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Question #43209

Difficulty: 8/10 | 🟢 Free

The Question

A thin symmetric biconvex glass lens is held at a constant depth $h$ below the free surface of a liquid. At an initial temperature $T_0$ , the liquid has density $\rho_0$ , and the lens has a focal length $f_0$ . The entire system is then heated uniformly to a temperature $T_0 + \Delta T$ . The glass has a coefficient of linear thermal expansion $\alpha$ and a bulk modulus $K$ . The liquid has a coefficient of volume expansion $\gamma$ . Assuming the refractive indices of both the glass and the liquid remain strictly independent of temperature and pressure, which of the following expressions gives the new focal length $f$ of the lens? (Assume $\alpha \Delta T \ll 1$ , $\gamma \Delta T \ll 1$ , and $\rho_0 g h \ll K$ )

Options

A.
$f_0 \left[ 1 + \left( \alpha + \frac{\rho_0 g h \gamma}{3K} \right) \Delta T \right]$
B.
$f_0 \left[ 1 + \left( \alpha - \frac{\rho_0 g h \gamma}{3K} \right) \Delta T \right]$
C.
$f_0 \left[ 1 + \left( 3\alpha + \frac{\rho_0 g h \gamma}{K} \right) \Delta T \right]$
D.
$f_0 \left[ 1 + \alpha \Delta T \right]$

Check Your Answer

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

Lens Maker’s Formula Constraints|Archimedes' Buoyancy Dynamics|Internal Energy Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of geometric optics, fluid statics, and solid mechanics (elastostatics).

The primary trap is assuming that the focal length only changes due to the direct thermal expansion of the lens ( $\alpha \Delta T$ ), leading students to choose Option D.

Students often fail to recognize that the surrounding fluid's thermal expansion decreases its density, which in turn reduces the hydrostatic pressure at a fixed depth.

This reduction in pressure allows the lens to undergo an elastic volumetric expansion, further increasing its radius of curvature and focal length.

Option B catches students who misapply the sign of the pressure change, while Option C catches those who incorrectly use volumetric expansion instead of linear expansion for the radius.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Lens Maker’s Formula Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free