Field and Wave Electromagnetics 2nd Edition by David Cheng

Exercise 13

In the d-e motor illustrated in Fig. 6- 32 we noted that a current I sent through the loop in a magnetic field B produces a torque that makes the loop rotate. As the loop rotates, the amount of the magnetic flux linking with the loop changes, giving rise to an induced emf. Energy must be expended by an external electric source to counter this emf and establish the current in the loop. Prove that this electric energy is equal to the mechanical work done by the rotating loop. (Hint : Consider the normal of the loop at an arbitary angle ex with B, and let it rotate by an angle .)