Chapter 2

Maxwell's equations

Magnetic energy stored in a coil

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 coil of length , radius and axis , is made of the winding of joined circular whorls by length unit.

Let us assimilate the coil to an infinite solenoid, and suppose the steady-state.

Question

The current travels through the coil.

Determine the magnetic field created by the coil.

Hint

The point of this exercice is to review electromagnetic energy. Follow the guidelines...

Can you write the local and then integral electromagnetic energy ?

Solution

The magnetic field is (see lesson) :

Question

What is the magnetic energy of the coil ? Deduce the value of the inductance L of the coil.

Hint

Solution

Magnetic energy can be expressed in two ways :

Hence :

Question

The coil is put in a series circuit.

There is a resistor and an electromotive force generator of constant value .

Determine the expression of the current in the coil in function of time.

Hint

Solution

We use Kirchoff' laws :

Classically :

Question

Compute the magnetic and electric fields created by the coil in any point at the moment .

Hint

Solution

Let's write :

Inside :

Outside the coil, the field is equal to zero.

The electric field is orthogonal to the radius (study the symmetries).

It depends on and time.

Let us use Stoke's theorem on a circle :

  • If  :

  • If  :

Question

Determine the density of magnetic and electric energy.

What can be noticed about the ratio of these two energies ? Conclude

Hint

Solution

The density of magnetic energy is :

The density of electric energy is, for example in where it is maximum : (using )

Let's calculate the ratio :

, with .

The electric energy is negligible.

In a steady-state approximation, a coil is almost only magnetic.

Question

What is the expression of Poynting's vector through a surface delimiting the volume of the coil ? Comment on that.

Hint

Solution

Let's compute Poynting's vector in  :

The flux entering the coil is then :

This flux indeed corresponds to the variation of the magnetic energy stored by the coil by time unit.

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