Ampère's theorem

The magnetic field produced by a wire can be derived from a general theorem, analogous to Gauss’s theorem but applied to a closed path instead of a surface.

Law — Ampère's theorem

The circulation of $\vec{B}$ along a closed path is proportional to the total enclosed current: $\ev{\Gamma_B = \oint \vec{B}\cdot \dd\vec{s} = \mu_0\,i_\text{concatenata}}$

“Enclosed” means: the sum of the currents passing through any surface bounded by the path. The power of the theorem, as with Gauss’s theorem in electrostatics, lies in the fact that when there is symmetry the circulation can be calculated without integrating: $\vec{B}$ is constant along the chosen path and can be taken outside the integral sign.

Derivation — The field of the wire from Ampère

Choosing a circle of radius $r$ centred on the wire as the path, by symmetry $|\vec{B}|$ is constant along the circle and tangent to it, so $\Gamma_B = 2\pi r\,|\vec{B}|$. The theorem imposes $2\pi r\,|\vec{B}| = \mu_0 i \quad\Longrightarrow\quad |\vec{B}| = \frac{\mu_0 i}{2\pi r}$ We recover exactly the field of the straight wire, now proven rather than postulated.

Ampère’s theorem is the key tool for calculating the field of symmetric configurations: wire, coaxial cable, flat sheet, solenoid and toroid. In its complete form, with the addition of Maxwell’s displacement current, it becomes one of the four fundamental equations of electromagnetism.

Collegamenti

Argomenti: Magnetismo Concetti: Teorema di Ampère Competenze: Analisi dimensionale

Esercizi collegati: Dimensional check of the solenoid field · Field inside a solenoid · Problem — B field in a charging capacitor