Chapter 1

# Gauss' law

### Presentation of electricity and magnetism video (Référence : "Physique collégiale")

A presentation of electrostatics and magnetism

### Fondamental :

From the Maxwell-Gauss equation, we can calculate the flux of the electric field exiting through a closed surface  :

Where represents the charges inside the closed surface .

Gauss' law

Gauss' law is still true in a time-depending regime, even though the electric charges can be moving.

In steady-state, the sources of the electric field are the charges  characterized by their density .

The field lines diverge from positive charges like a fluid coming out of a true source. They disappear on negative charges like a fluid in a well.

It is still true in a non steady-state, although is not the only source of electric field anymore. Thus electric field maps are not necessarily similar (See the consequence of the Maxwell-Faraday equation).

### Complément : Gauss' law for the gravitationnal field

We can observe the formal analogy between the electric Coulomb field created by a punctual charge :

And the Newton gravitational field created by a punctual mass :

represents the universal gravitational constant.

We can then associate the charge to the mass and the constant to .

Thus, Gauss' law remains true for the integral form of the gravitational field :

And its local form :

Where is the density at M, the point considered.

### Attention : Gauss' law for the gravitationnal field

And its local form :

### Méthode : Classic uses of Gauss' law

In highly symmetrical problems, Gauss' law is an easy way to compute the electric field. It is useful in these classic situations and must be remembered.

• Infinite plane evenly charged in surface (with constante ) :

above the plane and below the plane

Where (Oz) is perpendicular to the charged plane.

• Sphere of center O and radius equal to R, charged in volume ( and is the total charge, ) :

If  :

If  :

Here the spherical base is used.

Electric fiel of a sphere (Yves Pelletier, http://web.ncf.ca/ch865)
• Infinite wire linearly charged ( ) :

Here the cylindrical base is used.