Toroid

A toroid is a solenoid bent around until its two ends join, forming a closed ring; the field turns out to be confined almost entirely within the body of the ring. Three regions are checked using three different circular Amperian loops, all centred on the ring's central axis:

• Open space interior to the toroid (point $P$, loop 1 of radius $r_1$): this loop encloses no current, $I_{enclosed}=0$, so $\oint\vec B\cdot d\vec l = 0$, which forces $B_P = 0$.
• Open space exterior to the toroid (point $Q$, loop 3 of radius $r_3$): for every turn, the current coming out of the page is matched by the same turn's current going back into the page (the winding threads back and forth through the loop), so the net enclosed current is again zero, giving $B_Q = 0$.
• Inside the body of the toroid (point $S$, loop 2 of radius $r_2$, length $L_2=2\pi r_2$): this loop encloses all $N$ turns once, $I_{enclosed}=NI$, so $B_S(2\pi r_2) = \mu_0 NI$, giving $\boxed{B_S = \mu_0\,\frac{N}{2\pi r_2}\,I = \mu_0\,n\,I}, \qquad n=\frac{N}{2\pi r_2}$ …