Skin Depth effect in the DC limit

Question

I've got a question that's been bothering me for a while now. For an ohmic conductor the skin depth scales as $f^{-1/2}$ and approaches infinity in the DC limit $\omega\to0$. However we know from electrostatics that electric fields cannot penetrate conductors, which seems to contradict the infinite skin depth in the static limit. How do I reconcile these two facts?

After thinking about this, it seems like it must be that the $\omega\to0$ limit is not the same as the electrostatic limit. But this seems problematic too, let me explain with a thought experiment.

Let's say I have an initially electrostatic situation involving a point charge at rest next to a conductor. Initially the electric field is zero inside of the conductor. Now let's say I very gently shake the point charge in order to produce low frequency radiation. Using the skin depth idea these low frequency waves should penetrate fully into the conductor. But here's the problem, what if I were to shake the particle with some extremely long period (say the age of the universe). Wouldn't the particle then be effectively at rest? And if so what happens to the electric field inside of the conductor, is it zero because the system is effectively electrostatic, or would it be non-zero since we're dealing with very low frequency waves?

Answer 1

The skin effect is describing the penetration depth $\delta$ for the the transverse field. And that still is correct at DC, the transverse $E$-field, parallel to the surface, is the field that makes current flow in conductors. This field and also the current, can penetrate the whole conductor, for DC the skin-depth is really infinite.

For the normal component of the $E$-field the penetration is indeed very small, it is not described by $\delta$, but for AC and RF it also isn't! In both cases it depends more on the density of positive and negative carriers that have to form the surface charge layer to absorb the normal field. (See: depletion layers, inversion layers, and accumulation layers.)

Answer 2

Current involves the movement of charge. The skin effect formula applies to DC current flow. This is not the same as a static charge situation. If you apply Maxwell's equation to static charge, there will be no magnetic flux, as the net current is zero (based on Ampere's law), and hence there can't be a skin effect. And, as mentioned before, current flow happens due to the transverse electric field caused by the potential difference, which is much greater than the normal electric field due to static charge.