A circular coil having $200$ turns, $2.5 \times 10^{-4} \mathrm{~m}^2$ area and carrying $100 \mu \mathrm{A}$ current is placed in a uniform magnetic field of $1 \mathrm{~T}$. Initially the magnetic dipole moment $(\vec{M})$ was directed along $\vec{B}$. Amount of work, required to rotate the coil through $90^{\circ}$ from its initial orientation such that $\overrightarrow{\mathrm{M}}$ becomes perpendicular to $\vec{B}$, is. . . . $\mu \mathrm{J}$.
JEE MAIN 2024, Diffcult
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We know
$\mathrm{W}_{\text {ext }} =\Delta \mathrm{U}+\Delta \mathrm{KE} \quad(\mathrm{P} . \mathrm{E} .=-\overrightarrow{\mathrm{M}} \cdot \overrightarrow{\mathrm{B}})$
$=-\overrightarrow{\mathrm{M}} \cdot \overrightarrow{\mathrm{B}}_{\mathrm{f}}+\overrightarrow{\mathrm{M}} \cdot \overrightarrow{\mathrm{B}}_{\mathrm{i}}+0$
$=-\mathrm{MB} \cos 90+\mathrm{MB} \cos 0$
$=\mathrm{MB}$
$=\mathrm{NIAB}$
$=200 \times 100 \times 10^{-6} \times \frac{5}{2} \times 10^{-4} \times 1=5 \mu$
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