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Question #43031

Difficulty: 8/10 | 🟢 Free

The Question

A rigid planar conducting loop of arbitrary shape has area $A$ and moment of inertia $J$ about a fixed axis of rotation in its plane. The loop is placed in a radial magnetic field of constant magnitude $B$ , such that the magnetic field lines are always parallel to the plane of the loop and exert a magnetic torque $\tau = i A B$ when a current $i$ flows through it. At $t=0$ , the stationary loop is connected to a circuit containing a DC voltage source $V_0$ , a resistor of resistance $R$ , and an inductor of inductance $L$ in series. What is the minimum value of the resistance $R$ such that the angular velocity of the loop approaches its steady-state value without any oscillations?

Options

A.
$2 A B \sqrt{\frac{L}{J}}$
B.
$A B \sqrt{\frac{L}{J}}$
C.
$2 A B \sqrt{\frac{J}{L}}$
D.
$\frac{A B}{2} \sqrt{\frac{L}{J}}$

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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|Transient Current in RL Circuit|Motional EMF|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to model a coupled electromechanical system and recognize isomorphic differential equations.

A common trap is treating the mechanical and electrical transients as decoupled, or ignoring the inductor's effect on the mechanical motion.

If a student ignores the $L \frac{di}{dt}$ term, the system reduces to a first-order differential equation, falsely implying the angular velocity will never oscillate regardless of $R$ .

The core misconception is failing to recognize that the motional back-EMF creates a dynamic feedback loop with the inductor.

By expressing the current in terms of the rate of change of back-EMF, the rotating loop is revealed to be mathematically equivalent to a capacitor, transforming the problem into a standard RLC damping condition.

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