7 questions
A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \( m \) and radius \( r \) and it is in a uniform vertical magnetic field \( B_0 \), as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity \( g \), on two conducting supports at \( P \) and \( Q \). When a current \( I \) is passed through the loop, the loop turns about the line \( PQ \) by an angle \( \theta \) given by
A special metal S conducts electricity without any resistance. A closed wire loop , made of S, does not allow any chnage 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.
When the dipole \( m \) is placed at a distance \( r \) from the centre of the loop (as shown in the figure), the current induced in the loop will be proportional to:
