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Question #42135

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The Question

A Young's Double Slit Experiment (YDSE) apparatus is constructed entirely from a solid isotropic material with a coefficient of linear thermal expansion $\alpha$ . The slits are separated by a distance $d$ and the screen is rigidly attached at a distance $D$ from the slit plane. The apparatus is illuminated by a coherent light source of wavelength $\lambda$ (which remains unaffected by the temperature of the apparatus). Initially, at temperature $T$ , the $n^{\text{th}}$ order bright fringe is observed at a distance $y_n$ from the central maximum. If the entire apparatus is uniformly heated to a temperature $T + \Delta T$ , what is the shift $\Delta y_n$ in the position of the $n^{\text{th}}$ order bright fringe? (Assume $\alpha \Delta T \ll 1$ and retain leading-order terms in $\alpha \Delta T$ )

Options

A.
$-\frac{y_n^3}{D^2} \alpha \Delta T$
B.
0
C.
$\frac{y_n^3}{D^2} \alpha \Delta T$
D.
$y_n \alpha \Delta T$

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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

Expansion of Cavities in Solids|Young's Double Slit Geometry|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the small-angle approximation in wave optics under thermal expansion.

A common trap is for students to mechanically apply the standard fringe position formula $y_n = \frac{n\lambda D}{d}$ .

If they do this, substituting $D' = D(1+\alpha\Delta T)$ and $d' = d(1+\alpha\Delta T)$ yields $y_n' = y_n$ , incorrectly implying a shift of zero (Option B).

However, the small-angle approximation masks the true geometric path difference for higher-order fringes.

The exact condition $d\sin\theta = n\lambda$ must be used.

Because the wavelength $\lambda$ does not expand with the apparatus, the expansion of the slit separation $d$ causes the diffraction angle $\theta$ to decrease.

A rigorous binomial expansion of $\tan\theta$ reveals a non-intuitive, non-zero inward shift proportional to $y_n^3/D^2$ .

Quick Facts About This Question

Question Information

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