Surface charges and surface current on conductors carrying steady currents

[physicsworks]

It is known that on a wire carrying steady current there are surface charges (and hence electric field outside the wire, but let's forget about it). This surface charges play an important role to maintain a uniform electric filed along the whole wire. There are only few geometries for which one can calculate the surface charges distribution. Let us focus on a long cylindrical wire of radius a carrying a steady current I. Current returns along a perfectly conducting grounded coaxial cylinder radius b (see, for example, Prob. 7.57 in Griffiths). Ok, in this special case we have the surface charge density to be a linear function of z:
\sigma(z)=\frac{\varepsilon_0 I \lambda}{\pi a^3 \ln{a/b}}z
where z is measured along the axis of the cylinder, k = const,\lambda is the resistivity (I renamed \rho in the Book not to be confused with the volume charge density \rho).

How about the surface current \mathbf{K} (in Griffiths's notation)?

I mean the charges on the surface certainly move, but in such a way that the spatial distribution of the charges does not change: \nabla \cdot \mathbf{j}=0. This state is stable: if somewhere we have less charges than required, the condition \nabla \cdot \mathbf{j}=0 is violated in such a way to correct this issue.

But we can not write for the surface current just
\mathbf{K}=\sigma v \hat{\mathbf{z}}

because \sigma changes along the wire and we will get \mathbf{K} which is not a constant---surely, a nonsense.

Any ideas?

 

[physicsworks]

I think there will be something more difficult than \sigma v, but the problem is WHAT? Maybe Griffiths can explain... I'll send an e-mail to him.