Physics Optics and Modern - How to Solve: Refraction at Plane Surface (Snell’s Law, Total Internal Reflection, Prism) – IIT JEE Guide

How to Solve: Refraction at Plane Surface (Snell’s Law, Total Internal Reflection, Prism) – IIT JEE Guide

Introduction Mastering refraction at plane surfaces unlocks 10-15 marks in IIT JEE (Main + Advanced) every year—questions on Snell’s Law, total internal reflection, and prisms appear directly in both papers. This is your high-yield topic to secure easy marks while others struggle with sign conventions and critical angles.

WHAT YOU NEED TO KNOW FIRST

Before diving in, ensure you understand:
1. Basic optics: Definitions of incident ray, refracted ray, normal, angle of incidence (i), angle of refraction (r).
2. Relative refractive index (μ): μ = speed of light in medium 1 / speed of light in medium 2.
3. Trigonometry: Sine and cosine of angles (0°, 30°, 45°, 60°, 90°) must be memorized.

KEY TERMS & FORMULAS

1. Snell’s Law (Law of Refraction)

Formula: μ₁ sin i = μ₂ sin r - μ₁ = Refractive index of medium 1 (incident medium) - μ₂ = Refractive index of medium 2 (refracting medium) - i = Angle of incidence (measured from normal) - r = Angle of refraction (measured from normal) MEMORISE THIS – This is the core formula for all refraction problems.

2. Relative Refractive Index (μ₂₁)

Formula: μ₂₁ = μ₂ / μ₁ = v₁ / v₂ = λ₁ / λ₂ - μ₂₁ = Refractive index of medium 2 with respect to medium 1 - v₁, v₂ = Speed of light in medium 1 and 2 - λ₁, λ₂ = Wavelength of light in medium 1 and 2 Given on exam sheet (but understand it deeply).

3. Total Internal Reflection (TIR)

Conditions for TIR:
1. Light must travel from denser to rarer medium (μ₁ > μ₂).
2. Angle of incidence i > critical angle (C).

Critical Angle (C) Formula: sin C = μ₂ / μ₁ (where μ₁ > μ₂) MEMORISE THIS – Used in optical fibers, mirages, and prism problems.

4. Prism (Deviation & Dispersion)

Prism Formula (Angle of Deviation δ): δ = i + e – A - i = Angle of incidence - e = Angle of emergence - A = Angle of prism (given) MEMORISE THIS – Used in minimum deviation problems.

Minimum Deviation (δₘ): μ = sin[(A + δₘ)/2] / sin(A/2) MEMORISE THIS – Most important prism formula for JEE.

STEP-BY-STEP METHOD

Step 1: Identify the Problem Type

• Snell’s Law? → Use μ₁ sin i = μ₂ sin r.
• Total Internal Reflection? → Check if i > C and μ₁ > μ₂.
• Prism? → Use δ = i + e – A or minimum deviation formula.

Step 2: Draw the Diagram

• Always draw the normal (perpendicular to the surface).
• Label i, r, μ₁, μ₂ clearly.
• For prisms, draw incident ray, refracted ray inside prism, emergent ray.

Step 3: Apply the Correct Formula

• Snell’s Law: Plug in known values, solve for unknown.
• TIR: Find C = sin⁻¹(μ₂/μ₁), then check if i > C.
• Prism: Use δ = i + e – A or μ = sin[(A + δₘ)/2] / sin(A/2).

Step 4: Solve for the Unknown

• Use trigonometry (sin⁻¹, cos⁻¹) if needed.
• For prisms, minimum deviation occurs when i = e (symmetrical path).

Step 5: Check Units & Validity

• Angles must be in degrees (not radians).
• μ must be > 1 for denser medium.
• TIR only happens when μ₁ > μ₂.

WORKED EXAMPLES

Example 1 – Basic (Snell’s Law)

Problem: A ray of light passes from air (μ = 1) into water (μ = 4/3) at an angle of incidence of 30°. Find the angle of refraction.

Solution:
1. Identify: Snell’s Law problem.
2. Draw diagram: Air → Water, i = 30°.
3. Apply Snell’s Law: μ₁ sin i = μ₂ sin r 1 × sin 30° = (4/3) × sin r 0.5 = (4/3) sin r sin r = (0.5 × 3)/4 = 3/8 = 0.375
4. Solve for r: r = sin⁻¹(0.375) ≈ 22.02°

What we did and why: We used Snell’s Law to relate the angles and refractive indices. The key was correctly identifying μ₁ and μ₂ (air first, then water).

Example 2 – Medium (TIR)

Problem: A light ray travels from glass (μ = 1.5) to air. What is the critical angle for total internal reflection?

Solution:
1. Identify: TIR problem (glass → air, μ₁ > μ₂).
2. Draw diagram: Glass (μ = 1.5) → Air (μ = 1).
3. Apply critical angle formula: sin C = μ₂ / μ₁ = 1 / 1.5 = 2/3 ≈ 0.6667
4. Solve for C: C = sin⁻¹(2/3) ≈ 41.81°

What we did and why: We used the critical angle formula for TIR. The key was recognizing that μ₁ > μ₂ (glass is denser than air).

Example 3 – Exam-Style (Prism + Minimum Deviation)

Problem: A prism of angle 60° has a refractive index of √2. Find the angle of minimum deviation.

Solution:
1. Identify: Prism problem (minimum deviation).
2. Draw diagram: Prism with A = 60°, μ = √2.
3. Apply minimum deviation formula: μ = sin[(A + δₘ)/2] / sin(A/2) √2 = sin[(60° + δₘ)/2] / sin(30°) √2 = sin[(60° + δₘ)/2] / 0.5 sin[(60° + δₘ)/2] = √2 × 0.5 = √2/2 ≈ 0.7071
4. Solve for δₘ: (60° + δₘ)/2 = sin⁻¹(0.7071) = 45° 60° + δₘ = 90° δₘ = 30°

What we did and why: We used the minimum deviation formula for prisms. The key was recognizing that sin⁻¹(√2/2) = 45° (a common angle).

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP (Night Before Exam)

"Listen up—this is your 60-second refraction cheat sheet for JEE!

1. Snell’s Law: μ₁ sin i = μ₂ sin r. Memorize it. If light bends towards the normal, μ increases. If it bends away, μ decreases.
2. Total Internal Reflection (TIR): Only happens when light goes from denser to rarer medium (μ₁ > μ₂). Critical angle C = sin⁻¹(μ₂/μ₁). If i > C, TIR occurs.
3. Prism: Deviation δ = i + e – A. At minimum deviation, i = e, and μ = sin[(A + δₘ)/2] / sin(A/2). This formula is gold—memorize it!
4. Common angles: sin 30° = 0.5, sin 45° = √2/2, sin 60° = √3/2. Know these cold.
5. Exam traps: Watch for μ given as λ₀/λ, multiple refractions in prisms, and TIR conditions.

Final tip: Always draw the diagram first. Label i, r, μ₁, μ₂ clearly. If stuck, go back to Snell’s Law. You’ve got this—now go ace that exam!