Chapter 11

# Reflection of an electomagnetic wave on a non perfect metal

Take 20 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

Ohmic conductor of conductivity occupies the half-space , the space being vacuum.

An incident wave of the form : propagates in a vacuum.

It gives birth to a transmitted wave of the form (see the lecture notes on the skin effect) : With : And a reflected wave of the form : ## Question

Determine the corresponding magnetic fields.

### Solution

The structure of a progressive harmonic plane wave gives to the incident wave and the reflected wave, the following expressions of magnetic fields : And : For the transmitted wave, the structure of relationship is written as : ## Question

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 .

### Solution

The boundary condition of the electrical and magnetic fields in leads to the following two equations :

• • so These two equations allow to deduce the complex transmission coefficient : The first order in : Hence : ## Question

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.

### Hint

• One can calculate the average value of the Poynting vector with the expressions of the complex fields using the relationship : • Use local Ohm's law ( )

### Solution

• 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 : Hence : 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 : So : 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.

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Electromagnetic waves in metals, skin effect
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Group velocity of a wave packet