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A small circular loop of area A and resistance \( R \) is fixed on a horizontal \( x y \)-plane with the center of the loop always on the axis \( \hat{h} \) of a long solenoid. The solenoid has \( m \) turns per unit length and carries current I counterclockwise as shown in the figure. The magnetic field due to the solenoid is in \( \hat{h} \) direction. List-l gives time dependences of \( \hat{h} \) in terms of a constant angular frequency \( \omega \). List-Il gives the torques experienced by the circular loop at time \( t=\frac{\pi}{6 \omega} \) Let \( a=\frac{\left.A^{2} \mu_{0}^{2} m^{2}\right|^{2} \omega}{2 R} \).
List-II
(I) \( \frac{1}{\sqrt{2}}(\sin \omega t \hat{\jmath}+\cos \omega t \hat{k}) \)
(P) 0
(II) \( \frac{1}{\sqrt{2}}(\sin \omega t i+\cos \omega t \hat{\jmath}) \)
(Q) \( -\frac{a_{i}}{4} \)
(III) \( \frac{1}{\sqrt{2}}(\sin \omega t \hat{\imath}+\cos \omega t \hat{k}) \)
(R) \( \frac{3 a}{4} \hat{\imath} \)
- (IV) \( \frac{1}{\sqrt{2}}(\cos \omega t \hat{\jmath}+\sin \omega t \hat{k}) \)
(S) \( \frac{a_{j}^{4}}{4} \)
(T) \( -\frac{3 \alpha_{i}}{4} \)
- Which one of the following options is correct?
(a) I \( \rightarrow \) Q, II \( \rightarrow \) P, III \( \rightarrow \) S, IV \( \rightarrow \) R
(b) I \( \rightarrow \) Q, II \( \rightarrow \) P, III \( \rightarrow \) S, IV \( \rightarrow \) T
(c) I \( \rightarrow \) S, II \( \rightarrow \mathrm{T} \), III \( \rightarrow \mathrm{Q} \), IV \( \rightarrow \) P
(d) I \( \rightarrow \) T, II \( \rightarrow \) Q, III \( \rightarrow \) P, IV \( \rightarrow \) R
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