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Question #43892

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The Question

A thin convex lens of focal length $f_0$ is mounted at the free end of a horizontal bimetallic cantilever of length $L$ . The cantilever consists of two identical metallic strips, each of thickness $d$ , bonded together. The top strip has a coefficient of linear expansion $\alpha_1$ and the bottom strip has $\alpha_2$ (where $\alpha_1 > \alpha_2$ ). A small specimen is placed on a fixed horizontal surface directly below the lens at a distance $u_0 = \frac{f_0}{2}$ . The lens material has a coefficient of linear expansion $\alpha_g$ . If the temperature of the entire system is increased by $\Delta T$ , what is the first-order change in the linear magnification ( $\Delta m$ ) of the image? (Assume the refractive index of the lens remains constant and the bending of the cantilever is small).

Options

A.
$-2 \Delta T \left( \alpha_g + \frac{L^2 (\alpha_1 - \alpha_2)}{f_0 d} \right)$
B.
$-2 \Delta T \left( \alpha_g - \frac{L^2 (\alpha_1 - \alpha_2)}{f_0 d} \right)$
C.
$-\Delta T \left( \alpha_g + \frac{L^2 (\alpha_1 - \alpha_2)}{2 f_0 d} \right)$
D.
$- \frac{2 L^2 (\alpha_1 - \alpha_2) \Delta T}{f_0 d}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Optical Instrument Magnification|Bimetallic Strip Curvature|Thermal Expansion of Area and Volume

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple mechanical deformation with thermal optical effects.

The primary trap lies in failing to recognize that both the focal length shift (due to the lens's thermal expansion) and the geometric translation of the optical axis (due to the bimetallic strip's curvature) simultaneously affect the magnification.

A common mistake is misassigning the sign of the object distance change; students often assume thermal expansion strictly increases distances, failing to realize that the downward bending of the cantilever decreases the lens-to-specimen distance, leading to a sign error between the coupled terms.

Quick Facts About This Question

Question Information

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