Derivation of Eq. (27.26) for a Circular Current Loop A wire ring lies in the xy-plane with its center at the origin. The ring carries a counter clockwise current I (Fig). A uniform magnetic field B is in the. + x-direction, B = Bxi (The result is easily extended to B in an arbitrary direction.)
(a) In Fig show that the element dl = R dθ (-sinθi + cosθj), and find DF = Idl x B.
(b) Integrate df around the loop to show that the net force is zero.
(c) From part (a), find dT = r x df, where r = R (cosθi + sinθj) is the vector from the center of the loop to the element dt. (Note that dt is perpendicular to r.)
(d) Integrate d1 over the loop to find the total torque 'I on the loop. Show that the result can be written as T = µ x B, where µ = IA. (Note: fcos2x dx = ½ x + ¼sin2x, fsin2x dx = ½x €“ ¼sin 2x, and fsin xcosx dx = ½ sin2x.)

Transcribed Image Text:

de