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Question #42639

Difficulty: 8/10 | 🟢 Free

The Question

A continuous harmonic transverse wave of wavelength $\lambda$ is generated at $x=0$ on a taut string of linear mass density $\mu_1$ . A detector is fixed at $x=L$ . The string is joined smoothly at $x=L+d$ to a second semi-infinite string of linear mass density $\mu_2$ under the same tension. The detector measures the time-averaged intensity of the superposition of the incident wave and the first reflected wave. Let $d_1$ be the minimum non-zero distance $d$ to observe a local maximum in intensity at the detector when $\mu_2 = 4\mu_1$ . Let $d_2$ be the minimum non-zero distance $d$ to observe a local maximum in intensity when $\mu_2 = \frac{\mu_1}{4}$ . Which of the following represents the correct pair $(d_1, d_2)$ ?

Options

A.
$\left( \frac{\lambda}{4}, \frac{\lambda}{2} \right)$
B.
$\left( \frac{\lambda}{2}, \frac{\lambda}{4} \right)$
C.
$\left( \frac{\lambda}{2}, \lambda \right)$
D.
$\left( \lambda, \frac{\lambda}{2} \right)$

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

Phase Difference vs. Path Difference|Wave Intensity and Medium|Boundary Condition Reflection

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate spatial path differences with boundary-induced phase jumps.

A common trap is calculating the geometric path difference as just $\Delta x = d$ instead of the round-trip path difference $\Delta x = 2d$ .

Furthermore, students often blindly apply a $\pi$ phase shift for all reflections or fail to evaluate the impedance ratio $Z = \sqrt{T\mu}$ .

By forcing the evaluation of both a 'hard' boundary ( $\mu_2 > \mu_1$ ) and a 'soft' boundary ( $\mu_2 < \mu_1$ ), the problem creates a lethal collision between geometric phase accumulation and discontinuous boundary conditions, strictly penalizing rote memorization of interference formulas.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Phase Difference vs. Path Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free