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Question #43927

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical resonance tube of length $L$ and radius $r$ is open at one end and closed at the other by a light, movable conducting diaphragm of mass $m$ . The diaphragm is attached to a coil of total wire length $l$ placed in a uniform radial magnetic field $B$ . The coil is connected in series with a resistor $R$ and an inductor $L_0$ . The air column is excited into its fundamental acoustic resonance, causing the diaphragm to oscillate. If the external acoustic excitation is suddenly removed, the diaphragm's oscillation amplitude undergoes a transient exponential decay. Assuming purely electromagnetic damping, what is the time constant $\tau$ of this amplitude decay? (Let $c$ be the speed of sound in air. Include the end correction for the open end).

Options

A.
$\frac{m \left[ 4 R^2 (L + 0.6r)^2 + \pi^2 c^2 L_0^2 \right]}{2 B^2 l^2 R (L + 0.6r)^2}$
B.
$\frac{m \left[ 4 R^2 L^2 + \pi^2 c^2 L_0^2 \right]}{2 B^2 l^2 R L^2}$
C.
$\frac{2 m R}{B^2 l^2}$
D.
$\frac{m \left[ 4 R^2 (L + 0.6r)^2 + \pi^2 c^2 L_0^2 \right]}{4 B^2 l^2 R (L + 0.6r)^2}$

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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

End Correction|Transient Current in RL Circuit|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple acoustic resonance with transient AC electrical impedance.

A common trap is treating the mechanical and electrical systems as isolated, or ignoring the inductor's reactance at the acoustic frequency, which leads to a simple DC damping factor (Option C).

Another critical failure point is forgetting the end correction for the open end of the tube, which shifts the resonance frequency and alters the inductive reactance (Option B).

Finally, students often confuse the amplitude decay time constant with the energy decay time constant, missing a factor of 2 (Option D).

The core physical insight is recognizing that the back-EMF creates a velocity-dependent force whose phase shift modifies the effective damping of the acoustic oscillator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	End Correction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free