A small circular loop of area A and resistance R is fixed on a horizontal -plane with the center of the loop always on the axis of a long solenoid. The solenoid has turns per unit length and carries current counterclockwise as shown in the figure. [

Question

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 $\hat{n}$ 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 $\hat{n}$ direction. List-I gives time dependences of $\hat{n}$ in terms of a constant angular frequency $ω$ . List-II gives the torques experienced by the circular loop at time $t = \frac{π}{6 ω} .$ Let $α = \frac{A^{2} μ_{0}^{2} m^{2} I^{2} ω}{2 R} .$ [2022]

	List-I		List-II
(I)	$\frac{1}{\sqrt{2}} (\sin ω t \hat{j} + \cos ω t \hat{k})$	(P)	$0$
(II)	$\frac{1}{\sqrt{2}} (\sin ω t \hat{i} + \cos ω t \hat{j})$	(Q)	$- \frac{α}{4} \hat{i}$
(III)	$\frac{1}{\sqrt{2}} (\sin ω t \hat{i} + \cos ω t \hat{k})$	(R)	$\frac{3 α}{4} \hat{i}$
(IV)	$\frac{1}{\sqrt{2}} (\cos ω t \hat{j} + \sin ω t \hat{k})$	(S)	$\frac{α}{4} \hat{j}$
		(T)	$- \frac{3 α}{4} \hat{i}$

Which one of the following options is correct?

Smart Achievers · Accepted Answer

(3)

(I) $\hat{n}$

$ϕ = \vec{B} \cdot \vec{A} = \frac{μ_{0} m I}{\sqrt{2}} \cos (ω t) A$

$ε = - \frac{d ϕ}{d t} = \frac{μ_{0} m I ω A}{\sqrt{2}} \sin (ω t)$

$i = \frac{ε}{R} = \frac{μ_{0} m I ω A}{\sqrt{2} R} \sin (ω t)$

So, $\vec{M} = i \vec{A} = \frac{μ_{0} m I ω A^{2}}{\sqrt{2} R} \sin (ω t) \hat{k}$

$\vec{τ} = \vec{M} \times \vec{B} = \frac{μ_{0}^{2} m^{2} I^{2} ω A^{2}}{2 R} \sin^{2} (ω t) (- \hat{i})$

$For t = \frac{π}{6 ω}$

$- \frac{3 α}{4} \hat{i}$

$= \frac{A^{2} μ_{0}^{2} m^{2} I^{2} ω}{2 R} \times \frac{1}{4} (- \hat{i}) = \frac{α}{4} (- \hat{i})$ . So, $(I) \to Q$

$(II) Solving as (I), ϕ = 0, ε = 0, i = 0, \vec{M} = 0 .$ So, $(II) \to P$

$Similarly, (III) \to S$

$(IV) \to R$

Solution

1 (I) → (Q); (II) → (P); (III) → (S); (IV) → (T)

2 (I) → (S); (II) → (T); (III) → (Q); (IV) → (P)

3 (I) → (Q); (II) → (P); (III) → (S); (IV) → (R)

4 (I) → (T); (II) → (Q); (III) → (P); (IV) → (R)

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