Chapter 1

# Ampère's circuital law

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

A presentation of electrostatics and magnetism

### 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 Right-hand rule.

Ampere's circuital law

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 must-know 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

Visualization of magnetic field lines (Vidéo d'Alain Le Rille, enseignant en CPGE au Lycée Janson de Sailly, Paris)
Magnetic field in a torus (www-fusion-magnetique.cea.fr)

Magnetic field created by a torus (the one from the picture above), composed of whorls crossed by current  :

The cylindar base is used.

Video : magnetic field of a torus (Alain Le Rille, Enseignant en CPGE au lycée Janson de Sailly, Paris)

### Simulation :

A JAVA animation by JJ Rousseau on the field of a torus : click HERE

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Local Ohm's law
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Uniform rotation of a cylinder evenly charged in volume