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Question #42885

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The Question

An organ pipe of length $L$ , closed at one end, resonates in its fundamental acoustic mode. The open end of the pipe is fitted with a very light, flexible conducting membrane of area $S$ . The membrane oscillates with a displacement $y(t) = A \sin(\omega_0 t)$ , where $\omega_0$ is the fundamental angular frequency of the pipe and $A$ is the displacement amplitude.

A fixed rigid conducting plate is placed at a distance $d$ ( $d \gg A$ ) from the equilibrium position of the membrane, forming a parallel-plate capacitor $C(t)$ . This capacitor is connected in series with an ideal DC voltage source $V_0$ , an inductor $L_0$ , and a resistor $R$ . The electrical circuit is tuned such that its natural resonant frequency matches the acoustic frequency $\omega_0$ .

Assuming ideal conditions and neglecting fringing fields, what is the amplitude $I_0$ of the steady-state AC current induced in the circuit, and what is its phase relationship with the membrane's displacement $y(t)$ ?

Options

A.
$I_0 = \frac{V_0 A}{R d}$ , $i(t)$ is in phase with $y(t)$
B.
$I_0 = \frac{V_0 A \omega_0 C_0}{d}$ , $i(t)$ leads $y(t)$ by $\frac{\pi}{2}$
C.
$I_0 = \frac{V_0 A}{R d}$ , $i(t)$ leads $y(t)$ by $\frac{\pi}{2}$
D.
$I_0 = \frac{V_0 A \omega_0 C_0}{d}$ , $i(t)$ is in phase with $y(t)$

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Correct Answer

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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

Harmonic Content in Pipes|Frequency Sensitivity of Z|Vector Analysis

Why This Question Is Tricky

Common Mistakes

Students often misapply the relation $Q = CV$ by assuming the voltage across the variable capacitor remains constant at the DC source value $V_0$ .

This leads them to simply differentiate the charge $q(t) = C(t)V_0$ to find the current $i(t) = dq/dt$ , which incorrectly yields an amplitude dependent on $\omega_0 C_0$ and a current that leads the displacement by $\pi/2$ (Option B).

This ignores the dynamic impedance of the series RLC circuit.

In reality, the capacitance variation acts as an internal parametric AC voltage source.

Because the circuit is at resonance, the inductive and capacitive reactances cancel, making the circuit purely resistive.

Consequently, the induced current is determined solely by the resistance and is perfectly in phase with the effective driving voltage, which itself is in phase with the mechanical displacement.

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