Ampère's law

Ampère's law in its static form says that the line integral of the magnetic field B around any closed loop equals μ₀ times the total electric current threading the loop: ∮ B·dℓ = μ₀ I_enc. The "enclosed current" is the integral of current density J·dA over any surface bounded by the loop — and any surface gives the same answer, because charge is conserved.

Ampère's law is to magnetostatics what Gauss's law is to electrostatics. Both are exact integral statements of Maxwell's equations; both let you compute fields the easy way for symmetric current distributions; both fail to determine the field uniquely if the symmetry is broken. The trick is always the same: pick an Amperian loop whose geometry matches the symmetry of the current, so that B is constant in magnitude and parallel to dℓ along the loop. The textbook applications are an infinite straight wire (circular loop concentric with the wire, gives B = μ₀ I / (2πr) immediately), a long solenoid (rectangular loop straddling the windings, gives B = μ₀ n I where n is turns per length), and a toroid (circular loop running through the doughnut hole, gives B = μ₀ N I / (2πr)).

Maxwell completed Ampère's law in 1865 by adding a "displacement current" term: ∮ B·dℓ = μ₀ (I_enc + ε₀ d/dt ∫ E·dA). The extra piece says that a changing electric field acts like a current as far as B is concerned — which is what makes electromagnetic waves possible. For purely steady currents the displacement-current term vanishes and the law reduces to its 1820 form, which is what's used in magnetostatics.