Skin depth

Stan Zurek, Skin depth, Encyclopedia Magnetica,
http://www.e-magnetica.pl/doku.php/skin_depth

Skin depth (often denoted by lower-case Greek letter delta δ) - such a depth of penetration of alternating current or electromagnetic field, at which the amplitude is reduced to a value of 1/e (around 37%), as compared to the amplitude at the surface of the conductor.

Mechanism of skin effect

^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}

Skin effect involves inducing eddy currents due to the electromagnetic field caused by the current flowing in the the conductor.

Because the magnetic field acts around the axis of the conductor the eddy currents are induced in such a way that near the wire axis they flow in the direction opposite to the main current, thus lowering the current density.

Near the surface, the direction of eddy currents coincides with the main current, increasing the local current density.

Skin depth is a function of frequency of current, as well as magnetic permeability and electric resistivity (or conductivity) of the conductor in question.

For good conductors (like most metals) the function can be defined as:¹⁾

$$ \delta = \sqrt{\frac{2 · \rho }{\omega · \mu_r · \mu_0}} = \frac{1}{\sqrt{\pi · f · \mu_r · \mu_0 · \sigma}} $$	(m)
where: π - mathematical constant, f - frequency (Hz), μ_r - relative permeability of the conductor, μ₀ - magnetic permeability of free space (H/m), σ - electric conductivity (S/m), ρ = 1/σ - electric resistivity (Ω·m), ω = 2·π·f - pulsation frequency (rad/s)