Skin Effect Explanation

Question

I do not understand some things about the Skin Effect.

Its Wikipedia definition is:

Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor, and decreases with greater depths in the conductor. The electric current flows mainly at the "skin" of the conductor, between the outer surface and a level called the skin depth. The skin effect causes the effective resistance of the conductor to increase at higher frequencies where the skin depth is smaller, thus reducing the effective cross-section of the conductor.

Now I have two questions:

1. Can you explain me step by step the “vectorial” reason of the skin effect? If I look at this image and at the electric and magnetic field lines, I do not see the cause of the skin effect.

   

2. To quantify the skin effect we use the skin depth δ. But from basic electromagnetism courses I know that it is defined as the depth below the surface of the conductor at which an electromagnetic incident wave has been reduced by $1/e$, and it is $0$ for a perfect electric conductor, which acts as a perfect reflecting surface.

So, I do not understand why the skin depth defined in this way is related to the skin effect, according to the initial definition of the last one. Which is the bond between these two phenomena (the penetration of an incident EM wave in a real conductor, and the tendency of an AC current to flow near the surface of a conductor).

Answer 1

1) In the figure, for any radius $R$, the induced currents $I_\text{w}$ are aligned with the main current $I$, and therefore reinforce each other, for all points with $r>R$ and reduce each other for $r<R$. This means that, given the currents actually flow in the directions as given in the figure, the net current intensity will decrease in the middle of the wire and get shifted to outer parts.

Since the relation between $I_\text{w}$ and $I$ include $\omega$ as a multiplicand, as the AC frequency increases, the current pushes itself increasingly to the skin (thanks to the time derivative in Faraday's law).

2) Actually, what you remember as skin depth is called the penetration depth.

But, there are some similarities: Maxwell's equations are of the same form ($\rho\ne0$, optionally $\epsilon>1,\ \mu\ne1$) for any non-ideal conductor except the external fields. The equations are solved to reveal that under this assumptions, the fields inside the conductor show a exponential decay (this is really fun to derive).

For an external electromagnetic wave, the incident wave comes from outside the wire and tries to penetrate the medium. For conduction, however, the system is externally driven by a current. Therefore, current inside a wire goes under skin effect, and a EMW has a penetration depth.