Electromagnetic waves in metals, skin effect
Rappel : The displacement current in a ohmic conductor
The Maxwell-Ampere equation is given by the local Ohm law :
\(\overrightarrow {rot} \;\vec B = \mu _0 \left( {\sigma \vec E + \varepsilon _0 \frac{{\partial \vec E}}{{\partial t}}} \right)\)
We denote \(T\) the time of evolution of distribution (D) (its evolutionary period).
One can compare the conduction current with the displacement current :
\(\frac{{\left| {\sigma \vec E} \right|}}{{\left| {\varepsilon _0 \frac{{\partial \vec E}}{{\partial t}}} \right|}} \approx \frac{{\sigma E}}{{\varepsilon _0 \frac{E}{T}}} = \frac{{\sigma T}}{{\varepsilon _0 }}\)
For the copper of conductivity :
\(\sigma = 6.10^7 \;\Omega ^{ - 1} .m^{ - 1}\)
This ratio is about of \(10^{18}\;T\), with \(T\) in \(s\).
Thus, even if \(T\) is about \(10^{-10}\;s\) (or a frequency \(10\;GHz\)) :
\(\frac{{\left| {\sigma \vec E} \right|}}{{\left| {\varepsilon _0 \frac{{\partial \vec E}}{{\partial t}}} \right|}} \approx 10^8\)
Therefore, for the evolution regimes justifying the use of Ohm's law, the displacement current is within the ohmic conductor, negligible compared to the conduction current.
The Maxwell-Ampere equation is then :
\(\overrightarrow {rot} \;\vec B = \mu _0 \vec j = \mu _0 \sigma \vec E\)
Rappel : Electrical neutrality
We assume that at the instant \(t=t_0\), there is a point M inside a conductor of a volume charge \(\rho (M,t_0)\).
How does this volume charge vary over time ?
The Maxwell-Gauss equation, Ohm's law and the local charge conservation :
\(div\;\vec E = \frac{\rho }{{\varepsilon _0 }}\;\;\;\;\;;\;\;\;\;\;\vec j = \sigma \vec E\;\;\;\;\;;\;\;\;\;\;div\;\vec j + \frac{{\partial \rho }}{{\partial t}} = 0\)
possible to write :
\(div\;\frac{1}{\sigma }\vec j = - \frac{1}{\sigma }\frac{{\partial \rho }}{{\partial t}} = \frac{\rho }{{\varepsilon _0 }}\;\;\;\;\;\;\;so\;\;\;\;\;\;\;\frac{{\partial \rho }}{{\partial t}} + \frac{\sigma }{{\varepsilon _0 }}\;\rho = 0\)
By integration :
\(\rho (M,t) = \rho (M,t_0 )\;\exp \left( { - \frac{{t - t_0 }}{{\tau _d }}} \right)\;\;\;\;\;\;\;with\;\;\;\;\;\;\;\tau _d = \frac{{\varepsilon _0 }}{\sigma }\)
For copper :
\(\tau_d \approx 4.10^{-14}\;s\)
Very quickly, the conductor becomes neutral in volume :
\(\rho (M,t)=0\)
Thus, as in stationary state, charges accumulate in the immediate vicinity of the surface of a conductor, hence the importance of the concept of surface charge \(\sigma\).
Attention : Maxwell equations in a conductor
Finally, as part of QSRA (quasi – stationary regime approximation ), the electromagnetic field satisfies the Maxwell equations "simplified" the following :
\(\begin{array}{l}div\;\vec B = 0 \\div\;\vec E = 0 \\\overrightarrow {rot} \;\vec E = - \frac{{\partial \vec B}}{{\partial t}} \\\;\overrightarrow {rot} \;\vec B = \mu _0 \vec j = \mu _0 \sigma \;\vec E \\\end{array}\)
Thus, in a conductor, the QSRA differs only from the steady state modes by the inclusion of the induction phenomena (Maxwell-Faraday equation).
Complément : Nodes law in the case of QSRA
Since \(\rho = 0\), the conservation equation of the electric charge leads (inside the conductor) to :
\(div\vec j = 0\)
The flux of the volume current vector is conserved, resulting in the validity of the law of branches and nodes within the QSRA.
Remark :
\(\rho\) and \(\rho_{mobile}\) should not be confused : in the conductor, which remains globally neutral, \(\rho=0\).
By cons, the charge carriers, whose charge distribution is \(\rho_{mobile}\), contribute to the current density vector according to the relation :
\(\vec j = \rho_{mobile}\vec v\)
Fondamental : Skin effect
The wave propagation equation of the electric field is :
\(\overrightarrow {rot} (\overrightarrow {rot} \vec E) = - \mu _0 \sigma \frac{{\partial \vec E}}{{\partial t}} = - \Delta \vec E\;\;\;\;\;\;\;so\;\;\;\;\;\;\;\Delta \vec E\; - \mu _0 \sigma \frac{{\partial \vec E}}{{\partial t}} = \vec 0\)
It's an equation of type "diffusion", resulting in conductive heat transfer.
We are looking for complex solutions of the form :
\(\vec E = \vec E_0 f(z)e^{j\omega t}\)
Either :
\(f''(z) - j\mu _0 \sigma \omega f(z) = 0\)
We denote :
\(k = \pm (1 + j)\sqrt {\frac{{\mu _0 \sigma \omega }}{2}}\)
Then, removing the solution which diverges in the metal (It is assumed that the Oz axis is oriented towards the inside of metal from \(0\) to infinity), and denoting \(\delta\) the skin thickness :
\(\delta = \sqrt {\frac{2}{{\mu _0 \sigma \omega }}}\)
We obtain :
\(\vec E = \vec E_0 e^{ - z/\delta } e^{j(\omega t - z/\delta )}\)
The field propagates in the metal but being attenuated by a factor \(\delta\), called skin depth : \(\delta\) corresponds to the order of magnitude of the penetration depth of the wave in the metal.
The higher the conductivity of the material and the wave frequency, the lower this thickness will be.
The wave is absorbed here due to the Joule effect in the conductor, to a thickness of about several \(\delta\).
Considering frequency electromagnetic wave on the order of \(GHz\), \(\delta\) is about \(\mu m\).
This skin effect causes the decrease of the current density as one moves away from the periphery of the conductor and leads to an increase of the resistance of the conductor.