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Physics Optics Reading Time: 18 min 6 Solved Examples

Solving the Prism: A Step-by-Step Guide to Deviation and Refractive Index Problems

Master the prism formula and minimum deviation concept to solve any JEE optics problem with confidence.

2-3
Questions per JEE
4-8
Marks Weightage
3
Key Formulas
95%
Success Rate

Why Prism Problems are Scoring Opportunities

Prism problems in JEE Physics follow predictable patterns and use standard formulas. Once you understand the core concepts, you can solve them quickly and accurately, making them high-value questions.

🎯 JEE Analysis

Prism problems appear in 2-3 questions per JEE paper with 4-8 marks weightage. They're often considered "sure-shot" marks by prepared students.

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Basic Concepts Prism Formula Min Deviation Practice

1. Basic Prism Concepts and Terminology

Essential Prism Terminology

Key Parameters

  • A - Angle of Prism
  • δ - Angle of Deviation
  • δₘ - Minimum Angle of Deviation
  • μ - Refractive Index
  • i - Angle of Incidence
  • e - Angle of Emergence
  • r₁, r₂ - Angles of Refraction

Fundamental Relations

  • $A = r_1 + r_2$
  • $δ = i + e - A$
  • At minimum deviation: $i = e$ and $r_1 = r_2$
  • $r_1 = r_2 = \frac{A}{2}$ at δₘ
  • $i = e = \frac{A + δ_m}{2}$ at δₘ

Prism Geometry

[Prism Diagram: Showing A, δ, i, e, r₁, r₂]
A = r₁ + r₂
δ = i + e - A

Remember: The angle of prism (A) equals the sum of the two refraction angles inside the prism.

2. The Prism Formula and Its Applications

The Master Formula

$$μ = \frac{\sin\left(\frac{A + δ_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$

This formula applies specifically at minimum deviation condition

Derivation Insight

At minimum deviation:

  • $i = e = \frac{A + δ_m}{2}$
  • $r_1 = r_2 = \frac{A}{2}$
  • Using Snell's Law at first surface: $μ = \frac{\sin i}{\sin r_1}$
  • Substitute: $μ = \frac{\sin\left(\frac{A + δ_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

Example 1: Basic Refractive Index Calculation

Problem: A prism with angle 60° produces minimum deviation of 40°. Find refractive index.

Solution

Step 1: Identify given values

$A = 60°$, $δ_m = 40°$

Step 2: Apply prism formula

$μ = \frac{\sin\left(\frac{60° + 40°}{2}\right)}{\sin\left(\frac{60°}{2}\right)} = \frac{\sin(50°)}{\sin(30°)}$

Step 3: Calculate

$μ = \frac{0.7660}{0.5} = 1.532$

Answer: μ = 1.532

3. Understanding Minimum Deviation

What is Minimum Deviation?

Minimum deviation (δₘ) occurs when the light ray passes symmetrically through the prism. This is a special condition where:

  • Angle of incidence = Angle of emergence ($i = e$)
  • First refraction angle = Second refraction angle ($r_1 = r_2$)
  • The light ray inside the prism is parallel to the base
  • Deviation is minimum for that prism-material combination

Example 2: Finding Minimum Deviation

Problem: A prism of refractive index 1.5 and angle 60° is used. Find the minimum deviation.

Solution

Step 1: Write the prism formula

$μ = \frac{\sin\left(\frac{A + δ_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

Step 2: Substitute known values

$1.5 = \frac{\sin\left(\frac{60° + δ_m}{2}\right)}{\sin(30°)}$

Step 3: Solve for δₘ

$\sin\left(\frac{60° + δ_m}{2}\right) = 1.5 × \sin(30°) = 1.5 × 0.5 = 0.75$

$\frac{60° + δ_m}{2} = \sin^{-1}(0.75) ≈ 48.59°$

$60° + δ_m = 97.18°$

$δ_m = 37.18°$

Answer: δₘ ≈ 37.2°

💡 Important Characteristics at Minimum Deviation

  • Symmetric path: i = e and r₁ = r₂ = A/2
  • Unique value: For given A and μ, δₘ is fixed
  • Experimental use: Used to determine refractive index accurately
  • Dispersion: Different colors have different δₘ values

4. Advanced Problem Solving Techniques

Example 3: Finding Prism Angle

Problem: A prism produces minimum deviation of 50° for a light ray. If refractive index is 1.6, find the prism angle.

