Chapter 2

# 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 .

### 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 ?

### 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.

### 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

### 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 ?

### Solution

• Maxwell's equations are :

(Gauss' law for electricity, MG)

(Gauss' law for magnetism)

(Ampère's law, MA)

• In a metallic conductor :

## Question

Prove the integral form of Gauss' law of electricity

### Solution

Let us apply the divergence theorem :

So :

## Question

Prove the integral form of Ampere's circuital law.

### Solution

Let us apply Stoke's theorem :

So :

## Question

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

### 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.

### 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 .

### 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 ?

### 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.

### 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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