Reflection of an electomagnetic wave on a non perfect metal
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Determine the corresponding magnetic fields.
The structure of a progressive harmonic plane wave gives to the incident wave and the reflected wave, the following expressions of magnetic fields :
For the transmitted wave, the structure of relationship is written as :
Assume there are no surface currents.
By writing the boundary condition of the electrical and magnetic fields, to establish the expression of as a function of .
Ensure that for (what is assumed hereafter), we have :
limiting the calculations to the order in .
The boundary condition of the electrical and magnetic fields in leads to the following two equations :
These two equations allow to deduce the complex transmission coefficient :
The first order in :
In fact, the conductor has a surface in the plane .
Calculate the time average flux of the Poynting vector in .
What does this magnitude represent ?
Show that the time average of the power dissipated by Joule effect in a volume of conductive element is :
Deduce the temporal average power dissipated by Joule effect throughout the conductor.
Compare with the results of the previous question.
The electric field is transmitted :
Using complex expressions of and , we get :
Thus, for :
Similarly, the transmitted magnetic field is, at :
The mean value of the transmitted Poynting vector, at , is :
The flux of this vector through the conductor surface is the electromagnetic power transmitted to the metal :
The average value of the power density dissipated by Joule effect within the metal (the depth ) is :
And the time average of the power dissipated by Joule effect in a volume of conductive element is :
The total average power dissipated in the conductor will be :
And, finally :
We find the same expression as before :
In time average, the electromagnetic power transmitted from the wave to the metal is completely dissipated by Joule effect.