12.4 Ampère’s Law

Ampère’s Law is the magnetic analogue of Gauss’s law for highly symmetric current distributions. It states

This means the circulation of the magnetic field around a closed loop equals ${\mu_0}$ times the net current enclosed by the loop. Ampère’s law is most useful when symmetry allows B to be constant in magnitude along parts of the loop and parallel to ${d\mathbf{\ell}}$.

For a long straight wire, choosing a circular Amperian loop gives

so

For an ideal solenoid, Ampère’s law gives

inside, where n is turns per unit length, and approximately zero outside. For a toroid, the field is

inside the core region. Ampère’s law is therefore the fastest route to magnetic fields in systems with cylindrical or loop symmetry.