Applications of Ampere’s Circuital Law > Magnetic Field of a Toroidal Solenoid

The magnetic field produced by a toroid is confined almost entirely within the core of the ring, unlike a straight solenoid where field lines exist along the full length.

A toroid is a hollow circular ring (like an anchor ring) on which a large number of turns of insulated wire are closely wound. It is essentially a straight solenoid bent into a closed circular shape, forming an "endless solenoid."

• Consider a toroid of average radius r, with N total turns, carrying a steady current I.
• Inside the core, the magnetic field lines form concentric circles, tangential to any circular path drawn through the core.
• To apply Ampere's Law, choose an Amperian loop — a circle of radius r passing through the point where the field is to be calculated, concentric with the toroid's axis.

Step 1 - Apply Ampere's Circuital Law: \[\oint\vec{B}\cdot d\vec{l}=\mu_0I_{enc}\]​

Step 2 - Field is tangential and constant in magnitude along the loop: Since B is tangential to the circular Amperian loop of radius r and has the same magnitude at every point on it, B(2πr) = μ₀NI

Step 3 - Solve for B: B = \[\frac{\mu_0NI}{2\pi r}\]

Step 4 — Express in terms of turns per unit length: Let n = \[\frac {N}{2πr}\]​ (number of turns per unit length of the toroid).

Think of a toroid like a circular garden hose coiled into a ring — water (current) flows through the coiled hose, and the "pressure field" (magnetic field) stays trapped within the ring-shaped hose itself, never leaking to the empty space inside the ring's hole or outside the ring.

Practical Application: Toroidal transformers and inductors use this principle to confine magnetic flux, reducing electromagnetic interference — commonly seen in power adapters and audio equipment.