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Question #43538

Difficulty: 8/10 | 🟢 Free

The Question

A plano-convex lens made of a material with refractive index $n = 1.5$ has a radius of curvature $R = 20 \text{ cm}$ on its convex side. The convex surface is silvered. An object is placed on the principal axis at a distance $u = 20 \text{ cm}$ from the flat surface. The lens is subjected to a uniform temperature increase at a constant rate of $\frac{dT}{dt} = 10 \text{ }^\circ\text{C/s}$ . The coefficient of linear expansion of the lens material is $\alpha = 2 \times 10^{-5} \text{ }^\circ\text{C}^{-1}$ , and the temperature coefficient of its refractive index is $\frac{dn}{dT} = -3 \times 10^{-5} \text{ }^\circ\text{C}^{-1}$ . Assuming the object remains stationary, what is the speed of the image formed by the system?

Options

A.
$60 \text{ \mu m/s}$
B.
$30 \text{ \mu m/s}$
C.
$0 \text{ \mu m/s}$
D.
$80 \text{ \mu m/s}$

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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

Thermal Expansion of Area and Volume|Lens Silvering Logic|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple thermal expansion with geometric optics in a compound system, specifically targeting the misconception that the refractive index remains constant during thermal expansion.

Students often memorize the fractional change in focal length as simply $\frac{df}{f} = \alpha \Delta T$ , ignoring the opto-thermal effect ( $\frac{dn}{dT}$ ).

Because the equivalent focal length of this silvered lens is $F = -\frac{R}{2n}$ , the negative temperature coefficient of the refractive index actually compounds the rate of change of the focal length.

Failing to apply the quotient rule to account for the time-varying refractive index leads to exactly half the correct velocity (Option B), while ignoring the lens effect entirely and treating it as a simple mirror leads to Option D.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Thermal Expansion of Area and Volume
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free