Solution

Step 1: Use prism formula

$μ = \frac{\sin\left(\frac{A + δ_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

Step 2: Substitute values

$1.6 = \frac{\sin\left(\frac{A + 50°}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

Step 3: Let $x = \frac{A}{2}$, then:

$1.6 = \frac{\sin(x + 25°)}{\sin x}$

$1.6 \sin x = \sin(x + 25°) = \sin x \cos 25° + \cos x \sin 25°$

Step 4: Rearrange and solve

$1.6 \sin x - \sin x \cos 25° = \cos x \sin 25°$

$\sin x (1.6 - \cos 25°) = \cos x \sin 25°$

$\tan x = \frac{\sin 25°}{1.6 - \cos 25°}$

$\tan x = \frac{0.4226}{1.6 - 0.9063} = \frac{0.4226}{0.6937} ≈ 0.609$

$x ≈ 31.3°$

$A = 2x ≈ 62.6°$

Answer: A ≈ 62.6°

Example 4: Grazing Incidence and Emergence

Problem: For a prism with A = 60° and μ = 1.5, find the maximum possible deviation.

Solution

Step 1: Maximum deviation occurs at grazing incidence or emergence

Step 2: For grazing incidence, i = 90°

Using Snell's Law at first surface: $\sin i = μ \sin r_1$

$\sin 90° = 1.5 \sin r_1$ ⇒ $\sin r_1 = \frac{1}{1.5} = 0.6667$

$r_1 = \sin^{-1}(0.6667) ≈ 41.81°$

Step 3: Find r₂ using A = r₁ + r₂

$r_2 = A - r_1 = 60° - 41.81° = 18.19°$

Step 4: Find angle of emergence e

$\sin e = μ \sin r_2 = 1.5 × \sin 18.19° = 1.5 × 0.3123 = 0.4685$

$e = \sin^{-1}(0.4685) ≈ 27.93°$

Step 5: Calculate deviation δ = i + e - A

$δ = 90° + 27.93° - 60° = 57.93°$

Answer: Maximum δ ≈ 57.9°

📋 Prism Problem Solving Framework

Step-by-Step Approach

  1. Identify given parameters (A, μ, δ, i, e, r₁, r₂)
  2. Check if at minimum deviation (if δₘ given or implied)
  3. Apply relevant formulas based on available data
  4. Use Snell's Law at refraction surfaces
  5. Apply geometry relations (A = r₁ + r₂, δ = i + e - A)
  6. Solve systematically for unknown variables

Key Formulas to Remember

  • Prism formula: $μ = \frac{\sin\left(\frac{A + δ_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
  • Geometry: $A = r_1 + r_2$
  • Deviation: $δ = i + e - A$
  • At δₘ: $i = e = \frac{A + δ_m}{2}$, $r_1 = r_2 = \frac{A}{2}$
  • Snell's Law: $μ = \frac{\sin i}{\sin r}$

5. Practice Problems

Test Your Understanding

Problem 1: A prism with A = 70° has μ = 1.6. Calculate the minimum deviation.

Hint: Use the prism formula directly

Problem 2: For a prism, A = 60° and δₘ = 45°. Light is incident at 55°. Find the deviation.

Hint: First find μ, then apply general deviation formula

Problem 3: A prism deviates a ray by 40° at minimum deviation. If A = 60°, find i and e.

Hint: At minimum deviation, i = e = (A + δₘ)/2

Problem 4: Calculate the refractive index of a prism for which A = 60° and i = e = 50°.

Hint: i = e indicates minimum deviation condition

Problem Solving Tips

  • Always check if the problem involves minimum deviation
  • Remember the symmetric conditions at δₘ
  • Use the prism formula only at minimum deviation
  • For general cases, use Snell's Law at both surfaces

⚠️ Common Mistakes to Avoid

❌
Using prism formula for non-minimum deviation

The formula μ = sin((A+δₘ)/2)/sin(A/2) works ONLY at minimum deviation

❌
Forgetting A = r₁ + r₂ relation

This geometric relation is fundamental to all prism problems

❌
Mixing degrees and radians

JEE prism problems typically use degrees for angles

❌
Assuming i = e when not at minimum deviation

This symmetry occurs only at minimum deviation

🎯 JEE Exam Strategy

⚡
Quick Identification

Recognize minimum deviation problems quickly - they're the fastest to solve

🔢
Numerical Accuracy

Keep trigonometric values handy: sin 30°=0.5, sin 45°=0.707, sin 60°=0.866

📝
Step-wise Solution

Show key steps clearly for partial credit in case of calculation errors

⏱️
Time Management

Prism problems should take 3-5 minutes maximum

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