See also skin effect.

Skin depth is a term used for the depth at which the amplitude of an electromagnetic wave attenuates to 1/e of its original value. It also has applications in numerous other areas, such as seismic exploration.

The skin depth can be calculated from the relative permittivity and conductivity of the material and frequency of the wave. First, find the material's complex permittivity, $\backslash varepsilon\_c$

$\backslash varepsilon\_c=\backslash right)\}$

where:

$\backslash varepsilon$ = permittivity of the material of propagation

$\backslash omega$ = angular frequency of the wave

$\backslash sigma$ = conductivity of the material of propagation

Thus, the propagation constant, k, will also be a complex number, and can be separated into real and imaginary parts.

$k\_c\; =\; \{\backslash omega\}\backslash sqrt\{\backslash mu\backslash varepsilon\_c\}\; =\; \backslash beta\; +\; j\backslash alpha$

$\backslash alpha\; =\; \{\backslash omega\}\backslash sqrt)^2\}\; -\; 1\backslash right)\}$

$\backslash beta\; =\; \{\backslash omega\}\backslash sqrt)^2\}\; +\; 1\backslash right)\}$

where:

$\backslash mu$ = permeability of the material

$\backslash alpha$ = attenuation constant of the propagating wave

The skin depth is defined as $\backslash delta\; =\; \{1\backslash over\{Im\backslash \{k\_c\backslash \}\}\}\; =\; \{1\backslash over\{\backslash alpha\}\}$.

It can be seen that the imaginary part of the complex permittivity increases with conductivity, implying that the attenuation constant also increases with in conductive materials. Therefore, a high frequency wave will only flow through a very small region of the conductor (much smaller than in the case of a lower frequency current), and will therefore encounter more electrical resistance (due to the decreased surface area).

The term "skin depth" traditionally assumes ω real. This is not necessarily the case; the imaginary part of ω characterizes' the waves attenuation in time. This would make the above definitions for α and β complex, and so they would need to be redefined so that $Im\backslash \{k\_c\backslash \}\; =\; \backslash alpha$.

The same equations also apply to a lossy dielectric. Defining

$\backslash varepsilon\_c=\{\backslash left(\{\backslash varepsilon\text{'}\}\; -\; j\{\backslash varepsilon\text{'}\text{'}\}\backslash right)\}$

replace $\backslash varepsilon$ with $\backslash varepsilon\text{'}$, and $\{\backslash sigma\backslash over\{\backslash omega\backslash varepsilon\}\}$with $\backslash varepsilon\text{'}\text{'}\backslash over\{\backslash varepsilon\text{'}\}$

References