What does a toroid consist of ? Show that for an ideal toroid of closely wound turns, the magnetic field (i) inside the toroid is constant, and (ii) in the open space inside and exterior to the toroid is zero.

Explanation

A toroid consists of an anchor ring of mean radius R, over which a large number of turns (say N) of an insulated metallic wire is wound.
(i) When a current I is passed through the toroid, the magnetic field produced will be same at all points on the central axis of the right (shown by dotted curve in the figure) and directed along tangent to the ring. Hence,
`oint vecB . vecdl = oint B dl = B oint dl = B (2 pi R)`
and according to Ampere.s circuital law
`oint vecB . vecdl = mu_(0)` (total current enclosed) = `mu_0 (N I)`
`:. " " B. 2 pi R = mu_0 N I`
`implies " " B = (mu_0 N I)/(2piR) = mu_0 n I`
Where `n = (N)/(2 pi R)` = Number of turns per unit length of toroid.
Obviously this magnetic field is constnat at all point inside the toroid.
(ii) If we apply Ampere.s law to find magnetic field either in (a) open space inside the toroid , or (b) open space exterior to the toroid, then for any closed loop of radius .r. , we have
`oint vecB . vecdl = B(2 pi r) = mu_0 (I) = mu_0 (0)`