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Question #43248

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The Question

In a Young's Double Slit Experiment, the entire apparatus is kept in a vacuum. A transparent cylindrical tube of length $L$ is placed immediately in front of slit $S_1$ . The tube contains an ideal gas initially at temperature $T_0$ , and its initial refractive index is $n_0$ . The gas is heated isobarically such that its temperature increases linearly with time as $T(t) = T_0 + \gamma t$ , where $\gamma$ is a positive constant. The refractive index $n$ of the gas satisfies the Gladstone-Dale relation $(n - 1) \propto \rho$ , where $\rho$ is the density of the gas. Ignore the thermal expansion of the tube. The slit separation is $d$ , the screen distance is $D$ , and the wavelength of the light in vacuum is $\lambda$ . Which of the following expressions correctly gives the instantaneous fringe width $\beta(t)$ on the screen and the speed $v(t)$ of the central maximum?

Options

A.
$\beta(t) = \frac{\lambda D}{d}, \quad v(t) = \frac{D L (n_0 - 1) T_0 \gamma}{d (T_0 + \gamma t)^2}$
B.
$\beta(t) = \frac{\lambda D}{d} \left( \frac{T_0 + \gamma t}{T_0} \right), \quad v(t) = \frac{D L (n_0 - 1) \gamma}{d (T_0 + \gamma t)}$
C.
$\beta(t) = \frac{\lambda D}{d \left[ 1 + (n_0 - 1)\frac{T_0}{T_0 + \gamma t} \right]}, \quad v(t) = \frac{D L (n_0 - 1) T_0 \gamma}{d (T_0 + \gamma t)^2}$
D.
$\beta(t) = \frac{\lambda D}{d}, \quad v(t) = \frac{D L (n_0 - 1) \gamma}{d T_0}$

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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

Fringe Width Invariance|Medium Dependency|Differential Calculus

Why This Question Is Tricky

Common Mistakes

This question tests two major conceptual traps in wave optics and thermodynamics.

First, students often misapply the medium dependency of fringe width, assuming that a changing refractive index in any part of the setup alters the fringe width $\beta$ .

They fail to recognize that a localized medium (the tube) only introduces a spatially uniform phase shift, translating the pattern without changing $\beta$ , which is governed solely by the vacuum wavelength where the beams overlap.

Second, students frequently misinterpret the Gladstone-Dale relation in conjunction with the ideal gas law, incorrectly assuming $n(t) - 1 \propto T$ instead of $n(t) - 1 \propto 1/T$ for an isobaric process, which leads to an incorrect constant velocity for the shifting fringes.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Fringe Width Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free