Prism - Dispersion, Deviation & Rainbow Physics

Master prism formulas, angular deviation, dispersion, and rainbow formation for JEE Main and Advanced

9 min read

Prism Optics - Splitting Light

The Movie Hook 🎬

Remember Pink Floyd’s Dark Side of the Moon album cover with the prism creating a rainbow? Or the rainbow in The Wizard of Oz? Or how Thor: Ragnarok uses the Bifrost as a cosmic prism? These aren’t just artistic choices - they show the beautiful physics of dispersion that separates white light into colors!

The Big Picture

A prism is a transparent optical element with flat, polished surfaces that refract light. Key phenomena:

  • Deviation: Bending of light through prism
  • Dispersion: Separation of white light into colors
  • Rainbow formation: Nature’s prism show
  • Spectrometers: Measuring wavelengths

Connection Alert: Builds on Refraction, relates to Wave Optics.

Core Concepts

1. Prism Terminology

Key Terms:

  • Refracting surfaces: Two plane surfaces at an angle
  • Refracting edge: Intersection of two refracting surfaces
  • Refracting angle (A): Angle between two refracting surfaces
  • Principal section: Section perpendicular to refracting edge
  • Angle of incidence (i): Angle of incoming ray with normal
  • Angle of emergence (e): Angle of outgoing ray with normal
  • Angle of deviation (δ): Angle between incident and emergent rays

Memory Trick: “Prisms Refract All Rays Differently” (PRARD)

2. Prism Formula (Deviation)

For a ray passing through prism:

$$\boxed{\delta = i + e - A}$$

Where:

  • δ = angle of deviation
  • i = angle of incidence
  • e = angle of emergence
  • A = angle of prism (refracting angle)

At both surfaces, Snell’s law applies:

At first surface:

$$n_1 \sin i = n \sin r_1$$

At second surface:

$$n \sin r_2 = n_1 \sin e$$

Relation between angles:

$$\boxed{A = r_1 + r_2}$$

Where r₁ and r₂ are angles of refraction at first and second surfaces.

3. Minimum Deviation

Condition for minimum deviation:

  • Ray passes symmetrically through prism
  • i = e (angle of incidence = angle of emergence)
  • r₁ = r₂ = r

At minimum deviation:

$$\boxed{\delta_m = 2i - A}$$

Or:

$$\boxed{i = \frac{A + \delta_m}{2}}$$

Also:

$$r = \frac{A}{2}$$

Refractive index from minimum deviation:

$$\boxed{n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}}$$

This is THE most important prism formula for JEE!

Memory Trick: “At Minimum, I Equal Everything” (i = e, r₁ = r₂)

4. Thin Prism (Small Angle Approximation)

For small angles (A < 10°):

$$\boxed{\delta = (n-1)A}$$

This is independent of i for thin prisms!

Derivation: For small angles, sin θ ≈ θ

From Snell’s law and geometry:

$$\delta \approx (n-1)A$$

Application: Used in lens aberration corrections, optical instruments.

5. Dispersion

Dispersion: Splitting of white light into constituent colors due to variation of refractive index with wavelength.

Cauchy’s Equation:

$$\boxed{n = A + \frac{B}{\lambda^2}}$$

Shows that n decreases as λ increases.

Order of colors (increasing wavelength): VIBGYOR: Violet, Indigo, Blue, Green, Yellow, Orange, Red

Refractive indices:

$$n_V > n_I > n_B > n_G > n_Y > n_O > n_R$$

Deviations:

$$\delta_V > \delta_I > \delta_B > \delta_G > \delta_Y > \delta_O > \delta_R$$
Note
Key Insight: Violet light bends most (highest n), red light bends least (lowest n).

6. Angular Dispersion

Angular dispersion (θ): Angular separation between extreme colors (violet and red)

$$\boxed{\theta = \delta_V - \delta_R}$$

For thin prism:

$$\boxed{\theta = (n_V - n_R)A}$$

Memory Trick: “Very Red Difference” (θ = δ_V - δ_R)

7. Dispersive Power

Dispersive power (ω): Measure of prism’s ability to disperse light

$$\boxed{\omega = \frac{\theta}{\delta_Y} = \frac{\delta_V - \delta_R}{\delta_Y}}$$

Or using refractive indices:

$$\boxed{\omega = \frac{n_V - n_R}{n_Y - 1}}$$

Where n_Y is refractive index for yellow light (mean color).

