Question

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 centre at the origin. A magnetic dipole of moment $m$ is brought along the axis of this loop from infinity to a point at distance $r(>>$ a) from the centre 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 $\frac{{\mu }_0}{2\pi }\frac{m}{r^3}$, 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 $\frac{k{m}_1{m}_2}{r^4}$, 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

• A. $m/r^5$
• B. $m^2/r^5$
• C. $m^2/r^6$
• D. $m^2/r^7$

Answer 1

Answer: (C)

$dW=F.dx=\frac{-Km\left(i_1\pi a^2\right)}{r^4}(dr)$
$i_1=\frac{ma}{\pi r^3}$
$W=\underset{\infty }{\overset{r}{\int }}\frac{km\left(\frac{ma}{\pi r^3}\times \pi a^2\right)dr}{r^3}=k{m}^2{a}^3\int \frac{dr}{r^7}$
$W\alpha \frac{m^2}{r^6}$