skin effect

Most of the AC current in a conductor travels along the skin of conductor, with about 37% of the current density at a distance of $δ$ from the edge of the conductor. This depth ( $δ$ ) is dependent on the resistivity ( $ρ$ ) and the magnetic permeability ( $μ$ ) of the material and varies based on the angular frequency ( $ω$ ) of the AC current flowing through it.

explanation

A main current $I$ flowing through a conductor induces a magnetic field $H$ . If the current increases, as in this figure, the resulting increase in $H$ induces separate, circulating eddy currents $I_{W}$ which partially cancel the current flow in the center and reinforce it near the skin.

skin depth equation

This equation is only accurate for frequencies below $\frac{1}{ρ ϵ}$ , which is $\approx 10^{18}$ Hz in copper wire.

δ = \sqrt{\frac{2 ρ}{ω μ}} = \sqrt{\frac{2 ρ}{2 π f μ}}

where:
$δ =$ the depth at which about 37% $or (\frac{1}{e})$ of the current in the conductor is concentrated, as measured from the outer edge.
$ρ =$ the resistivity of the material
$μ =$ the magnetic permeability of the material
$ϵ =$ the permittivity of the conductor
The angular frequency of the current, $ω = 2 π f$ where $f$ is the frequency in Hz.

practical limitations from the skin effect

the impedance $(Z)$ of a wire increases dramatically with an increase in the frequency of an AC current. For a conductor with a diameter $(D)$ which is much greater than the skin depth $(δ)$ , the effective cross sectional area of is approximately that of a hollow tube with a wall thickness of $δ$ .

$∴$ you can approximate the effective resistance $(R)$ of a wire a given length $(l)$ and resistivity $(ρ)$ as:

\begin{aligned} R & \approx \frac{l ρ}{π (D - δ) δ} \approx \frac{l ρ}{π D δ} \\ for D ≫ δ \end{aligned}

The last column in the table below shows the approximate frequency for which the skin depth in a solid core AWG annealed copper wire radius of the wire.
I.e. at or above the $δ_{m a x}$ frequency, the skin effect becomes a factor in the impedance (or effective resistance) of the wire.
Below this frequency, the skin effect will have a negligible impact on the observed resistance.

AWG	Diameter (mm)	A (mm^2)	Diameter (in)	A (in^2)	Area (kcmil)	δmax (Hz)
4	5.189	21.147	0.2043	0.03278	41.74	649
6	4.115	13.299	0.162	0.02061	26.24	1032
8	3.264	8.367	0.1285	0.01297	16.51	1640
10	2.588	5.260	0.1019	0.00816	10.38	2608
12	2.053	3.310	0.0808	0.00513	6.53	4145
14	1.628	2.082	0.0641	0.00323	4.11	6591
16	1.291	1.309	0.0508	0.00203	2.58	10481
18	1.024	0.824	0.0403	0.00128	1.62	16660
20	0.812	0.518	0.032	0.00080	1.02	26495
22	0.644	0.326	0.0253	0.00050	0.64	42122
24	0.511	0.205	0.0201	0.00032	0.40	66901
26	0.4049	0.129	0.0159	0.00020	0.25	106557

The $δ_{m a x}$ equation and constants (as derived from the approximate #skin depth equation):

f_{δ_{m a x}} = \frac{4 ρ_{c u}}{d^{2} π μ_{0} μ_{r}}

where:
$d =$ wire diameter
$ρ_{c u} = {(58 \cdot 10^{6} \frac{S}{m})}^{- 1}$ the resistivity of annealed copper
$μ_{0} = 4 π \cdot 10^{- 7}$ vacuum permeability
$μ_{r} = 0.999994$ the relative permeability of copper