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Question #43721

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The Question

In one arm of an interferometer, a transparent cylindrical tube of length $L$ contains an ideal gas. The tube is connected to a reservoir such that the gas pressure remains constant at $P_0$ . Initially, the gas is at a uniform temperature $T_0$ , and its refractive index is $n_0$ . The Gladstone-Dale relation states that the quantity $(n - 1)$ is directly proportional to the density of the gas, where $n$ is the refractive index.

A heating element is activated, creating a steady-state linear temperature gradient along the tube's axis. The temperature is $T_0$ at the entrance ( $x=0$ ) and $T_1$ at the exit ( $x=L$ ). If light of vacuum wavelength $\lambda$ is used, what is the number of fringes $\Delta N$ that shift past a reference point on the screen due to this heating process?

Options

A.
$\frac{(n_0 - 1) L}{\lambda} \left[ 1 - \frac{T_0}{T_1 - T_0} \ln\left(\frac{T_1}{T_0}\right) \right]$
B.
$\frac{(n_0 - 1) L}{\lambda} \left[ \frac{T_1 - T_0}{T_1 + T_0} \right]$
C.
$\frac{(n_0 - 1) L}{\lambda} \left[ \frac{T_1 - T_0}{2 T_1} \right]$
D.
$\frac{(n_0 - 1) L}{\lambda} \left[ 1 - \frac{T_1 - T_0}{T_0 \ln(T_1/T_0)} \right]$

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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

Optical Path Difference|Variation of Density with Temperature|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from a discrete optical path difference to a continuous spatial integral when the medium's properties are non-uniform.

The primary trap lies in assuming that a linear temperature gradient results in a linear variation of the refractive index, or simply using the average temperature $T_{avg} = (T_0 + T_1)/2$ to calculate the new optical path (Option B).

Because the Gladstone-Dale relation couples the refractive index to density (which is inversely proportional to temperature at constant pressure), the refractive index varies as $1/x$ .

Students who fail to set up the differential element $dx$ and integrate the inverse function will incorrectly choose options based on arithmetic means, completely missing the logarithmic dependence characteristic of this thermodynamic-optical coupling.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Optical Path Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free