Before You Begin

Question #43724

Difficulty: 8/10 | 🟢 Free

The Question

In a Young's double slit experiment, the distance between the slits is $d$ and the distance to the screen is $D$ . A transparent isotropic solid slab of thickness $t$ , refractive index $\mu$ , coefficient of linear thermal expansion $\alpha$ , and Poisson's ratio $\nu$ is placed in front of the upper slit. The slab is tightly fitted into a rigid, non-expanding lateral frame so that it cannot expand in the directions parallel to the screen, but is free to expand in the direction of light propagation. The temperature of the slab is increased by a small amount $\Delta T$ . If the refractive index of the slab varies with temperature as $\frac{d\mu}{dT} = \gamma$ , which of the following expressions gives the additional upward shift of the central maximum on the screen?

Options

A.
$\frac{D t \Delta T}{d} \left[ (\mu-1)\alpha \left( \frac{1+\nu}{1-\nu} \right) + \gamma \right]$
B.
$\frac{D t \Delta T}{d} \left[ (\mu-1)\alpha + \gamma \right]$
C.
$\frac{D t \Delta T}{d} \left[ 3(\mu-1)\alpha + \gamma \right]$
D.
$\frac{D t \Delta T}{d} \left[ (\mu-1)\alpha \left( \frac{1-\nu}{1+\nu} \right) + \gamma \right]$

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

Central Fringe Shift|Expansion of Cavities in Solids|Temperature Coefficient (Alpha)

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of wave optics with thermo-elasticity.

Students commonly fall into two traps: (1) They apply the standard unconstrained linear expansion formula ( $\Delta t = \alpha t \Delta T$ ) ignoring the lateral constraints. (2) They assume that since lateral expansion is prevented, the entire volume expansion ( $3\alpha$ ) is forced into the thickness direction ( $\Delta t = 3\alpha t \Delta T$ ), which is only true for perfectly incompressible materials ( $\nu = 0.5$ ).

The correct approach requires applying generalized Hooke's law to find the exact uniaxial strain under lateral constraint, which depends on Poisson's ratio, and then coupling this geometric change with the thermo-optic variation using the product rule on the optical path difference.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Central Fringe Shift
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free