Electrostatics and Magnetostatics
Ampère's circuital law
Presentation of electricity and magnetism video (Référence : "Physique collégiale")
Fondamental :
Stokes' theorem, which we use without demonstration, is the equivalent of Ostrogradsky's theorem.
Stokes' theorem :
Let be an oriented closed loop and a given surface which is supported by (as if it was a hat and was the extremity).
The normal vector of the surface is oriented with the Righthand rule.
Stokes' theorem is :
This theorem allows us to write Ampère's circuital law in its integral form.
Maxwell Ampère equation in static regime is :
We compute the line integral of the magnetic field around a closed loop at a given moment. A surface is supported by .
Let's use the Maxwell Ampère equation :
We recognize :
The intensity going through .
Hence Ampère's circuital law is :
Attention : Flux and circulation of B (Ampère's circuital law)

The flux piercing through a closed surface of is equal to zero :

Ampère's circuital law :
Méthode : Classic use of Ampere's circuital law
Ampère's circuital law helps computing easily a magnetic field in a highly symmetrical problem, especially in these mustknow classic examples :

Infinite wire (without thickness) crossed by a current :
The cylindrical base is used.

Infinitely long solenoid, composed of whorls by length unit, crossed by a current :
A JAVA animation by JJ.Rousseau on the field of a solenoid : click HERE
Magnetic field created by a torus (the one from the picture above), composed of whorls crossed by current :
The cylindar base is used.
Simulation :
A JAVA animation by JJ Rousseau on the field of a torus : click HERE