Field and Wave Electromagnetics 2nd Edition by David Cheng

Exercise 57

Consider an infinitely long solenoid with n turns per unit length around a ferromagnetic core of cross-sectional area S. When a current is sent through the coil to create a magnetic field, a voltage v 1 = - n d /dt is induced per unit length, which opposes the current change. Power P 1 = - v 1 1 per unit length must be supplied to overcome this induced voltage in order to increase the current to I.
a) Prove that the work per unit volume required to produce a final magnetic flux density B f is

b) Assuming that the current is changed in a periodic manner such that B is reduced from B f to - B f and then is increased again to B f , prove that the work done per unit volume for such a cycle of change in the ferromagnetic core is represented by the area of the hysteresis loop of the core material.