Q.32 Physics · Section A ⭐ 360° Option Elimination Active

NEET 2024 Physics Question 32 Solution

Topic: Ray Optics and Optical Instruments · Official NTA Booklet Code T1

$NEET 2024 Physics Question 32: Refractive Index of a Right-Angled Prism with Internal Grazing Emergence$

Question

A light ray enters through a right angled prism at point P with the angle of incidence 30° as shown in figure. It travels through the prism parallel to its base BC and emerges along the face AC. The refractive index of the prism is:

✓

Verified Against NTA Final Answer Key

Option (4) : √5 / 2

📚 Ray Optics and Optical Instruments Medium

📋 Given Parameters & Problem Setup

Prism Geometry Right-angled prism at vertex A (A = 90°)

Angle of incidence at face AB i = 30° at point P

Internal ray path Travels parallel to base BC

Emergence Emerges grazing along face AC (r2 = critical angle θ_c)

⚡

The 30-Second Exam Hall Shortcut: ⚡ Right-Angled Prism Geometry Hack

⏱️ Save ~50 seconds

A = 90° -> r1 + θ_c = 90° -> sin θ_c = cos r1 = 1/μ. Snell at entrance: sin 30° = μ sin r1 -> sin r1 = 1/(2μ). sin² + cos² = 1 gives μ = √5 / 2!

1 At entrance face AB: 1 · sin 30° = μ sin r1 => sin r1 = 1 / (2μ).

2 At face AC, the ray emerges along the face (grazing emergence): r2 = θ_c (critical angle) => sin θ_c = 1 / μ.

3 For a prism, r1 + r2 = A. Here prism angle A = 90°, so r1 + θ_c = 90° => θ_c = 90° - r1.

4 Therefore, sin θ_c = cos r1 = 1 / μ.

5 Apply identity sin² r1 + cos² r1 = 1: [1 / (2μ)]² + [1 / μ]² = 1 => 1 / (4μ²) + 1 / μ² = 5 / (4μ²) = 1.

6 μ² = 5 / 4 => μ = √5 / 2. Pick Option (4) in under 20 seconds!

🛡️

Why You Can 100% Trust This Shortcut: Snell’s Law & Prism Boundary Geometry (sin² r1 + cos² r1 = 1 with r1 + r2 = A = 90°).

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Snell’s Law at the First Face AB

Light enters the prism from air (n = 1) at point P on face AB with angle of incidence i = 30°. By Snell’s law:

1 \cdot \sin 30^\circ = \mu \sin r_1 \implies \frac{1}{2} = \mu \sin r_1 \implies \sin r_1 = \frac{1}{2\mu}

Step 2

Condition for Grazing Emergence at Face AC

The ray emerges along the face AC, which means the angle of refraction in air is 90°. Therefore, the internal angle of incidence r2 at face AC must be identically equal to the critical angle θ_c for the prism-air interface:

\sin r_2 = \sin \theta_c = \frac{1}{\mu}

Step 3

Prism Vertex Angle Geometry

For any triangular prism of refracting angle A, the internal angles satisfy r1 + r2 = A. The figure indicates that vertex A is a right angle (A = 90°):

r_1 + r_2 = A = 90^\circ \implies r_2 = 90^\circ - r_1

Step 4

Trigonometric Relation Between Angles

Substituting r2 = 90° - r1 into the critical angle formula:

\sin(90^\circ - r_1) = \cos r_1 = \frac{1}{\mu}

Step 5

Solving for Refractive Index μ

Using the fundamental trigonometric identity sin² r1 + cos² r1 = 1:

\left(\frac{1}{2\mu}\right)^2 + \left(\frac{1}{\mu}\right)^2 = 1 \implies \frac{1}{4\mu^2} + \frac{1}{\mu^2} = 1 \implies \frac{5}{4\mu^2} = 1

Step 6

Final Extraction

Inverting and taking the positive square root:

\mu^2 = \frac{5}{4} \implies \mu = \frac{\sqrt{5}}{2}

⭐

360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 (1)

✗ Incorrect

Verdict & Scientific Reason:

Less than 1 (refractive index of glass cannot be < 1).

⚠️

Student Fallacy (Why candidates pick this trap): Using 60° prism geometry and dividing by 4.

2 (2)

✗ Incorrect

Verdict & Scientific Reason:

Also less than 1 (0.866).

⚠️

Student Fallacy (Why candidates pick this trap): Physical impossibility: μ of glass must exceed 1.

3 (3)

✗ Incorrect

Verdict & Scientific Reason:

Less than 1 (0.559).

⚠️

Student Fallacy (Why candidates pick this trap): Forgetting to take square root of 4 in the denominator.

4 (4)

✓ 100% Correct

Verdict & Scientific Reason:

Physically valid (√5/2 ≈ 1.118) and mathematically exact.

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Why this is airtight: None.

⚠️

NTA Trap Alert: Eliminating Sub-Unity Refractive Indices

Remarkably, Options (1), (2), and (3) all have values LESS than 1 (√3/4 ≈ 0.43, √3/2 ≈ 0.866, √5/4 ≈ 0.559).

🛡️ How to avoid the -1 mark: The refractive index of any optical medium in air must be μ > 1. Option (4) is the ONLY option with μ > 1 (√5/2 ≈ 1.118)! You could solve this without even writing a single equation!

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Core NEET Syllabus Standard

Physics Core Optics · Ray Optics and Optical Instruments Standard Benchmark

Core Scientific Principle Tested:

"In a right-angled prism (A = 90°), r1 + r2 = 90°, so sin² r1 + sin² r2 = 1. Grazing emergence means sin r2 = 1/μ."

Syllabus Focus: Refraction Through a Prism and Total Internal Reflection

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

Check options first: Only Option 4 exceeds 1 (√5/2 = 1.12)!

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NEET 2024 Physics Question 32 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Right-Angled Prism Geometry Hack

Step-by-Step Rigorous Workflow & Mathematical Proof

Snell’s Law at the First Face AB

Condition for Grazing Emergence at Face AC

Prism Vertex Angle Geometry

Trigonometric Relation Between Angles

Solving for Refractive Index μ

Final Extraction

360° Option Elimination Engine Key Feature

NTA Trap Alert: Eliminating Sub-Unity Refractive Indices

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

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