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Question #42863

Difficulty: 8/10 | 🟢 Free

The Question

A uniform flexible wire of length $L$ and mass per unit length $\mu$ is suspended from a rigid support at its upper end, while its lower end is free. The wire carries a steady current $I$ . A large fixed vertical sheet is placed nearby, carrying a uniform vertical surface current density $K$ . This creates a uniform horizontal magnetic field $\vec{B}$ across the region of the wire. The wire reaches a steady equilibrium state. A small transverse wave pulse is generated at the free lower end. What is the time $\tau$ taken by the pulse to travel along the wire and reach the rigid support? (Assume the magnetic field of the flexible wire does not affect the sheet, and $g$ is the acceleration due to gravity).

Options

A.
$2 \sqrt{ \frac{L}{ \left[ g^2 + \left( \frac{\mu_0 I K}{2\mu} \right)^2 \right]^{1/2} } }$
B.
$2 \sqrt{ \frac{L}{ g + \frac{\mu_0 I K}{2\mu} } }$
C.
$2 \sqrt{ \frac{L}{g} }$
D.
$2 \sqrt{ \frac{2\mu L}{\mu_0 I K} }$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Fundamental Frequency in Strings|Force Between Parallel Conductors|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze wave mechanics in a system with a non-uniform tension caused by multiple distributed loads.

A common trap is assuming the magnetic force simply adds algebraically to gravity (Option B) or ignoring it because it acts transversely to the initial vertical orientation (Option C).

Students must recognize that the transverse magnetic force and gravity act orthogonally, creating an effective gravitational field $g_{eff}$ that dictates both the equilibrium slanted shape of the wire and the linearly varying tension $T(s) = \mu g_{eff} s$ .

The wave speed then depends on this vector-summed effective tension, requiring integration over the length of the wire.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Fundamental Frequency in Strings
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free