Using relevant equations for E and J, show that the current in a steady current I in a cylindrical conductor with uniform conductivity $\sigma$ is uniformly distributed across its cross-section.
I think the relevant equations are the divergence of the E field from the Maxwell equations and $\sigma$E=J but calculating the divergence of J using the symmetry of the problem doesn't seem to work at all. I think I might be confused about the definition of the variables or using something I should be.
Answer
Since the cylinder is an ideal conductor, the electric field inside the cylinder must be parallel to the axis of the conductor, and thus no charge should be moving radially inward or outward.
Now consider the electric field at a certain radius r from the central axis of the conductor. By your equation, $ \sigma E = J$. Sigma and E are constants no matter what the radius is, because of the problem's specifications and the fact that the cylinder is a conductor, so J must be constant. That is the general idea behind the proof, but you might need more rigor especially regarding the first paragraph.
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