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Question #43298

Difficulty: 8/10 | 🟢 Free

The Question

A rigid square loop of side $L$ is hinged along the $y$ -axis. At $t=0$ , the loop lies in the $xy$ -plane with its vertices at $(0,0,0)$ , $(L,0,0)$ , $(L,L,0)$ , and $(0,L,0)$ . The loop rotates about the $y$ -axis with a constant angular velocity $\omega$ such that it moves into the region $z > 0$ . A static, non-uniform magnetic field exists in the region, given by $\vec{B} = \beta x^2 \hat{i} - 2\beta x y \hat{j}$ , where $\beta$ is a positive constant. Find the magnitude of the induced electromotive force (EMF) in the loop at time $t = \frac{\pi}{3\omega}$ .

Options

A.
$\frac{5}{24} \beta L^4 \omega$
B.
$\frac{5}{32} \beta L^4 \omega$
C.
$\frac{1}{6} \beta L^4 \omega$
D.
$0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Magnetic Flux Scalar Sign|Ray Optics Sign Convention Dynamics|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of Faraday's Law in a non-uniform, non-Cartesian-aligned magnetic field, specifically targeting the 'Orientation Flip' trap.

Students frequently commit one of three lethal errors: (1) Evaluating the magnetic field at the geometric centroid of the loop and multiplying by the area, which only works for linear fields (yielding the trap answer 5/32). (2) Integrating over the spatial coordinate $x$ from $0$ to $L$ instead of the material coordinate $r$ , forgetting that the loop's projected width on the $x$ -axis shrinks dynamically (yielding 1/6). (3) Using a static area vector $\vec{A} = L^2 \hat{k}$ , ignoring the time-evolution of the surface normal $\hat{n}(t)$ , which incorrectly yields zero flux.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Magnetic Flux Scalar Sign
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free