A special metal \( S \) conducts electricity without any resistance. A closed wire loop, made of \( S \), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \( a \), with its center at the origin. A magnetic dipole of moment \( m \) is brought along the axis of this loop from infinity to a point at distance \( r \, (\gg a) \) from the center of the loop with its north pole always facing the loop, as shown in the figure below.
The magnitude of magnetic field of a dipole \( m \), at a point on its axis at distance \( r \), is \( \dfrac{\mu_{0} m}{2\pi r^{3}} \), where \( \mu_{0} \) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \( m_{1} \) and \( m_{2} \), separated by a distance \( r \) on the common axis, with their north poles facing each other, is \( \dfrac{k\, m_{1} m_{2}}{r^{4}} \), where \( k \) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.
The work done in bringing the dipole from infinity to a distance \( r \) from the center of the loop by the given process is proportional to:
