Ampère's law

Where Gauss's law relates a field's outward flux through a closed surface to the charge enclosed, Ampère's law relates the magnetic field's circulation around a closed loop to the current passing through it. Circulation here means summing the field's component along the loop's direction all the way around — and exactly as with Gauss's law, that sum depends only on what passes through the loop, never on the loop's exact size or shape. A tight circular loop and a large irregular one, both encircling the same current once, give identical circulation.

This shape-independence is what makes the law computationally useful in symmetric situations, the same way Gauss's law is: for an infinite straight wire, choosing a circular loop centred on the wire makes the field constant by symmetry along the whole loop, and a hard integral becomes simple algebra.

The law as Ampère first wrote it had a hole, found by considering a charging capacitor. Current flows in the wire leading to each plate, but nothing crosses the gap between the plates — yet a compass placed near that gap still deflects, showing a real circulating magnetic field with no current threading the loop that field circles. Maxwell resolved this by recognising that the changing electric field between the plates, as the capacitor charges, acts as a source of magnetic field in exactly the way an ordinary current does — the displacement current. Adding that term completes the law and is what makes it symmetric with Faraday's law, where a changing magnetic field similarly produces a circulating electric field. That symmetry between the corrected Ampère's law and Faraday's law is what allows changing electric and magnetic fields to sustain one another with no charges or currents present at all — an electromagnetic wave.