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Skin effect
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== Examples == [[File:Skin depth by Zureks-en.svg|thumb|350px|Skin depth vs. frequency for some materials at room temperature, red vertical line denotes 50 Hz frequency:{{ubl |Mn-Zn – magnetically soft [[Ferrite (magnet)|ferrite]] |Al – metallic aluminum |Cu – metallic copper |steel 410 – magnetic [[stainless steel]] |Fe-Si – [[grain-oriented electrical steel]] |Fe-Ni – high-permeability [[permalloy]] (80%Ni-20%Fe) }} ]] We can derive a practical formula for skin depth as follows: <math display="block">\begin{align} \delta &= \frac{1}{\alpha} = \sqrt{{2\rho }\over{(2 \pi f) (\mu_0 \mu_r)}} \\ &= \frac{1}{\sqrt{\pi f \mu \sigma}} \approx 503\,\sqrt{\frac{\rho}{\mu_r f}} \approx 503\,\frac{1}{\sqrt{\mu_r f \sigma}}, \end{align}</math> where {{unbulleted list | style = padding-left: 1.5em | <math>\delta = </math> the skin depth in meters | <math>\alpha = </math> the attenuation in <math>\frac{Np}{m}</math> | <math>\mu_0 = </math> the permeability of free space | <math>\mu_r = </math> the relative permeability of the medium (for copper, <math>\mu_r</math> = {{val|1.00}}) | <math>\mu = </math> the permeability of the medium | <math>\rho = </math> the resistivity of the medium in Ω·m, also equal to the reciprocal of its conductivity: <math>\rho = \frac{1}{\sigma}</math> (for copper, ρ = {{val|1.68|e=-8|u=Ω·m}}) | <math>\sigma = </math> the conductivity of the medium (for copper, <math>\sigma \approx </math> {{val|58.5|e=6|u=S/m}}) | <math>f = </math> the frequency of the current in Hz }} Gold is a good conductor with a resistivity of {{val|2.44|e=-8|u=Ω·m}} and is essentially nonmagnetic: <math>\mu_r = </math> 1, so its skin depth at a frequency of 50 Hz is given by <math display="block">\delta = 503 \,\sqrt{\frac{2.44 \cdot 10^{-8}}{1 \cdot 50}}= 11.1\,\mathrm{mm} </math> Lead, in contrast, is a relatively poor conductor (among metals) with a resistivity of {{val|2.2|e=-7|u=Ω·m}}, about 9 times that of gold. Its skin depth at 50 Hz is likewise found to be about 33 mm, or <math>\sqrt{9} = 3 </math> times that of gold. Highly magnetic materials have a reduced skin depth owing to their large permeability <math>\mu_r</math> as was pointed out above for the case of iron, despite its poorer conductivity. A practical consequence is seen by users of [[induction cooker]]s, where some types of [[stainless steel]] cookware are unusable because they are not ferromagnetic. At very high frequencies skin depth for good conductors becomes tiny. For instance, skin depths of some common metals at a frequency of 10 GHz (microwave region) are less than a [[micrometre|micrometer]]: :{| class="wikitable" |+ Skin depths at microwave frequencies |- style="vertical-align:top;" ! Conductor !! Skin depth<br/>([[μm]]) |- | Aluminum || style="text-align:center;"| 0.820 |- | Copper || style="text-align:center;"| 0.652 |- | Gold || style="text-align:center;"| 0.753 |- | Silver || style="text-align:center;"| 0.634 |} Thus at microwave frequencies, most of the current flows in an extremely thin region near the surface. Ohmic losses of waveguides at microwave frequencies are therefore only dependent on the surface coating of the material. A layer of silver 3 [[μm]] thick evaporated on a piece of glass is thus an excellent conductor at such frequencies. In copper, skin depth can be seen to fall according to the square root of frequency: :{| class="wikitable" style="text-align:right;" |+ Skin depth in copper |- style="vertical-align:top;" ! Frequency !! Skin depth<br/>(μm) |- | 50 Hz || 9220 |- | 60 Hz || 8420 |- | 10 kHz || 652 |- | 100 kHz || 206 |- | 1 MHz || 65.2 |- | 10 MHz || 20.6 |- | 100 MHz || 6.52 |- | 1 GHz || 2.06 |} In ''Engineering Electromagnetics'', Hayt points out that in a power station a [[busbar]] for alternating current at 60 Hz with a radius larger than one-third of an inch (8 mm) is a waste of copper,<ref>{{Harvtxt|Hayt|1981|pp=401}}</ref> and in practice bus bars for heavy AC current are rarely more than half an inch (12 mm) thick except for mechanical reasons.
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