# 3.12: Skin Depth

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.