Chapter 3

Electromagnetic induction

Test your understanding of the lesson

Question

  • Give the volume expression of the Laplace's force applied to a conductor in a magnetic field .

  • What is the  Laplace's force which applies to an element of conductor of length , travelled by intensity when immersed in a magnetic field  ?

Hint

Solution

  • Volume expression of the Laplace's forces :

  • Laplace's Force :

Question

Write Lenz's and Faraday's laws.

Hint

Solution

  • Lenz's law :

    The induced current has such a sign that the induced flux created opposes the variations of inductor flux.

    Or :

    The electromotive force tends because of its consequences to oppose the cause which made it appear.

  • Faraday's law :

    The electromotive force induced in a closed, steady circuit in the Galilean laboratory is opposed to the time derivative of the magnetic field through the circuit :

Question

Prove Faraday's law from one of Maxwell's equations.

Hint

Solution

Maxwell-Faraday equation :

Let be a closed wire circuit.

Stokes's theorem :

Hence :

Question

Define self-inductance and mutual inductance .

Hint

Solution

  • Self inductance :

    A closed thread-like circuit is travelled by an intensity .

    The flux of the circuit's own magnetic field through an outline oriented by the positive sense of the chosen current, or, proper flux, is proportional to :

    is the self inductance of the circuit , also known as induction coefficient.

  • Mutual inductance :

    Two thread-like circuits and are travelled by two intensities and .

    The flux of the magnetic field through the closed outline , oriented with the positive sign of the current is proportional to  :

    Likewise, the flux of the magnetic field through the closed outline , oriented with the positive sign of the current is proportional to .

    is the mutual inductance of the two circuits.

Question

What is the relation between the Laplace's force and the power of the electromotive force of induction ?

Hint

Solution

The power of the electromotive force of induction is compensated by that of the Laplace's force applied to the circuit :

Question

What is the self-inductance of a solenoid ?

Hint

Solution

The magnetic field inside an “infinite” solenoid is :

The proper flux through the whorls over the length is :

Hence the self-inductance :

Order of magnitude :

For a coil of length made of a whorls of diameter , the self-inductance is around : Henry is a big unit.

Question

What is the magnetic energy of two coupled circuits ?

Let , and be the coefficients of self and mutual inductance.

Hint

Solution

The magnetic energy of a system of two circuits of self-inductance and and mutual inductance , which are travelled by the two respective currents and is :

Question

  • What is the relation between the voltage of the primary winding and the voltage of the secondary for a perfect transformer ?

  • What is an isolation transformer ?

Hint

Solution

  • Let and  be the number of whorls in the primary and secondary windings :

  • In the case , .

    The secondary voltage difference is the same as in the primary, yet it is isolated from the primary.

    Mass problems are avoided.

Question

Define Foucault's currents (or Eddy's currents).

Hint

Solution

The currents induced in massive mobile conductors placed in permanent magnetic fields are called Foucault's currents.

They cause Laplace's forces which tend to oppose the movement which made them appear.

This is the principle of electromagnetic braking, used in particular for heavy weights, haulers and high speed trains.

Foucault's currents are also used in induction cookers.

Question

A coil made of whorls is shortcut and placed in a magnetic field oriented along the axis of the coil.

The value of the magnetic field changes at the speed of .

What is the thermal power released in the coil, considering its section to be and its internal resistance to be  ?

Hint

Solution

The electromotive force is given by Faraday's law of induction :

The intensity is :

And the dissipated power is :

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