- #1

- 96

- 0

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

[tex]cz/R = (-E\epsilon_0 / (R\ln{(L/R)}))z[/tex]

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

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

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

[tex]cz/R = (-E\epsilon_0 / (R\ln{(L/R)}))z[/tex]

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: