Question

List-I contains four conducting loops lying in the \(XY\) plane, as shown in the figures. The loops are rotating about \(Z\)-axis passing through the point \(O\) with time period \(T\) in clockwise direction. The region \(x>0\) contains a uniform magnetic field \(B\) in the \(+z\) direction. List-II contains the qualitative variation of the induced current \(i(t)\) for each of these loops. Choose the option which describes the correct match between the entries in List-I to those in List-II.

• A. \(P \to 5,\ Q \to 4,\ R \to 1,\ S \to 3\)
• B. \(P \to 3,\ Q \to 2,\ R \to 5,\ S \to 4\)
• C. \(P \to 3,\ Q \to 2,\ R \to 1,\ S \to 4\)
• D. \(P \to 5,\ Q \to 1,\ R \to 2,\ S \to 3\)

Hint:

Induced current depends on: \[ i\propto-\frac{d\Phi}{dt} \] If enclosed area changes uniformly: \[ i=\text{constant} \] If enclosed area changes sinusoidally: \[ i\sim\sin(\omega t) \]

Answer 1

The Correct Option is D

Step 1: Use Faraday's law.
Induced current: \[ i\propto-\frac{d\Phi}{dt} \] where magnetic flux: \[ \Phi=BA \] Only the part of the loop inside region: \[ x>0 \] contributes to flux. As loops rotate uniformly: \[ \theta=\omega t \] Hence induced current depends on how enclosed area changes with time.

Step 2: Case (P).
Loop is a semicircle symmetric about the \(y\)-axis. Area entering magnetic field changes smoothly and sinusoidally. Thus: \[ i(t) \] changes linearly near zero and reverses sign after: \[ T/2 \] This matches graph: \[ (5) \] Hence: \[ P\to5 \]

Step 3: Case (Q).
The loop is sector-shaped and only enters/leaves field suddenly at certain intervals. Flux remains constant for half rotations and abruptly changes sign. Thus induced current is piecewise constant. This matches: \[ (1) \] Hence: \[ Q\to1 \]

Step 4: Case (R).
This loop produces alternating entry and exit through the boundary. Hence current alternates sign periodically in shorter intervals. This matches: \[ (2) \] Hence: \[ R\to2 \]

Step 5: Case (S).
The area inside field increases continuously during first half and decreases during second half. Thus induced current remains positive during first half and negative during second half. This matches: \[ (3) \] Hence: \[ S\to3 \]

Step 6: Final matching.
\[ P\to5,\qquad Q\to1,\qquad R\to2,\qquad S\to3 \] Hence correct option is: \[ \boxed{\mathrm{(D)}} \]