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The answers in this post suggest charges move through the bulk of conductors, not only their surface, when dealing with direct currents.

However, suppose we have a steady direct current, this steady state condition implies $\nabla .\textbf{J}=0$ within the conductor. Furthermore, conductors, by definition, satisfy $\,\textbf{J}=\sigma \textbf{E}$. From these two equations, we find $$\nabla. \textbf{E} =0$$

again, within the conductor. And since $\nabla. \textbf{E} =\frac{\rho}{\epsilon _{0}}$, we will ultimately have $$\rho =0$$

inside the conductor. This is in clear contradiction with the statement that charges move inside conductors for direct currents.

P.S. The reasoning above can be found in Purcell's Electricity and Magnetism, page 131.

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    $\begingroup$ How does charge density being zero contradict existence of current ? Electrons constitute current and charge over any volume is balanced by equal amounts of elections and kernels $\endgroup$ – Kutsit Sep 23 at 11:33
  • $\begingroup$ You're right; a null charge density implies only either the balance or the nonexistence of charge in that region, the second option is obviously to be ruled out. The equation tell us charge does not accumulate inside the conductor. $\endgroup$ – Hilbert Sep 23 at 11:47

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