Solenoids & Toroids

Wind wire into a long tight coil and you get a remarkably uniform magnetic field inside and almost none outside: the electromagnet in its simplest form.

The solenoid#

A solenoid is insulated wire wound closely into a helix, usually long compared with its width. Think of it as a stack of circular loops. Two coaxial loops with equal currents give fields that add near the axis but oppose well off it. Stack many and the fields outside cancel more and more while those inside reinforce: a tightly wound long solenoid has a uniform field inside and zero outside.

Apply Ampère’s law to a rectangle PQRS with side PQ (length l) inside, parallel to the axis, and the opposite side outside. B⃗·dl⃗ is Bl along PQ and zero on the other three sides (B is zero outside, and perpendicular to the two short sides inside). If there are n turns per unit length, nl turns, each carrying i, pass through the rectangle:

Strictly this is for an infinitely long solenoid, but it holds well in the middle part of any solenoid more than five or six times as long as its diameter.

The toroid#

Bend a solenoid into a ring and join its ends (or wind wire closely round a ring) to get a toroid. Take a circle of radius r inside the winding, concentric with the ring. By symmetry B is tangent to it and the same all round, so ∮B⃗·dl⃗ = 2πrB. With N turns in all, the current through the circle is Ni: