Consider an infinitely long solenoid with n turns per unit length around a ferromagnetic c...

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 v1 = - n dΦ/dt is induced per unit length, which opposes the current change. Power P1 =  - v11 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 Bf is

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b) Assuming that the current is changed in a periodic manner such that B is reduced from Bf to  -Bf and then is increased again to Bf , 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.