DC current in a wire I'm sure that this question was addressed here before, but I failed to find any other instances, so with your permission I ask the question myself.
I'm experiencing a very disturbing glitch, there is a fallacy somewhere in my logic, but I can't put a finger on it.
I was considering a cross section of a circular wire (radius $R$) carrying DC current with magnitude I. Let's say for now that this wire is a perfect conductor.
If we calculate the magnetic intensity adjacent to the wire's surface, we get $H = I/2\pi R$.
Since the fields inside the wire are zero, one could easily calculate the surface current density $K$:
$\oint K\,R\,\mathrm{d}\phi = \oint H\,R\,\mathrm{d}\phi \Rightarrow K = H = I/2\,\pi\,R$. ($\mathrm{d}\phi$ is the differential of the polar angle). Since $K = I/2\,\pi\,R$, that means that the surface currents hog all the current that flows through the wire. Thus, inside the wire the current density is zero.
Please someone explain to me how one arrives at the conclusion that DC current is uniformly distributed across a wire's cross section, taking Maxwell's equations as a starting point.
Thank you in advance.
 A: The flaw in your reasoning is in assuming that there are no magnetic fields inside the conductor.  Equilibrium electric fields cannot exist in a perfect conductor, but magnetic fields can as long as they are not changing in time.  Empirically, the charge density is quite uniform through the wire's cross-section, so the current does indeed flow through the bulk, and there are concentric loops of magnetic field: see here.
Explaining this theoretically is quite complicated, because there are two competing effects: the fact that parallel currents attract tends to concentrate the currents at the center of the wire, but is counterbalanced by the fact that the moving charges' electrical repulsion resists such charge clustering.  Things only get more complicated when you consider the relativistic Lorentz contraction of the moving charges.  See here for a simple model that tries to incorporate these effects.
A: There's something called the skin effect which is roughly what you described in your question. The skin effect forces charges to aggregate on the surface of a conductor. How far from the surface the charges exist is called the "skin depth", which has a low-frequency approximation of $\delta = \sqrt{\frac{2\rho}{\omega \mu}} $. For a perfect electrical conductor (PEC) the resistivity ($\rho$) is 0 so its skin depth is 0, meaning the charges really are clinging to the surface. For DC, one can say that the wavelength is infinite, which means the frequency is 0. Plugging directly into that formula will cause a division by 0 error, but without analyzing we can see that skin depth goes up as the frequency goes down. So the skin depth for DC is sort of infinity, but this really just means that the skin will cover the whole cross-sectional area.
Edit: for the DC the equation is actually in indeterminate form (as pointed out by @tparker). Regardless, the skin effect is what determines where the charges flow.
