Chapter 2

Maxwell's equations

To test the understanding of the lesson

Question

Write the electric current density vector in an environment that contains carriers of charge by volume unit, which move at the velocity .

Hint

Solution

The electric current density vector is :

Question

How can we define the intensity in the presence of a volume current ? And of a surface current ?

Hint

Solution

  • Volume current : the intensity of current is the flux of the vector through a surface S :

    If the vector is uniform and perpendicular to the surface, then :

  • Surface current : the integral becomes linear :

    If the vector is uniform and perpendicular to the segment of length  :

Question

Give the local expression of the charge conservation principle.

Hint

Solution

It is a classical conservation equation :

is the total density of charges, and not only the density of the mobile charges which appears in the definition of .

Question

Demonstrate Kirchhoff's law in steady-state.

Hint

Solution

In steady state, . The flux of is thus conserved :

Or, equivalently, on a field tube (or current tube) at a nodal point :

Hence we have demonstrated Kirchhof's law :

Question

  • Write all four equations of Maxwell in the general case.

  • How can they be simplified in a conductor ?

Hint

Solution

  • Maxwell's equations are :

    (Gauss' law for electricity, MG)

    (Gauss' law for magnetism)

    (Faraday's law of induction, MF)

    (Ampère's law, MA)

  • In a metallic conductor :

Question

Prove the integral form of Gauss' law of electricity

Hint

Solution

Let us apply the divergence theorem :

So :

Question

Prove the integral form of Ampere's circuital law.

Hint

Solution

Let us apply Stoke's theorem :

So :

Question

Why is it said that the flux of the magnetic field is conserved ?

Hint

Solution

The Gauss' law of magnetism, , leads to :

Which means the flux of the magnetic field is conserved.

Question

  • Give the density of electric energy of an electromagnetic field.

  • Give the density of magnetic energy of an electromagnetic field.

Hint

Solution

  • Density of electric energy of an electromagnetic field :

  • Density of magnetic energy of an electromagnetic field :

  • is the total density of energy of the electromagnetic field.

Question

  • Define Poynting's vector in electromagnetism, writen .

  • Give the density of power received by the matter from an electromagnetic field. What is it in the case of a metal conductor ?

  • In the presence of volume current , write the local then global electromagnetic energy of the electromagnetic field .

Hint

Solution

  • Poynting's vector in electromagnetism :

  • Density of power received by the matter from an electromagnetic field :

    For a metal conductor (for which Ohm's local law is verified, ) :

  • Local conservation of electromagnetic energy :

    The integral form of electromagnetic energy conservation is :

Question

  • Give the magnetic energy instantly stored in a coil of inductance , travelled by the intensity .

  • A system is made of two circuits which have respectively the self-inductances and and a mutual inductance .

    Both circuits are crossed respectively by two currents and .

    What is the magnetic energy of the system ?

Hint

Solution

  • The magnetic energy instantly stored in a coil of inductance , crossed by the intensity is :

  • The magnetic energy of the system is :

Question

  • What is the energy stored by a capacitor ?

  • Define the capacitance of a capacitor. Define it in the case of a plane capacitor.

Hint

Solution

  • Electric energy instantly stored in a capacitor of capacitance under the voltage :

  • The capacitance of a capacitor is defined by :

    For a plane capacitor :

    Where is the surface of the frames and the distance between them.

    If a dielectric of relative permittivity is put between the two frames, the capacitance becomes :

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