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 [tex]a[/tex] carrying a steady current [tex]I[/tex]. Current returns along a perfectly conducting grounded coaxial cylinder radius [tex]b[/tex] (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 [tex]z[/tex]:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\sigma(z)=\frac{\varepsilon_0 I \lambda}{\pi a^3 \ln{a/b}}z[/tex]

where [tex]z[/tex] is measured along the axis of the cylinder, [tex]k = const[/tex],[tex]\lambda[/tex] is the resistivity (I renamed [tex]\rho[/tex] in the Book not to be confused with the volume charge density [tex]\rho[/tex]).

How about the surface current [tex]\mathbf{K}[/tex] (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: [tex]\nabla \cdot \mathbf{j}=0[/tex]. This state is stable: if somewhere we have less charges than required, the condition [tex]\nabla \cdot \mathbf{j}=0[/tex] is violated in such a way to correct this issue.

But we can not write for the surface current just

[tex]\mathbf{K}=\sigma v \hat{\mathbf{z}}[/tex]

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

Any ideas?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Surface charges and surface current on conductors carrying steady currents

**Physics Forums | Science Articles, Homework Help, Discussion**