Toroid

A toroid is a coil of wire wound around a doughnut-shaped core. If the coil is uniformly wound and the doughnut is closed (no gap), the magnetic field is entirely confined to the interior of the torus and runs in circles around the doughnut hole. The leakage field outside the toroid is negligibly small — far smaller than that of a comparable solenoid, because the field has nowhere to escape to.

Apply Ampère's law to a circular loop concentric with the doughnut hole, lying inside the torus, and you get BL = μ₀ N I, where N is the total number of turns wound on the toroid and L = 2πr is the loop circumference. So B = μ₀ N I / (2πr) inside, dropping smoothly with radius across the doughnut cross-section. Outside the torus (either through the hole or around the outer rim), the loop encloses zero net current — the windings come back through the inside as many times as they go out — and B = 0.

Toroidal coils are everywhere in power electronics. Every switching power supply contains toroidal inductors and transformers because the closed magnetic circuit means almost no stray flux to interfere with neighbouring components. Toroidal transformers in audio amplifiers run quieter and produce less external hum than equivalent E–I core designs. Tokamak fusion reactors confine deuterium-tritium plasma in a toroidal magnetic bottle for the same fundamental reason: a closed-loop B field has no end-points where the plasma could escape along field lines.