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Question #42884

Difficulty: 8/10 | 🟢 Free

The Question

A horizontal rectangular pipe of width $w$ , height $d$ , and initial length $L_0$ is closed at one end by a rigid wall. The other end is sealed by a movable, frictionless dielectric plunger of mass $M$ and dielectric constant $K$ . The plunger perfectly fits the pipe and acts as a rigid node for acoustic waves. A section of the pipe's top and bottom walls forms a parallel-plate capacitor connected to a constant DC voltage source $V$ . The plunger is partially inserted into this capacitor region such that an inward displacement of the plunger (compressing the gas) increases the length of the dielectric inside the capacitor. The pipe contains an ideal gas at initial pressure $P_0$ . When the voltage is applied, the electrostatic force draws the plunger inward. The plunger moves slowly, compressing the gas isothermally until a new mechanical equilibrium is reached. If $f_0$ is the initial fundamental frequency of the gas column, what is the new fundamental frequency $f'$ at equilibrium? (Ignore fringing fields).

Options

A.
$f_0 \left[ 1 + \frac{\epsilon_0 V^2 (K-1)}{2 P_0 d^2} \right]$
B.
$f_0 \left[ 1 - \frac{\epsilon_0 V^2 (K-1)}{2 P_0 d^2} \right]$
C.
$f_0 \left[ 1 + \frac{\epsilon_0 V^2 (K-1)}{P_0 d^2} \right]$
D.
$f_0 \left[ 1 + \frac{\epsilon_0 V^2}{2 P_0 d^2} \right]$

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

Harmonic Content in Pipes|Dielectric Slab Displacement|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the integration of electrostatics, thermodynamics, and acoustics.

The primary trap lies in the energy balance of the capacitor: students often incorrectly apply $F = -dU/dx$ without accounting for the work done by the constant-voltage battery ( $W_{batt} = 2\Delta U$ ), leading to a sign error (Option B) and the false conclusion that the dielectric is repelled outward.

Additionally, students must recognize that for an isothermal transition to a new equilibrium, the speed of sound remains constant because it depends only on temperature, not pressure.

Option D traps students who confuse the force on a dielectric with the standard electrostatic pressure on conducting plates, missing the $(K-1)$ dependence.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Harmonic Content in Pipes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free