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.