Before You Begin

Question #43551

Difficulty: 8/10 | 🟢 Free

The Question

In a Young's Double Slit Experiment (YDSE), a transparent tube of length $L$ is placed directly in front of one of the slits. The tube is connected to a large reservoir such that it is filled with an ideal gas maintained at a constant pressure. Initially, the gas is at a uniform absolute temperature $T_1$ , and the central maximum of the interference pattern is observed at point $O$ on the screen. The temperature at the end of the tube near the screen is then slowly increased to $T_2$ ( $T_2 > T_1$ ) while the end near the slit is maintained at $T_1$ . The gas reaches a steady state of thermal conduction. Assume the thermal conductivity of the gas is independent of temperature. The refractive index $n$ of the gas depends on its absolute temperature $T$ according to the relation $n - 1 = \frac{\alpha}{T}$ , where $\alpha$ is a positive constant. If the wavelength of the light in vacuum is $\lambda$ , which of the following expressions gives the number of fringes $\Delta N$ that shift past the point $O$ during this process?

Options

A.
$\frac{\alpha L}{\lambda} \left[ \frac{1}{T_1} - \frac{\ln(T_2/T_1)}{T_2 - T_1} \right]$
B.
$\frac{\alpha L}{\lambda} \left[ \frac{1}{T_1} - \frac{2}{T_1 + T_2} \right]$
C.
$\frac{\alpha L}{2\lambda} \left[ \frac{1}{T_1} - \frac{1}{T_2} \right]$
D.
$\frac{\alpha L}{\lambda (T_2 - T_1)} \ln\left(\frac{T_2}{T_1}\right)$

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

Microscope Tube Length|Wave Intensity and Medium|Phase Relationship

Why This Question Is Tricky

Common Mistakes

This question tests the integration of thermal conduction with wave optics, specifically targeting the misconception that optical path length (OPL) can be calculated using spatially averaged macroscopic properties.

Students often fall into the trap of either using the average temperature $T_{avg} = (T_1+T_2)/2$ to find an effective refractive index (Option B), or assuming the refractive index itself varies linearly between the two ends (Option C).

However, because $n$ depends inversely on $T$ , and $T$ varies linearly with distance $x$ due to steady-state conduction, the correct OPL requires a rigorous spatial integration $\int n(x) dx$ , yielding a logarithmic dependence.

This highlights the critical mathematical difference between the average of a function and the function of an average.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Microscope Tube Length
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free