In a uniform magnetic field of $0.049 \mathrm{~T}$, a magnetic needle performs $20$ complete oscillations in $5$ seconds as shown. The moment of inertia of the needle is $9.8 \times 10^{-5} \mathrm{~kg} \mathrm{~m}^2$. If the magnitude of magnetic moment of the needle is $x \times 10^{-5} \mathrm{Am}^2$, then the value of ' $x$ ' is :
NEET 2024, Medium
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Time period of Oscillation, $T=2 \pi \sqrt{\frac{1}{M B}}$
$\Rightarrow \frac{1}{4}=2 \pi \sqrt{\frac{9.8 \times 10^{-6}}{M \times 0.049}}$
$\Rightarrow \frac{1}{16}=4 \pi^2 \times \frac{9.8 \times 10^{-6}}{M \times 49 \times 10^{-3}}$
$\Rightarrow M=\frac{4 \pi^2 \times 9.8 \times 10^{-6}}{49 \times 10^{-3}} \times 16$
$\frac{4 \pi^2 \times 9.8 \times 16 \times 10^{-3}}{49}$
$\quad=12.8 \pi^2 \times 10^{-3} \times 10^{-2} \times 10^2$
$=1280 \pi^2 \times 10^{-5} \mathrm{Am}^2$
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