Field Inside a Toroid

N total turns, r = distance from toroid axis. Field is confined entirely inside the toroid body; B = 0 in the central cavity and outside. Follows from Ampere's law on concentric circular loops.

Setup

A toroid has $N$ total turns wound uniformly on a torus with inner radius $r_1$ and outer radius $r_2$, carrying current $I$. By azimuthal symmetry, $\vec{B} = B(r)\,\hat{\phi}$ inside the toroid.

Three Regions

Region 1 — Cavity ($r < r_1$):

Circular Amperian loop of radius $r$: no current threads through it.

$$B \cdot 2\pi r = 0 \implies B = 0$$

Region 2 — Inside toroid ($r_1 < r < r_2$):

All $N$ turns thread through the loop, each carrying current $I$:

$$B \cdot 2\pi r = \mu_0 N I$$ $$\boxed{B = \frac{\mu_0 N I}{2\pi r}}$$

Region 3 — Outside ($r > r_2$):

Each turn passes through the loop twice in opposite directions — net enclosed current is zero.

$$B \cdot 2\pi r = 0 \implies B = 0$$

Comparison with Solenoid

The field inside a toroid varies as $1/r$ — not uniform. For a thin toroid ($r_2 - r_1 \ll r$), with $n = N/2\pi r$:

$$B \approx \mu_0 n I$$

matching the solenoid result.