ALTERNATING-CURRENT CIRCUITS — Edward Purcell & David Morin
Physics — E&M · mutual-inductance-symmetry
Problem
Mutual-inductance symmetry ** In Section 7.7 we made use of the vector potential to prove that M12=M21. We can give a second proof, this time in the spirit of Exercise 3.64. Imagine increasing the currents in two circuits gradually from zero to the final values of I1f and I2f (``f'' for ``final''). Due to the induced emfs, some external agency has to supply power to increase (or maintain) the currents. The final currents can be brought about in many different ways. Two possible ways are of particular interest.
((a)) Keep I2 at zero while raising I1 gradually from zero to I1f. Then raise I2 from zero to I2f while holding I1 constant at I1f.
((b)) Carry out a similar program with the roles of 1 and 2 exchanged, that is, raise I2 from zero to I2f first, and so on.
Compute the total work done by external agencies, for each of the two programs. Then complete the argument. See Crawford (1992) for further discussion.