Exercise 4.1.16: Toroidal coil

The toroid is invariant under rotation about its axis, so $H$ depends on $\rho$ and $z$ only. The winding is a sheet of current that flows along the meridians, and by the same argument as for the solenoid $H=H_{\varphi }(\rho ,z)a_{\varphi }$: the field lines are circles threading the winding.

Take $C$ to be the circle of radius $\rho$ coaxial with the toroid. For $a<\rho <b$ (inside the winding) it is pierced by all $N$ turns, each carrying $I$ in the same sense. For $\rho <a$ it encloses no current, and for $\rho >b$ every turn pierces it twice in opposite directions, net zero:

$NI=800$ A:

i.e. $B=2.0$ mT, $1.6$ mT and $1.33$ mT: the field is $50\%$ larger at the inner edge than at the outer one.

Units: A/m. A toroid is a solenoid bent into a ring: $NI/2\pi \rho$ is $nI$ with the local turn density $n=N/2\pi \rho$. Its great advantage is that it has no outside field at all, whatever $N$, which is why toroidal cores are used in transformers and chokes that must not disturb (or be disturbed by) their neighbours. A thin toroid ($b-a\ll a$) has an almost uniform field $NI/2\pi {\rho }_{\text{mean}}$.