State Ampere's circuital law

Ampere's circuital law: the line integral of the magnetic field intensity H around any closed path equals the net current enclosed by that path.

∮ H·dl = I_enc          (point form: ∇×H = J)

The direction of the current and the direction of the path follow the right-hand rule. The law is used to find H for symmetrical current distributions by choosing a closed path (Amperian loop) on which H is constant and tangential.

Example: infinitely long straight wire carrying current I

Take a circle of radius ρ around the wire as the Amperian path. By symmetry H has only an a_φ component of constant value on the circle.

∮ H·dl = H_φ (2πρ) = I
H = I/(2πρ)  a_φ     (A/m)

The same method gives the field of a coaxial cable, a solenoid (H = NI/l inside), and a toroid (H = NI/(2πρ)).