# 3.12: Skin Depth

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The electric and magnetic fields of a wave are diminished as the wave propagates through lossy media. The magnitude of these fields is proportional to

\[e^{-\alpha l} \nonumber\]

where \(\alpha\triangleq\mbox{Re}\left\{\gamma\right\}\) is the attenuation constant (SI base units of m\(^{-1}\)), \(\gamma\) is the propagation constant, and \(l\) is the distance traveled. Although the rate at which magnitude is reduced is completely described by \(\alpha\), particular values of \(\alpha\) typically do not necessarily provide an intuitive sense of this rate.

An alternative way to characterize attenuation is in terms of *skin depth* \(\delta_s\), which is defined as the distance at which the magnitude of the electric and magnetic fields is reduced by a factor of \(1/e\). In other words:

\[e^{-\alpha\delta_s} = e^{-1} \cong 0.368 \label{m0158_esddef}\]

Skin depth \(\delta_s\) is the distance over which the magnitude of the electric or magnetic field is reduced by a factor of \(1/e \cong 0.368\).

Since power is proportional to the square of the field magnitude, \(\delta_s\) may also be interpreted as the distance at which the power in the wave is reduced by a factor of \((1/e)^2\cong 0.135\). In yet other words: \(\delta_s\) is the distance at which \(\cong 86.5\)% of the power in the wave is lost.

This definition for skin depth makes \(\delta_s\) easy to compute: From Equation \ref{m0158_esddef}, it is simply

\[\boxed{ \delta_s = \frac{1}{\alpha} }\]

The concept of skin depth is most commonly applied to good conductors. For a good conductor, \(\alpha\approx \sqrt{\pi f\mu\sigma}\) (Section 3.11), so

\[\delta_s \approx \frac{1}{\sqrt{\pi f\mu\sigma}} ~~~\mbox{(good conductors)} \label{m0158_eSDGC}\]

Example \(\PageIndex{1}\): Skin depth of aluminum

Aluminum, a good conductor, exhibits \(\sigma\approx 3.7 \times 10^7\) S/m and \(\mu\approx\mu_0\) over a broad range of radio frequencies. Using Equation \ref{m0158_eSDGC}, we find \(\delta_s \sim 26~\mu\)m at 10 MHz. Aluminum sheet which is 1/16-in (\(\cong 1.59\) mm) thick can also be said to have a thickness of \(\sim 61\delta_s\) at 10 MHz. The reduction in the power density of an electromagnetic wave after traveling through this sheet will be

\[\sim\left(e^{-\alpha\left(61\delta_s\right)}\right)^2 = \left(e^{-\alpha\left(61/\alpha\right)}\right)^2 = e^{-122} \nonumber\]

which is effectively zero from a practical engineering perspective. Therefore, 1/16-in aluminum sheet provides excellent shielding from electromagnetic waves at 10 MHz.

At 1 kHz, the situation is significantly different. At this frequency, \(\delta_s\sim 2.6\) mm, so 1/16-in aluminum is only \(\sim 0.6\delta_s\) thick. In this case the power density is reduced by only

\[\sim\left(e^{-\alpha\left(0.6\delta_s\right)}\right)^2 = \left(e^{-\alpha\left(0.6/\alpha\right)}\right)^2 = e^{-1.2} \approx 0.3 \nonumber \]

This is a reduction of only \(\sim70\)% in power density. Therefore, 1/16-in aluminum sheet provides very little shielding at 1 kHz.