Question

An infinitesimally small bar magnet of dipole moment $M$ is pointing and moving with the speed $v$ in the $x-$direction. A small closed circular conducting loop of radius $a$ and negligible self - inductance lies in the $yz$-plane with its centre at $x=0$, and its axis coinciding with the $x$-axis. Find the induced current in the loop, if the resistance of the loop is $R$. Assume that the distance $x$ of the magnet from the centre of the loop is much greater than $a$.

• A. $\frac{3{\mu }_{0}M{a}^{2}v}{2x^{4}R}$
• B. $\frac{{\mu }_{0}M{a}^{2}v}{2x^{4}R}$
• C. $\frac{3{\mu }_{0}M{a}^{2}v}{x^{4}R}$
• D. $\frac{{\mu }_{0}M{a}^{2}v}{x^{4}R}$

Answer 1

The correct option is A $\frac{3{\mu }_{0}M{a}^{2}v}{2x^{4}R}$

Suppose the magnet is at a distance of $x$ from the centre of the loop.


The magnetic field due to the magnet at the centre of the loop,

$B=\frac{{\mu }_0}{4\pi }\frac{2M}{x^3}$

The magnetic flux due to the magnet,

$\varphi =BA=\frac{{\mu }_0}{4\pi }\frac{2M}{x^3}\left(\pi a^2\right)$

Induced emf in the loop,

$E=-\frac{d\varphi }{dt}=-\frac{d}{dt}\left[\frac{{\mu }_0}{4\pi }\frac{2M}{x^3}\right(\pi a^2\left)\right]$

$=\frac{{\mu }_{0}M{a}^2}{2}\left(\frac{3}{x^4}\frac{dx}{dt}\right)$

$=\frac{3{\mu }_{0}M{a}^{2}v}{2x^4}[\text{as}\frac{dx}{dt}=v]$

So, the induced current in the wire,

$i=\frac{E}{R}=\frac{3{\mu }_{0}M{a}^{2}v}{2x^{4}R}$

Hence, $\left(A\right)$ is the correct answer.