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Question #42858

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly flexible, insulating string of length $L$ , uniform mass per unit length $\mu$ , and uniform linear charge density $\lambda$ is stretched with tension $T$ along the central axis of a long, grounded, conducting cylindrical shell of inner radius $R$ ( $R \ll L$ ). The ends of the string are fixed at $x=0$ and $x=L$ on the axis. Assuming the transverse displacement $y(x,t)$ of the string is very small ( $y \ll R$ ), find the fundamental angular frequency $\omega_1$ of the transverse vibrations of the string. (Ignore gravity and edge effects of the cylinder).

Options

A.
$\sqrt{ \frac{\pi^2 T}{\mu L^2} - \frac{\lambda^2}{2\pi \epsilon_0 R^2 \mu} }$
B.
$\sqrt{ \frac{\pi^2 T}{\mu L^2} + \frac{\lambda^2}{2\pi \epsilon_0 R^2 \mu} }$
C.
$\frac{\pi}{L} \sqrt{ \frac{T}{\mu} }$
D.
$\sqrt{ \frac{\pi^2 T}{\mu L^2} - \frac{\lambda^2}{\pi \epsilon_0 R^2 \mu} }$

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

Standing Wave Nodes/Antinodes|Cavity Field Shielding|Potential of Concentric Shells

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of electrostatic boundary value problems (method of images for a cylindrical cavity) with mechanical wave dynamics.

The primary trap is neglecting the cavity field shielding effect, where displacing a charged string inside a grounded conducting shell induces an asymmetric surface charge distribution.

This induced charge creates a localized electric field gradient that exerts an outward, anti-restoring force on the string.

Students commonly fail to recognize this non-homogeneous modification to the wave equation (Option C) or incorrectly assign a positive sign to the electrostatic restoring term (Option B), fundamentally misunderstanding the destabilizing nature of the induced cavity field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Standing Wave Nodes/Antinodes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free