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Question #43066

Difficulty: 8/10 | 🟢 Free

The Question

A uniform conducting rod of mass $m$ and length $l$ is pivoted at one of its ends and can rotate freely in a horizontal plane. A uniform vertical magnetic field $B$ exists in the region. The pivot and the free end of the rod are connected via frictionless sliding contacts to a stationary circuit consisting of an inductor of inductance $L$ , a capacitor of capacitance $C$ , and a resistor of resistance $R$ in series. Initially, the rod is at rest and the capacitor is uncharged. At $t = 0$ , an impulsive torque $\int \tau dt = J$ is applied to the rod. Assuming the resistance of the rod and the contacts is negligible, what is the Quality Factor ( $Q$ ) of the subsequent transient electrical oscillations in the circuit?

Options

A.
$\frac{1}{R} \sqrt{\frac{L}{C}}$
B.
$\frac{1}{R} \sqrt{L \left( \frac{1}{C} + \frac{3 B^2 l^2}{4 m} \right)}$
C.
$\frac{1}{R} \sqrt{L \left( \frac{1}{C} + \frac{B^2 l^2}{2 m} \right)}$
D.
$\frac{1}{R} \sqrt{L \left( \frac{1}{C} - \frac{3 B^2 l^2}{4 m} \right)}$

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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

Quality Factor|Motional EMF in Rotating Rods|Impulsive Torque and Angular Momentum

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize electro-mechanical coupling and its effect on circuit parameters.

A common trap is treating the motional EMF as an independent voltage source, leading students to incorrectly assume the Quality Factor is simply $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$ (Option A).

However, by Lenz's Law, the induced current creates a magnetic torque that opposes the rod's rotation.

This back-action couples the mechanical inertia of the rod to the electrical circuit, effectively acting as an additional capacitor.

Students must integrate the rotational equation of motion to find the relationship between angular velocity and charge, and substitute this into Kirchhoff's Voltage Law to derive the effective stiffness (inverse capacitance) of the coupled oscillator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Quality Factor (Q-factor)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free