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 $\frac{ km _{1} m _{2}}{ r ^{4}}$, 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 center of the loop (as shown in the figure), the current induced in the loop will be proportional to

• A. $m / r ^{3}$
• B. $m ^{2} / r ^{2}$
• C. $m / r ^{2}$
• D. $m ^{2} / r$

Answer 1

The Correct Option is A

The problem involves a special metal \( S \), which conducts electricity without any resistance, placed in a closed wire loop. The loop generates a compensating flux in response to a changing magnetic flux through it, inducing a current that cannot decay due to the zero resistance of the material. This induced current generates a magnetic moment that repels the source of the magnetic field or flux. The system consists of a loop of radius \( a \), with the center at the origin, and a magnetic dipole of moment \( m \) brought along the axis of the loop from infinity to a distance \( r \) from the center of the loop (with \( r \gg a \)). The north pole of the dipole always faces the loop.

Step 1: Magnetic Field Due to a Dipole

The magnitude of the magnetic field \( B \) produced by a magnetic dipole of moment \( m \) at a point on its axis, at a distance \( r \), is given by the formula:

\( B = \frac{\mu_0}{2\pi} \frac{m}{r^3} \)

Where \( \mu_0 \) is the permeability of free space, and \( m \) is the magnetic moment of the dipole. This field is along the axis of the dipole, pointing towards the dipole’s south pole if the north pole is facing the loop.

Step 2: Induced Current in the Loop

The wire loop of radius \( a \) will induce a current to generate a compensating flux in response to the magnetic field produced by the dipole. According to Faraday's Law of Induction, the induced electromotive force (emf) is given by:

\( \text{emf} = -\frac{d\Phi}{dt} \)

Where \( \Phi \) is the magnetic flux through the loop. The induced current in the loop will be proportional to the emf, and because the loop has zero resistance, the current will be proportional to the magnetic field \( B \) at the location of the loop. Since the magnetic field is given by \( B = \frac{\mu_0}{2\pi} \frac{m}{r^3} \), the induced current will be proportional to this field.

Step 3: Current Induced in the Loop

The induced current \( I \) in the loop is proportional to the magnetic field at a distance \( r \) from the dipole’s center. Thus, the current induced in the loop will be proportional to:

\( I \propto \frac{m}{r^3} \)

Final Answer:

The induced current in the loop will be proportional to \( \frac{m}{r^3} \).