A solenoid is bent in such a way its ends are joined together to form a closed ring shape, is called a toroid which is shown in Figure 3.47. The magnetic field has constant magnitude inside the toroid whereas in the interior region (say, at point P) and exterior region (say, at point Q), the magnetic field is zero.
Let us calculate the magnetic field BP at point P. We construct an Amperian loop 1 of radius r1 around the point P as shown in Figure 3.48. For simplicity, we take circular loop so that the length of the loop is its circumference.
L1 = 2πr1
Ampère’s circuital law for the loop 1 is
Since, the loop1 encloses no current, Ienclosed = 0
This is possible only if the magnetic field at point P vanishes i.e.
Let us calculate the magnetic field BQ at point Q. We construct an Amperian loop 3 of radius r3 around the point Q as shown in Figure 3.48. The length of the loop is
L3 = 2πr3
Ampère’s circuital law for the loop 3 is
Since, in each turn of the toroid loop, current coming out of the plane of paper is cancelled by the current going into the plane of paper. Thus, Ienclosed = 0
This is possible only if the magnetic field at point Q vanishes i.e.
Let us calculate the magnetic field BS at point S by constructing an Amperian loop 2 of radius r2 around the point S as shown in Figure 3.48. The length of the loop is
L2 = 2πr2
Ampere’s circuital law for the loop 2 is
Let I be the current passing through the toroid and N be the number of turns of the toroid, then
The number of turns per unit length is n = N/2πr2 , then the magnetic field at point S is