Chapter 2

Maxwell's equations

Skin effect in a metallic conductor plunged in a solenoid

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.

A cylindrical solenoid of axis and radius is constituted of whorls by meter.

It is travelled by a variable current of intensity .

We admit the own magnetic field created by the solenoid is uniform inside ( ) :

And equal to zero outside ( ).

We also admit the electric field is ortho-radial :

Question

Determine the electric field inside the solenoid

Hint

  • Which equation of Maxwell explains why an electric field appears when the magnetic field depends on time ?

  • Think about Stoke's theorem.

Solution

Let's use the Faraday's law of induction and Stoke's theorem :

With the hypothesis of the exercise :

So :

Question

A long and massive cylinder with the same axis as the solenoid, of conductivity , height and radius is put inside the solenoid.

Determine the density of current created by the electric field .

What is the observable effect associated to these currents ?

Hint

Solution

Ohm's local law gives :

Heating due to Joule effect : let's compute the density of power

Principle of heating by induction (Vidéo d'Alain Le Rille, enseignant en CPGE au lycée Janson de Sailly, Paris)

Question

Deduce the magnetic field created by the currents on the axis.

Give the conduction under which this “induced” field is negligible in front of the one created by the solenoid.

Hint

Solution

The magnetic field created by the currents on the axis is along .

We can use Ampère's circuital law to calculate this field.

Skin effect in a metallic conductor plunged in a solenoid

The field outside the cylinder of radius is equal to zero.

Using the closed circuit given on the figure, we can write :

Thus :

Finally :

The ratio of these two magnetic fields is :

In other words, , which is the skin depth.

Question

If this condition is not verified, indicate without any justification the repartition of currents in the cylinder.

Hint

Solution

There is a skin effect. The current only exists on the periphery of the cylinder, on a few .

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