Question

A small bar magnet of dipole moment $M$ is moving with speed $v$ along $x$ direction towards a small closed circular conducting loop of radius ' $a$ ' with its centre $O$ at $x=0$ (see figure). Assume $x>>a$ and the coil has a resistance $R$. Then which of the following statement(s) is/are true?

• A. Magnetic field at the centre $O$ of the circular coil due to the bar magnet is $\frac{M}{x^3}$
• B. Induced EMF is proportional to $\frac{1}{x^4}$
• C. The magnetic moment $\mu$ due to induced current in the coil is proportional to $a^4$
• D. The heat produced is proportional to $\frac{1}{x^6}$

Answer 1

Answer: (B), (C)

Magnetic field at an axial point is $2\left(\frac{{\mu }_0}{4\pi }\right)\frac{M}{x^3}$
$\varphi =2\left(\frac{{\mu }_0}{4\pi }\right)\frac{M}{x^3}\cdot \pi a^2$
$EMF=\frac{-d\varphi }{dt}=-2\left(\frac{{\mu }_0}{4\pi }\right)M\pi a^2\left(\frac{-3}{x^4}\right)\frac{dx}{dt}$
$=6\left(\frac{{\mu }_0}{4\pi }\right)\frac{M\pi a^2}{x^4}V$
$EMF\propto \frac{1}{x^4}$
Current in coil $=\frac{EMF}{R}$
$i=6\left(\frac{{\mu }_0}{4\pi }\right)\frac{M\pi a^2}{x^4}\frac{V}{R}$
Magnetic moment $\mu =i\pi a^2,\mu \propto a^4$
Rate at which heat is produced $=i^{2}R.\propto \frac{1}{x^8}$