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 $2\left(\frac{\mu_{0}}{4 \pi}\right) \frac{M}{x^{3}}$
$\phi=2\left(\frac{\mu_{0}}{4 \pi}\right) \frac{M}{x^{3}} \cdot \pi a^{2}$
$E M F=\frac{-d \phi}{d t}=-2\left(\frac{\mu_{0}}{4 \pi}\right) M \pi a^{2}\left(\frac{-3}{x^{4}}\right) \frac{d x}{d t}$
$=6\left(\frac{\mu_{0}}{4 \pi}\right) \frac{M \pi a^{2}}{x^{4}} V $
$E M F \propto \frac{1}{x^{4}}$
Current in coil $=\frac{E M F}{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 $=i^{2} R . \propto \frac{1}{x^{8}}$