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Question #43029

Difficulty: 8/10 | 🟢 Free

The Question

A rigid circular wire loop of radius $R$ and mass $m$ carries a constant steady current $i$ . The loop is suspended vertically by a torsionless, insulating thread passing through its vertical diameter. A uniform horizontal magnetic field of magnitude $B$ is applied perpendicular to the initial plane of the loop. The loop is immersed in a viscous fluid that exerts a retarding torque $\tau_d = -b \omega$ , where $\omega$ is the angular velocity of the loop and $b$ is a positive damping constant. The loop is given a small angular displacement about the vertical axis and released. Assuming the damping is very small, what is the fractional increase in the time period of its oscillation, $\frac{\Delta T}{T_0}$ , compared to the undamped case?

Options

A.
$\frac{b^2}{4 \pi i B m R^4}$
B.
$\frac{b^2}{2 \pi i B m R^4}$
C.
$\frac{b^2}{\pi i B m R^4}$
D.
$\frac{b^2}{8 \pi i B m R^4}$

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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

Torque on Current Loop|Damped Oscillation Equation|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of electromagnetism and damped harmonic motion.

A common trap is misidentifying the moment of inertia of the loop.

Students often reflexively use $I = mR^2$ (moment of inertia about the central axis) instead of $I = mR^2/2$ (about the diameter), which leads to an incorrect inertial response and matches Option D.

Another trap is incorrectly defining the damping parameter in the differential equation as $\gamma = b/I$ instead of $b/2I$ , leading to Option C.

Finally, forgetting the factor of $1/2$ in the binomial expansion of the time period leads to Option B.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Torque on Current Loop
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free