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Ray Optics & Rainbows

Dispersion of Light

Observe how white light separates into its constituent spectral colors when passing through refractive media. Toggle between the glass prism bench, the room crystal window, and raindrop physics to explore the ROYGBIV spectrum.

Spectral Dispersion Laboratory ⓘ

Observe Cauchy's wavelength separation. Adjust materials and angles to change the rainbow caustics.

Dispersing

Spectral Telemetry

θ = (nv - nr)A ⇒ θ = (1.531 - 1.514) × 60° = 1.02°
Prism Glass Type
Crown Glass
Mean Index (ny)
1.520
Angular Dispersion (θ)
1.02°
Dispersive Power (ω)
0.033
Emergence Wavelengths
400nm - 700nm

Dispersion of White Light

When a beam of composite white light (such as sunlight) is passed through a glass prism, it splits into a beautiful band of colors. This phenomenon is known as the dispersion of light, and the resulting band of seven distinct colors is called the visible spectrum.

The colors appear in the order of Red, Orange, Yellow, Green, Blue, Indigo, and Violet (ROYGBIV). Sir Isaac Newton first demonstrated this splitting in 1666, and proved that prisms do not create the colors themselves but merely separate them.

What Causes Dispersion?

The speed of light in a vacuum is constant for all colors. However, when light enters a transparent medium like glass or water:

1. Wavelength-Dependent Speed

Different colors of light have different wavelengths. Inside glass, longer wavelengths (Red) travel faster than shorter wavelengths (Violet).

2. Variation in Refractive Index

Since refractive index is given by n = c/v, a higher speed means a lower index. Glass has a lower refractive index for red light and a higher refractive index for violet light (nviolet > nred).

Cauchy's Equation & Bending

The relationship between the refractive index and wavelength is described by Cauchy's dispersion formula:

Cauchy's Relation

n(λ) = B + C / λ2

Because of this inverse-square relationship, shorter wavelengths (λviolet ≈ 400nm) suffer greater deviation than longer wavelengths (λred ≈ 700nm).

Atmospheric Rainbows

Rainbows are a spectacular natural demonstration of light dispersion, produced by sunlight interacting with suspended raindrops:

  • Refraction & Dispersion: Sunlight enters the raindrop, bending and splitting into colors.
  • Internal Reflection: The light hits the back inner surface of the droplet and reflects.
  • Final Refraction: The rays emerge from the droplet, deviating red light at 42° and violet light at 40° relative to the sun-observer axis.

Step-by-Step Solved Problems

Practice math equations relating to dispersive power, Cauchy relations, and deviation angles.

Example 1 Problem Statement

A thin glass prism has a refracting angle of 4°. The refractive indices of the glass for red and violet light are 1.514 and 1.530 respectively. Calculate the angular dispersion produced by the prism.

View Step-by-Step Dispersion Solution
  1. Identify the given values: Apex/Refracting angle A = 4° (a thin prism), Refractive index for red light nr = 1.514, Refractive index for violet light nv = 1.530.
  2. Recall the formula for angular dispersion θ produced by a thin prism: θ = (nv - nr)A.
  3. Substitute the values into the formula: θ = (1.530 - 1.514) × 4°.
  4. Calculate the difference: 1.530 - 1.514 = 0.016.
  5. Multiply by the refracting angle: θ = 0.016 × 4° = 0.064°.

Final Derived Answer: Angular Dispersion θ = 0.064°.

Example 2 Problem Statement

For a flint glass prism, the refractive indices for red, yellow, and violet light are 1.622, 1.627, and 1.638 respectively. Find the dispersive power of the flint glass.

View Step-by-Step Dispersion Solution
  1. Identify the given refractive indices: Red nr = 1.622, Yellow (mean index) ny = 1.627, Violet nv = 1.638.
  2. Recall the formula for dispersive power ω: ω = (nv - nr) / (ny - 1).
  3. Calculate the numerator (angular dispersion factor): nv - nr = 1.638 - 1.622 = 0.016.
  4. Calculate the denominator (mean deviation factor): ny - 1 = 1.627 - 1 = 0.627.
  5. Divide the two factors: ω = 0.016 / 0.627 ≈ 0.0255.

Final Derived Answer: Dispersive Power ω ≈ 0.0255 (or 2.55%).

Example 3 Problem Statement

A ray of sunlight enters a spherical raindrop at an incidence angle of 60°. If the refractive index of water for red light is 1.331, calculate the angle of refraction inside the raindrop.

View Step-by-Step Dispersion Solution
  1. Identify the given values: Angle of incidence i = 60°, Refractive index of water for red light n = 1.331.
  2. Recall Snell's Law: sin(i) = n × sin(r).
  3. Rearrange the equation to solve for sin(r): sin(r) = sin(i) / n.
  4. Substitute the values: sin(r) = sin(60°) / 1.331.
  5. Evaluate sin(60°): sin(60°) ≈ 0.866.
  6. Calculate the sine of refraction: sin(r) = 0.866 / 1.331 ≈ 0.6506.
  7. Take the inverse sine to find r: r = arcsin(0.6506) ≈ 40.6°.

Final Derived Answer: Angle of Refraction inside the raindrop r ≈ 40.6°.

Self-Check Questions

Question 1

Explain why dispersion occurs when white light passes through a prism but not through a flat glass slab.

Show Answer & Explanation

A flat glass slab has parallel faces. The color splitting (dispersion) that occurs at the first face is exactly reversed (recombined) at the second parallel face because the rays emerge parallel to each other and recombine. In a prism, the faces are non-parallel (intersecting at apex angle A), so the deviations at both faces add up, leaving the colors separated as they emerge.

Question 2

Which color of the visible spectrum travels fastest inside a glass prism? Explain why.

Show Answer & Explanation

Red light travels the fastest inside a glass prism. The refractive index (n) of glass is inversely related to speed (n = c/v). Since glass has a lower refractive index for longer wavelengths (n_red < n_violet), red light experiences less resistance and travels faster than violet light.

Question 3

State Cauchy's empirical formula and describe what its constants represent.

Show Answer & Explanation

Cauchy's empirical formula is n(λ) = B + C/λ², which relates the refractive index (n) of a transparent medium to the wavelength (λ) of incident light. The constants B and C are parameters unique to the material, where B represents the base index of refraction for very long wavelengths, and C describes the material's dispersive strength.

Question 4

What is the difference between angular dispersion and dispersive power?

Show Answer & Explanation

Angular dispersion (θ) is the angular separation between the extreme colors (violet and red) emerging from a prism, measured in degrees (θ = δ_v - δ_r). Dispersive power (ω) is a dimensionless ratio that measures the material's ability to disperse light relative to its average bending ability: ω = (n_v - n_r) / (n_y - 1).

Question 5

Briefly describe the three steps that occur inside a water droplet to form a primary atmospheric rainbow.

Show Answer & Explanation

The three steps are: (1) Refraction and Dispersion: Sunlight enters the droplet and splits into colors. (2) Total Internal Reflection: Light travels to the back inner surface of the droplet and reflects. (3) Refraction and Further Dispersion: Light exits the droplet, bending further into separate colors towards the observer at angles of 40° (violet) to 42° (red).

Question 6

What does a high dispersive power indicate about a transparent material?

Show Answer & Explanation

A high dispersive power indicates that the material is highly effective at splitting composite light into its constituent spectrum colors. For example, flint glass has a higher dispersive power than crown glass, making it better suited for spectroscopes but prone to chromatic aberration in lenses.