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