Obtain an expression for magnetic field inside the toroid.

Explanation

Consider three circular Amperian loops 1, 2 & 3 as shown in the figure by dotted line.
Let the magnetic field along loop 1 be `B_(1)` in magnitude. The length of the length `intdl=2pir_(1)`. This current loop does not enclose current. Therefore `I_(e)=0 " and " B_(1)=0`. Therefore, the magnetic field at any point P in the open space inside the toroid is zero. Consider the Amperian loop 3. Let `B_(3)` be the magnetic field along the loop. Here `intdl=2pir_(3)`. The current coming out of the planes is cancelled exactly by the current going into it. Therefore `I_(e)=0 " and " B_(3)` at any point Q outside the toroid is zero.
Let the magnetic field inside the toroid be B. The length of Amperian in loop `2=intdl=2pir_(2)`. The current enclosed for N turns of toroidal coil is NI.
From Amperian Circuital law,
`" "Bintdl=mu_(0)I_("enclosed")`
i.e., `" "B(2pir_(2))=mu_(0)NI`
i.e., `" "B=(mu_(0) NI)/(2pir_(2))`
Let the number of turns per unit length be n. Then `N/(2pir_(2))=n`.
Thus, `" "B=mu_(0)nI`.