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(A, B, D) The magnetic field $B$ at an axial point $x$ due to a bar magnet of dipole moment $M$ is $B = \frac{\mu_0}{4\pi} \frac{2M}{x^3}$.Since $x >>

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The magnetic moment $\mu$ due to induced current in the coil is proportional to $a^4$

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(A, B, D) The magnetic field $B$ at an axial point $x$ due to a bar magnet of dipole moment $M$ is $B = \frac{\mu_0}{4\pi} \frac{2M}{x^3}$.Since $x >> a$, the magnetic flux $\phi$ through the coil of area $A = \pi a^2$ is $\phi = B \cdot A = \frac{\mu_0}{4\pi} \frac{2M}{x^3} \cdot \pi a^2$.The induced $EMF$ is given by $\epsilon = -\frac{d\phi}{dt} = -\frac{\mu_0}{4\pi} 2M \pi a^2 \frac{d}{dt}(x^{-3}) = -\frac{\mu_0}{4\pi} 2M \pi a^2 (-3x^{-4}) \frac{dx}{dt}$.Since $\frac{dx}{dt} = -v$ (as $x$ is decreasing), $\epsilon = \frac{\mu_0}{4\pi} \frac{6M \pi a^2 v}{x^4}$. Thus, $\epsilon \propto \frac{1}{x^4}$.The induced current $i = \frac{\epsilon}{R} \propto \frac{1}{x^4}$.The magnetic moment $\mu_{coil} = i \cdot A = i \cdot \pi a^2 \propto \frac{1}{x^4} \cdot a^2$. This does not match option $C$ as stated.The rate of heat production $P = i^2 R \propto (x^{-4})^2 = \frac{1}{x^8}$. Thus, option $D$ is correct.

$A$ small bar magnet of dipole moment $M$ is moving with speed $v$ along the $x$-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 $R$. Which of the following statement$(s)$ is/are true?

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