Vanishing surface charge density on current-carrying wire?

1. Jul 17, 2011

shoestring

In a paper by Coombes and Laue (http://docs.google.com/viewer?a=v&q...sig=AHIEtbQDQ3xeLo4PPx99CKlE3P6i4YWybw&pli=1") an expression, labelled (8), for the surface charge density cz/R on a long, straight, cylindrical current-carrying wire is given as

$$cz/R = (-E\epsilon_0 / (R\ln{(L/R)}))z$$

R i the radius, assumed to be << L.
E is the interior (longitudinal) electric field, driving the current.

The wire occupies the interval [-L,L] on the z axis, and the net charge is assumed to be zero. The surface charge density is a linear function of z, and for a given interior field E and radius R, the slope of the function depends inversely on ln(L/R), i.e. the slope gets weaker for larger L.

The expression is valid for z << L, i.e. the mid section of the wire.

In the paper it's pointed out that

"As L --> infinity, this surface charge density goes to zero at any given point z."

but then comes the interesting part, where it's stated that

"... an infinitely long wire in which a steady current is flowing has a vanishing surface charge density (8) and a uniform electric field both inside and outside the wire. It is particularly noteworthy that as L --> infinity the electric field outside the wire has a vanishing component normal to the wire."

something I find questionable.

The expression (8) is only assumed valid for z << L, so lets confine the argument to, say, the mid 1% of the wire and ignore the rest:

Now, no matter how big you choose L, you wont get a vanishing surface charge density on the mid section (which will always be 1/100 of the total length, i.e. it too will grow as L grows).

In other words, even if you have a pointwise convergence to zero as L grows to infinity, I wouldn't agree that the surface charge density on the wire goes to zero as L grows to infinity. No numerical value of L, no matter how large, will make the surface charge density function arbitrarily small on even the mid one percent of the wire. Quite the contrary, the maximum surface charge density on the (growing) mid 1 percent will go to infinity as L goes to infinity, so you shouldn't say that the surface charge density on the wire goes to zero. This isn't a case of uniform convergence to the zero function, only of pointwise convergence.

According to a critical comment on the paper (included in the link) the result of the paper can also be found in A. Sommerfeld's Electrodynamics (Academic, New York, 1952) which is described as "one of the classic treaties of the literature on electromagnetism." Nevertheless I think it's wrong, and that it's an obvious mistake to rely on pointwise convergence in this case. Any thoughts? Am I wrong?

Last edited by a moderator: Apr 26, 2017
2. Jul 18, 2011

shoestring

I didn't want to make the first post too long, so here are a few more thoughts on the issue:

The right way to calculate the surface charge density on a long, straight current-carrying wire (ignoring effects from charge distributions on other parts of the circuit) would be to calculate the surface charge density on a long (but finite) cylinder placed in a uniform electric field, with the cylinder's main axis along the direction of the field. If a conductor is placed in a uniform electric field, the induced surface charge will create a uniform interior field that inside the conductor exactly cancels the external field. For the cylinder that means that the surface charge will create exactly the kind of interior electric field that would drive a current through the cylinder.

Such calculations are difficult though, and the assumption that the surface charge density varies linearly along the cylinder turns out to create the uniform internal field you'd expect to drive the current through the wire, at least some distance from the endpoints.

That's what is done in the Coombes & Laue paper. They start with the assumption that the surface charge density varies linearly along the wire, and calculate the linear coefficient as a function of the interior field E and the radius R of the wire. Hopefully this gives a decent estimate of the surface charge density for |z| << L.

As has been said, the slope of the linear function describing the surface charge density gets weaker for larger values of L, so there'll be an increasing interval around the middle having a surface charge smaller than a certain value, but even if the absolute size of the interval grows, it'll constitute an increasingly smaller fraction of the wire. Because of that I think it's wrong to draw the conclusion that the surface charge density on the wire goes to zero as L goes to infinity.

The surface charge density is directly related to the outer radial electric field, so the surface charge density will also be related to the size and direction of the Poynting flow just outside the wire.

If it is as is concluded in the paper that the wire's surface charge density vanishes when L grows, then the Poynting flow just outside will be purely radial and directed towards the wire. If, on the other hand, there is a non-vanishing surface charge density, the Poynting flow near the wire will be larger and have a component parallel to the wire, at least almost everywhere along the wire.

Last edited: Jul 18, 2011