Solenoid

A solenoid is a coil of wire wound into a long cylindrical helix. When current flows through it, each turn produces its own field, and the net result is a strong, uniform magnetic field along the axis inside the coil and a much weaker, leaky field outside. For an idealised infinite solenoid, the inside field is exactly B = μ₀ n I, where n is the number of turns per unit length and I is the current — independent of the radius of the coil and of where you sit inside it.

Real solenoids are finite, but as long as the length is at least several times the diameter, the central region behaves close to ideal. Lab-grade solenoids reach a few hundred millitesla; superconducting solenoids in MRI machines and particle physics experiments reach 1.5 to 12 T routinely; the bore magnets at the LHC and ITER are technologically heroic versions of the same basic device. At the small end, every electric door bell, cigarette lighter, fuel injector, and starter motor is built around a small solenoid that pulls or pushes a ferromagnetic plunger when energised.

Solenoids are the cleanest application of Ampère's law. Apply it to a rectangular Amperian loop straddling the coil's wall — one side inside, one side outside, the two short sides perpendicular to the axis. Outside the field is approximately zero; inside it runs along the axis; the perpendicular short sides contribute nothing because B is perpendicular to dℓ there. The line integral collapses to BL = μ₀ (nL) I, giving B = μ₀ n I in two lines of algebra. No other geometry in undergraduate magnetism rewards Ampère's law so cleanly.