matters, but sometimes K matters.

The following argument in favor of H is given by McAllister. Let E(s) be the magnitude of the electric field along a field line, as a function of arc length along this field line. Gauss's law leads to the fact that E'/E=-2H, where H is the mean curvature of the equipotential passing through the field line at a given s. You can easily verify this in the cases where the conductor is a plane, a line, or a point.

The most common example used to argue that charge prefers to go to areas of greater curvature is the one where you have two spheres at an equal potential. Although K and H aren't independent variables for a sphere, it should be clear from the above that we're really talking about H here, not K. To confirm this, consider the case where you have two parallel, conducting cylinders of unequal radius at an equal potential. Clearly the charge densities are unequal, and this will relate to their H, not to their K, which is zero.

Nevertheless, there are cases in which the charge density is exactly proportional to K1/4! McAllister has shown that these include the following: ellipsoid, hyperboloid of two sheets, elliptic paraboloid, hyperboloid of one sheet, hyperbolic paraboloid. (These are the only cases where the potential depends on a single coordinate, the coordinates are orthogonal, and the Laplace equation is separable in these coordinates.)

Although this result by McAllister is limited to these five cases, I've observed in numerical simulations that there is quite a strong tendency for the charge to conform to the Gaussian curvature. In this figure http://www.lightandmatter.com/html_books/lm/ch21/ch21.html#fig:lightning-rod [Broken] , you can see that there's an extremely low density of charges in the region of the pear-shaped conductor where K is near zero, even though H is not near zero there.

There's a classic result that at an angular edge between two half-planes with exterior angle β, the surface charge density is $\propto R^{\pi/\beta-1}$. Note that it's impossible to make sense of this by attributing it to the Gaussian curvature, since K is an indeterminate form at an edge. (That is, K depends on the product of the two principal radii of curvature, and at an edge, the limiting value of this product depends on the rate at which one radius goes to zero while the other goes to infinity.) H blows up to infinity in a determinate way as you sharpen the edge, so that's better, but these solutions still have nonzero charge density away from the edge, where both H and K are zero.

I wish it were possible to come up with a simple conceptual explanation for all this, but there's quite a variety of phenomena to account for, and since none of the mathematical rules are universal, it seems unlikely that there's a really good conceptual explanation that applies in all cases. DaleSpam gave one here: https://www.physicsforums.com/showpost.php?p=1456578&postcount=2 . I don't quite understand the argument; maybe DaleSpam could explain in more detail. It seems like the curvature it would relate to would be H, not K, since the normal components of the field to which he's referring would tend to cancel for a saddle with H=0.

This may also be of interest: http://math.stackexchange.com/quest...-proportional-to-fourth-power-of-the-distance

The sequence of papers below shows that there is some controversy about this issue. It seems like a deep problem in the sense that there's an interplay of global and local.

Enze, http://iopscience.iop.org/0022-3727/19/1/005

Zhang, http://iopscience.iop.org/0022-3727/21/7/028

Fan, http://iopscience.iop.org/0022-3727/21/2/019

McAllister - I W McAllister 1990 J. Phys. D: Appl. Phys. 23 359 doi:10.1088/0022-3727/23/3/016

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