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 $\frac{\mu_0}{2 \pi} \frac{m}{r^3}$, 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.

($1$) When the dipole $m$ is placed at a distance $r$ from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to

$(A)$ $\frac{m}{r^3}$ $(B)$ $\frac{m^2}{r^2}$ $(C)$ $\frac{m}{r^2}$ $(D)$ $\frac{m^2}{r}$

($2$) 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

$(A)$ $\frac{m}{r^5}$ $(B)$ $\frac{m^2}{r^5}$ $(C)$ $\frac{m^2}{r^6}$ $(D)$ $\frac{m^2}{r^7}$

Give the answer or qution ($1$) and ($2$)

Q: 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 $\frac{\mu_0}{2 \pi} \frac{m}{r^3}$, 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. ($1$) When the dipole $m$ is placed at a distance $r$ from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to $(A)$ $\frac{m}{r^3}$ $(B)$ $\frac{m^2}{r^2}$ $(C)$ $\frac{m}{r^2}$ $(D)$ $\frac{m^2}{r}$ ($2$) 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 $(A)$ $\frac{m}{r^5}$ $(B)$ $\frac{m^2}{r^5}$ $(C)$ $\frac{m^2}{r^6}$ $(D)$ $\frac{m^2}{r^7}$ Give the answer or qution ($1$) and ($2$)

b $\phi= L 1=\frac{\mu_0 m }{2 \pi r ^3} \times \pi a ^2$ $\Rightarrow i =\frac{\mu_0 m \pi a ^2}{2 \pi r ^3 L }$ $\Rightarrow i \propto \frac{ m }{ r ^3}$ $m ^{\prime}=\pi a ^2 i =\frac{\mu_0 m ^2 a ^4}{2 \pi r ^3 L }$ $F =\frac{ km ^2 \pi^2 a ^4}{2 \pi r ^7 L }$ $W =\int Fdr \propto \int \frac{ m ^2 dr }{ r ^7}$ $W \propto \frac{ m ^2}{ r ^6}$

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 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 $\frac{\mu_0}{2 \pi} \frac{m}{r^3}$, 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.

($1$) When the dipole $m$ is placed at a distance $r$ from the center of the loop (as shown in the figure), the current induced in the loop will be proportional to

$(A)$ $\frac{m}{r^3}$ $(B)$ $\frac{m^2}{r^2}$ $(C)$ $\frac{m}{r^2}$ $(D)$ $\frac{m^2}{r}$

($2$) 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

$(A)$ $\frac{m}{r^5}$ $(B)$ $\frac{m^2}{r^5}$ $(C)$ $\frac{m^2}{r^6}$ $(D)$ $\frac{m^2}{r^7}$

Give the answer or qution ($1$) and ($2$)

A$A.B$
B$A.C$
C$A.D$
D$A.B,C$

IIT 2021, Advanced

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