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Question #43576

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The Question

A cylindrical vessel is filled with a transparent liquid up to a height $H$ . The temperature of the liquid at the bottom of the vessel is maintained at $T_0$ , and at the free surface (top) it is maintained at $T_0 + \Delta T$ , establishing a steady linear temperature gradient. The volumetric expansion coefficient of the liquid is $\gamma$ . The refractive index of the liquid at the bottom is $n_0$ . Assume that the refractive index $n$ and density $\rho$ of the liquid are related by the Gladstone-Dale relation, $(n - 1) \propto \rho$ . If the vessel is viewed normally from above in air ( $n_{\text{air}} = 1$ ), what is the apparent depth of the bottom of the vessel?

Options

A.
$\frac{H}{(n_0 - 1) \gamma \Delta T} \ln \left( \frac{n_0}{n_0 - (n_0 - 1) \gamma \Delta T} \right)$
B.
$\frac{H}{n_0 - \frac{1}{2}(n_0 - 1)\gamma \Delta T}$
C.
$\frac{H}{(n_0 - 1) \gamma \Delta T} \ln \left( \frac{n_0 + (n_0 - 1) \gamma \Delta T}{n_0} \right)$
D.
$\frac{H}{n_0 \gamma \Delta T} \ln \left( \frac{1}{1 - \gamma \Delta T} \right)$

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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|Apparent Depth/Shift|Variation of Density with Temperature

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize thermodynamics (thermal expansion and temperature gradients) with optics (variable refractive index and apparent depth).

The primary trap lies in the assumption of a constant or simply averaged refractive index (Option B), which incorrectly applies the static $D_{app} = H/n_{avg}$ formula to a continuously varying medium.

Furthermore, students must correctly apply the Gladstone-Dale relation to link density variations to the refractive index, rather than assuming $n$ itself scales directly with volume expansion (Option D).

The correct approach requires setting up a differential element and integrating the apparent depth contributions $\int dz/n(z)$ across the spatial gradient.

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