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Question #42385

Difficulty: 8/10 | 🟢 Free

The Question

A rigid rectangular coil of area $A$ and constant resistance $R$ is constrained to rotate about the $z$ -axis. A uniform, time-varying magnetic field $\vec{B}(t) = B_0 \sin(\omega_0 t) \hat{i}$ is present in the region. At $t=0$ , the area vector of the coil is aligned with the positive $x$ -axis ( $\hat{n} = \hat{i}$ ). An external agent applies a torque to keep the coil rotating at a constant angular velocity $\vec{\omega} = \omega_0 \hat{k}$ . What is the work done by the external agent during one complete revolution of the coil (from $t=0$ to $t=2\pi/\omega_0$ )? Ignore self-inductance.

Options

A.
$\frac{\pi B_0^2 A^2 \omega_0}{2R}$
B.
$\frac{\pi B_0^2 A^2 \omega_0}{R}$
C.
$-\frac{\pi B_0^2 A^2 \omega_0}{2R}$
D.
$0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Work-Energy Theorem for Rotation|Torque on Current Loop|Integration (Calculus of variations)

Why This Question Is Tricky

Common Mistakes

The core trap tests the misapplication of the Work-Energy theorem in non-conservative, time-varying fields.

Students commonly assume that since the coil rotates at a constant angular velocity ( $\Delta K_{rot} = 0$ ), the work done by the external agent must exactly equal the total Joule heat dissipated ( $W_{ext} = \int i^2 R dt = \pi B_0^2 A^2 \omega_0 / R$ ).

However, they fail to recognize that the time-varying magnetic field itself acts as a thermodynamic source, performing work on the system via transformer EMF.

The total induced EMF is a superposition of motional and transformer EMFs, meaning the external agent only supplies half the dissipated energy, while the changing magnetic field supplies the other half.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Work-Energy Theorem for Rotation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free