Properties:

  • Dimensionless quantity
  • Depends only on material, not on angle A
  • Crown glass: ω ≈ 0.03
  • Flint glass: ω ≈ 0.05

8. Deviation Without Dispersion (Achromatic Combination)

To get deviation without dispersion, combine two prisms:

Condition:

$$\boxed{\omega_1 A_1 + \omega_2 A_2 = 0}$$

Net deviation:

$$\boxed{\delta = (n_Y - 1)(A_1 + A_2)}$$

Usually: Two prisms of opposite orientation (one upright, one inverted)

Example: Crown glass prism + Flint glass prism (inverted)

9. Dispersion Without Deviation (Direct Vision Combination)

To get dispersion without deviation:

Condition:

$$\boxed{(n_{Y1} - 1)A_1 + (n_{Y2} - 1)A_2 = 0}$$

Net dispersion:

$$\boxed{\theta = \theta_1 + \theta_2}$$

Used in direct vision spectroscopes.

The Formula Cheat Sheet

PRISM ESSENTIALS:
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
1. Deviation:            δ = i + e - A
2. Angle relation:       A = r₁ + r₂
3. Min deviation:        n = sin[(A+δₘ)/2] / sin(A/2)
4. Thin prism:           δ = (n-1)A
5. Dispersion:           θ = δ_V - δ_R
6. Dispersive power:     ω = (n_V-n_R)/(n_Y-1)
7. Achromatic:           ω₁A₁ + ω₂A₂ = 0
━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

Rainbow Physics

Primary Rainbow

Formation:

  1. One internal reflection inside water droplet
  2. Refraction → Reflection → Refraction
  3. Red on top (42° from anti-solar point)
  4. Violet on bottom (40° from anti-solar point)

Angle: About 40-42° from anti-solar direction

Colors: ROYGBIV (reversed from prism!)

Secondary Rainbow

Formation:

  1. Two internal reflections inside droplet
  2. Refraction → Reflection → Reflection → Refraction
  3. Violet on top, Red on bottom (reversed!)
  4. Fainter than primary (more light lost)

Angle: About 50-53° from anti-solar point

Dark band between rainbows: Alexander’s dark band (no light scattered at these angles)

Note
Why is rainbow circular? Because all droplets at same angle from anti-solar point scatter light toward you - forms a cone!

Common Traps & Mistakes

Trap #1: Deviation Formula

❌ Wrong: δ = i - e + A ✅ Right: δ = i + e - A (both incident and emergent contribute!)

Trap #2: Minimum Deviation Condition

❌ Wrong: Using i = e in δ = i + e - A to get δ = 2i - A for all cases ✅ Right: δ = 2i - A is ONLY at minimum deviation!

Trap #3: Color Order in Rainbow

❌ Wrong: Thinking rainbow has same color order as prism ✅ Right: Primary rainbow has RED on top (opposite to prism spectrum on ground)

Trap #4: Dispersive Power

❌ Wrong: ω = (n_V - n_R)/n_Y ✅ Right: ω = (n_V - n_R)/(n_Y - 1) - don’t forget the -1!

Trap #5: Thin Prism Approximation

❌ Wrong: Using δ = (n-1)A for large angle prisms ✅ Right: Only valid for A < 10° approximately

Practice Problems

Level 1: JEE Main Warmup

Problem 1.1: A prism has refracting angle A = 60° and refractive index n = 1.5. Find the angle of minimum deviation.

Solution

Given: A = 60°, n = 1.5

Using minimum deviation formula:

$$n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$ $$1.5 = \frac{\sin\left(\frac{60° + \delta_m}{2}\right)}{\sin 30°}$$ $$1.5 = \frac{\sin\left(\frac{60° + \delta_m}{2}\right)}{0.5}$$ $$\sin\left(\frac{60° + \delta_m}{2}\right) = 0.75$$ $$\frac{60° + \delta_m}{2} = \sin^{-1}(0.75) = 48.6°$$ $$60° + \delta_m = 97.2°$$ $$\delta_m = 37.2°$$

Answer: δₘ = 37.2°

Problem 1.2: A thin prism of angle 5° and refractive index 1.5 is placed in front of a lens. Find the deviation produced.

Solution

For thin prism:

$$\delta = (n-1)A$$ $$\delta = (1.5 - 1) \times 5° = 0.5 \times 5° = 2.5°$$

Answer: δ = 2.5°

Note: Independent of angle of incidence for thin prisms!

Problem 1.3: For a prism, n_V = 1.52, n_R = 1.48, n_Y = 1.50. Find dispersive power.

