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Question #42696

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor is formed by two fixed semi-circular metal plates. A semi-circular dielectric slab is pivoted at the center of its straight edge and can rotate freely between the plates. The capacitance of the system as a function of the angle $\theta$ (in radians) of the slab inside the plates is given by $C(\theta) = C_0 (\pi + \beta \theta)$ , where $C_0$ and $\beta$ are positive constants. The capacitor is given a charge $Q$ and then electrically isolated. A torsional spring of stiffness $k$ exerts a restoring torque $\tau_s = -k\theta$ on the slab. The system is found to be in stable equilibrium at an angle $\theta = \theta_0$ ( $0 < \theta_0 < \pi$ ). Let $I$ be the moment of inertia of the slab about the pivot. Which of the following expressions represents the angular frequency $\omega$ of small oscillations of the slab about this equilibrium position?

Options

A.
$\sqrt{\frac{k}{I} \left( \frac{3 \beta \theta_0 + \pi}{\beta \theta_0 + \pi} \right)}$
B.
$\sqrt{\frac{k}{I}}$
C.
$\sqrt{\frac{k}{I} \left( \frac{\beta \theta_0 + \pi}{3 \beta \theta_0 + \pi} \right)}$
D.
$\sqrt{\frac{k}{I} \left( \frac{\pi - \beta \theta_0}{\beta \theta_0 + \pi} \right)}$

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

Dielectric Slab Displacement|Torsional Oscillations|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze non-linear electromechanical coupling.

A common and lethal misconception is assuming that the electrostatic torque on a dielectric slab is always constant.

While true for a capacitor connected to a constant voltage source (where $\tau_e = \frac{1}{2}V^2 \frac{dC}{d\theta}$ ), it is false for an isolated capacitor (constant charge $Q$ ).

For an isolated capacitor, the torque $\tau_e = \frac{Q^2}{2C^2}\frac{dC}{d\theta}$ depends strongly on $\theta$ .

Students who fall for this trap will assume the electrostatic force merely shifts the equilibrium position without altering the effective stiffness, leading them to incorrectly choose $\omega = \sqrt{k/I}$ .

The correct approach requires setting up the total potential energy and rigorously finding the second derivative to determine the true effective stiffness of the coupled system.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Dielectric Slab Displacement
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free