Ferromagnetic energy question

[alex.rechilie]

Consider a brick-shaped ferromagnetic of relative magnetic permeability µ >> 1, which has dimensions 2a × 2a × a and a narrow slit of width d and depth a sawed into it as shown in Figure. You may assume that µd >> a >> d. A circular loop of diameter a and inductance L, made of a superconducting material, is put into that slit; the loop carries electric current I. What is the mechanical work A needed to be done in order to pull the loop out of the slit and move it to a large distance from the ferromagnetic?

Remarks: (a) the inductance of the loop equals to L when it is far away from the ferromagnetic. (b) The current in the loop equals to I when the loop is inside the slit.

 

[Vailanor]

The mechanical work A required to pull the loop out of the slit can be calculated by taking into account the change in magnetic field energy due to the displacement of the loop. The magnetic field energy inside the ferromagnetic is given by U = (µL/2)I2, and outside the ferromagnetic it is U = (L/2)I2. Thus, the mechanical work A required to move the loop from inside the ferromagnetic to a large distance away is given by A = (µ - 1)LI2/2.