# Charge distribution on conductors?

You have seen that the excess charge on an isolated conductor moves entirely to the conductor’s surface. However, unless the conductor is spherical, the charge does not distribute itself uniformly. Put another way, the surface charge density s (charge per unit area) varies over the surface of any nonspherical conductor.

Why wouldn't the charge always distribute uniformly? I thought the charges would always want to maximize distance between themselves and so would spread out all over the conductor uniformly.

• To my mind you are trying to understand this the hard way (i.e. by reasoning about a great many charges in many different places all interacting together). I've written about the way I prefer (i.e. by reasoning about the electric field and it's relationship to charge) in an answer on another question. – dmckee Mar 12 '14 at 1:47

You have a constraint system. For every surface-element the tangent component of the electrical field strength must be zero (else the surface charge would start to move).

Consider a perpendicular edge of a cylindrical configuration as shown in the following picture.

Assume that the surface charge distribution would be uniform (constant charge per area).

There we have small surface charge elements $\sigma dA_1$ and $\sigma dA_2$ with $dA_1=dA_2$ generating a force on a small surface charge element $\sigma dA_3$ very close to the edge.

Thereby, $\sigma dA_1$ and $\sigma dA_2$ are just samples of the field generating surface charge elements over which we have to integrate. Nevertheless, we can discuss the main effect exemplary with their help. Both $\sigma dA_1$ and $\sigma dA_2$ have the same distance from $\sigma dA_3$ so that the absolute value of their field strength at the position of $\sigma dA_3$ is the same.

The tangent component $\vec F_{2\parallel}$ of the force $\vec F_2$ caused by $\sigma dA_2$ on $\sigma dA_3$ is small since $\vec F_2$ is directed almost normal to the surface.

On the other side the force $\vec F_1$ caused by $\sigma dA_1$ on $\sigma dA_3$ directly has tangent direction ($\vec F_1 = \vec F_{1\parallel}$).

Therefore, the sum of tangent forces $\vec F_{1\parallel}+\vec F_{2\parallel}$ on $\sigma dA_3$ will point towards the edge. In principle his holds also for the other charge elements contributing to the electrical field at the location of $\sigma dA_3$. Consequently, the surface charge $\sigma dA_3$ will move towards the edge until there is so much charge in the edge that for all surface charge elements the tangent component of the field strength is zero. ("The edge repells further charge.")

Eventually, at edges the surface charge density becomes infinite. Instead of a finite surface density you have a finite line charge density in the edge.

The effect at curved surfaces is similar but not so drastic.

The statement of the book that only spheres admit uniform charge distribution is only right if you restrict your considerations to bounded conductors and fast enough decaying fields at infinity.

If you admit infinite conductors then you also have uniform surface charge on a circular cylinder.

If you admitt outer charge distributions then you can adjust these outer charges such that the charge distribution on a considered smooth surface is uniform. Thereby, no restrictions are made on the shape of the surface.

The fact that the static charge does not spread uniformly is the basis for things like lightning rods. Sharp edges are places that static charges, particularly higher voltage ones, like to reside. This design also aids in dissipating dangerous voltages via the coronal (ionized air) discharge mechanism.

Sometimes, the uneven charge distribution is because those points are the most distant ones available, and electrons being of like charge, tend to repel each other.

However, the topology of the conductor is also important because charges prefer to reside on the outside surface(s) of a conductor as opposed to inside ones.

An interesting followup question might be whether charge actually is evenly distributed over the surface of a typical capacitor, such as an electrolytic type. We were taught that it was, but it probably wouldn't make very much difference to the functioning of such components if it were not. NPO (non-polarized) capacitors are available, but these are made by connecting two ordinary capacitors with a polarized dielectric in series such that the polarities of the pair are reversed.

• Thanks, but I can't really think of a scenario where "the uneven charge distribution is because those points are the most distant ones available, and electrons being of like charge". Could you please explicitly give me a scenario where an non-uniform distribution maximizes distance? – dfg Mar 11 '14 at 22:03

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