Solution
$$\omega = \frac{n_V - n_R}{n_Y - 1} = \frac{1.52 - 1.48}{1.50 - 1} = \frac{0.04}{0.50} = 0.08$$

Answer: ω = 0.08

Level 2: JEE Main/Advanced

Problem 2.1: A ray of light passes through a prism of angle 60° in minimum deviation position. If refractive index is √2, find: (a) Angle of incidence (b) Angle of minimum deviation (c) Angle of refraction inside prism

Solution

Given: A = 60°, n = √2

(a) Angle of incidence:

$$n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$ $$\sqrt{2} = \frac{\sin\left(\frac{60° + \delta_m}{2}\right)}{\sin 30°}$$ $$\sqrt{2} = \frac{\sin\left(\frac{60° + \delta_m}{2}\right)}{0.5}$$ $$\sin\left(\frac{60° + \delta_m}{2}\right) = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$$ $$\frac{60° + \delta_m}{2} = 45°$$ $$60° + \delta_m = 90°$$ $$\delta_m = 30°$$

(b) Minimum deviation: δₘ = 30°

For angle of incidence at minimum deviation:

$$i = \frac{A + \delta_m}{2} = \frac{60° + 30°}{2} = 45°$$

(c) Angle of refraction: At minimum deviation:

$$r = \frac{A}{2} = \frac{60°}{2} = 30°$$

Answers:

  • (a) i = 45°
  • (b) δₘ = 30°
  • (c) r = 30°

Problem 2.2: Two prisms of angles 6° each and refractive indices 1.5 and 1.6 are combined to produce dispersion without deviation. Are they in correct position or inverted with respect to each other?

Solution

For dispersion without deviation:

$$(n_1 - 1)A_1 + (n_2 - 1)A_2 = 0$$

Let’s check if they can be in same orientation:

$$(1.5 - 1) \times 6° + (1.6 - 1) \times 6° = 0.5 \times 6° + 0.6 \times 6°$$ $$= 3° + 3.6° = 6.6° \neq 0$$

So they must be inverted (opposite orientations).

Let A₂ be negative (inverted):

$$(1.5 - 1) \times 6° + (1.6 - 1) \times (-6°) = 3° - 3.6° = -0.6° \neq 0$$

This doesn’t work either with equal angles.

Actually, for dispersion without deviation, we need:

$$\frac{(n_1 - 1)A_1}{(n_2 - 1)A_2} = -1$$

This means one must be inverted and:

$$\frac{0.5 \times 6}{0.6 \times A_2} = -1$$ $$A_2 = -5°$$

Answer: Prisms must be inverted, but with different effective angles OR the question implies they can’t produce exactly zero deviation with equal angles.

Correct interpretation: With equal angles, they’re inverted and produce net deviation of 0.6°.

Level 3: JEE Advanced

Problem 3.1: A prism of refracting angle 4° is made of material with n_V = 1.524 and n_R = 1.516 for violet and red light. Find: (a) Angular dispersion (b) Dispersive power (assume n_Y = 1.520)

Solution

Given: A = 4°, n_V = 1.524, n_R = 1.516, n_Y = 1.520

(a) Angular dispersion:

For thin prism:

$$\theta = (n_V - n_R)A$$ $$\theta = (1.524 - 1.516) \times 4°$$ $$\theta = 0.008 \times 4° = 0.032° = 1.92'$$

(arc minutes)

(b) Dispersive power:

$$\omega = \frac{n_V - n_R}{n_Y - 1} = \frac{1.524 - 1.516}{1.520 - 1} = \frac{0.008}{0.520} = 0.0154$$

Answers:

  • (a) θ = 0.032° or 1.92 arc minutes
  • (b) ω = 0.0154

Quick Revision Cards

╔══════════════════════════════════════╗
║  PRISM QUICK FACTS                  ║
╚══════════════════════════════════════╝

🎯 δ = i + e - A (deviation)
🎯 A = r₁ + r₂ (angle relation)
🎯 At min: i = e, r₁ = r₂ = A/2
🎯 n = sin[(A+δₘ)/2] / sin(A/2)
🎯 Thin: δ = (n-1)A
🎯 Dispersion: θ = δ_V - δ_R
🎯 ω = (n_V - n_R)/(n_Y - 1)
🎯 VIBGYOR: V bends most!

Cross-Topic Connections

Final Tips

  1. Draw the prism with ray path - helps visualize angles
  2. Check TIR condition at second surface if angle seems large
  3. For minimum deviation: Remember i = e shortcut
  4. Thin prism: Only when A < 10°
  5. Color order: VIBGYOR (V highest n, R lowest n)

Pro Tip: In exam, if they give n for yellow/mean color, use that as n_Y for dispersive power!

Next up: Optical Instruments - using prisms and lenses together!