Electrostatics and Magnetostatics
Gauss' law
Presentation of electricity and magnetism video (Référence : "Physique collégiale")
Fondamental :
From the MaxwellGauss 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 is still true in a timedepending regime, even though the electric charges can be moving.
In steadystate, 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 steadystate, although is not the only source of electric field anymore. Thus electric field maps are not necessarily similar (See the consequence of the MaxwellFaraday equation).
Attention : Gauss' law
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.

Infinite wire linearly charged ( ) :
Here the cylindrical base is used.
Simulation : A JAVA animation about Gauss' law (JJ Rousseau, Université du Mans)
Gauss' law : click HERE