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Question #43215

Difficulty: 8/10 | 🟢 Free

The Question

A monochromatic plane wave of wavelength $\lambda$ is incident on a glass prism of refracting angle $A$ and refractive index $\mu$ . The prism has an isosceles triangular cross-section with a base of length $t$ . The prism is oriented in the minimum deviation position, and its entire entrance face is illuminated by the incident wave. The emergent wave from the second face undergoes Fraunhofer diffraction. Assuming the angular spread is small such that $\sin(\Delta \theta) \approx \Delta \theta$ , which of the following expressions gives the angular half-width $\Delta \theta$ of the central maximum of the emergent diffraction pattern? Assume the surrounding medium is vacuum.

Options

A.
$\frac{2 \lambda \sin(A/2)}{t \sqrt{1 - \mu^2 \sin^2(A/2)}}$
B.
$\frac{2 \lambda \sin(A/2)}{t}$
C.
$\frac{\lambda}{t}$
D.
$\frac{2 \lambda \cos(A/2)}{t \sqrt{\mu^2 - 1}}$

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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

Prism Minimum Deviation|Intensity Distribution in YDSE|Huygens Principle Wavefronts

Why This Question Is Tricky

Common Mistakes

Students typically treat the physical exit face of the prism as the diffracting aperture, incorrectly assigning the aperture width as the slant length $L$ or the base $t$ .

This ignores the fundamental wave-optics principle that the effective aperture for a propagating wavefront is the projection of the physical opening perpendicular to the direction of wave propagation ( $D = L \cos e$ ).

By failing to transition from the geometric boundary to the physical wavefront width, they miss the $\cos e$ projection factor, leading to dimensionally similar but physically flawed results.

This question forces the synthesis of Fermat's principle (minimum deviation symmetry) with the Huygens-Fresnel principle (projected wavefront aperture).

Quick Facts About This Question

Question Information

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