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Question #43214

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The Question

A hollow prism in air is formed by a frame of three thin rods pinned at their vertices. The base rod has a coefficient of linear expansion $\alpha_1$ and the two identical side rods have a coefficient of linear expansion $\alpha_2$ . The frame is filled with a transparent liquid of refractive index $\mu$ . At an initial temperature $T_0$ , the frame forms an equilateral triangle and the liquid has $\mu = \sqrt{3}$ . As the temperature changes, the liquid's refractive index varies with a thermo-optic coefficient $\frac{d\mu}{dT} = -1.0 \times 10^{-5} \, ^\circ\text{C}^{-1}$ . Given $\alpha_1 - \alpha_2 = 2\sqrt{3} \times 10^{-5} \, ^\circ\text{C}^{-1}$ . If the temperature of the system increases at a constant rate of $9.0 \, ^\circ\text{F/s}$ , what is the rate of change of the angle of minimum deviation $\delta_{min}$ with respect to time at $T_0$ ?

Options

A.
$3.0 \times 10^{-4} \, \text{rad/s}$
B.
$-1.0 \times 10^{-4} \, \text{rad/s}$
C.
$5.4 \times 10^{-4} \, \text{rad/s}$
D.
$5.0 \times 10^{-4} \, \text{rad/s}$

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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

Prism Minimum Deviation|Celsius-Fahrenheit Calibration Errors|Specific Heat Capacity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple thermal expansion with optical geometry.

A common lethal mistake is treating the prism's apex angle A as a constant during thermal changes, leading students to only differentiate the refractive index term (the thermo-optic effect).

In reality, the anisotropic expansion of the frame alters the apex angle, requiring implicit differentiation of the minimum deviation formula with respect to both mu and A.

Additionally, the problem includes a secondary trap involving the conversion of temperature rates from Fahrenheit to Celsius, forcing strict unit consistency.

Quick Facts About This Question

Question Information

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