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 steadystate.
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 selfinductances 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 :