§ 11. Dispersion of Light & Spectrum Formation in a Prism (Grade 9–11)
Wavelength dependence of refractive index n(λ) and spectral splitting. A narrow beam of white light in a glass prism splits into a rainbow spectrum because short violet waves have a lower phase velocity in glass than long red waves.
Theory Summary & Scientific Mechanism
Optical dispersion is the dependence of a medium’s refractive index n(λ) on wavelength λ. In transparent dielectrics such as crown or flint glass (normal dispersion), n(λ) increases as wavelength decreases according to Cauchy’s equation: n(λ) = A + (B)/(λ²) + (C)/(λ⁴), driven by ultraviolet electronic resonances of the material.
Tracing a ray through a triangular prism of apex angle A requires two Snell’s-law refractions: sin α₁ = nsin γ₁, γ₁ + γ₂ = A, and sin α₂ = nsin γ₂. The total angular deviation from the original beam direction is δ(λ) = α₁ + α₂ - A (or δ ≈ (n(λ) - 1)A for small apex angles).
Because short-wavelength violet light (λ ≈ 400 nm) travels slower in glass and experiences a higher refractive index than red light (nᵥiolet > nᵣed), violet rays deviate most strongly (δᵥiolet > δᵣed), fanning white light out into Newton’s continuous rainbow spectrum.
Everyday Analogy
A rainbow after rain or colorful sparkles from a crystal prism! White sunlight hides all rainbow colors inside. In glass, violet waves slow down more than red ones, so they bend more sharply and fan out into a spectrum.
Interactive Question & Derivation
A white light beam passes through a triangular glass prism. Rays of which color are refracted through the largest deviation angle δ and why?
- A: Red rays (λ ≈ 680 nm), because they have the longest wavelength
- B: Violet rays (λ ≈ 410 nm), because glass has a higher refractive index at shorter wavelengths (nᵥiol > nᵣed) and the lowest wavefront speed
- C: All colors refract at the exact same angle
Study Hint: Which color has the shorter wavelength λ and higher refractive index n(λ) in glass, red or violet, and therefore bends the most?
Proof & Explanation: Under normal dispersion, the refractive index increases monotonically as wavelength decreases according to Cauchy's law n(λ) = n₀ + B/λ². Since λᵥiol nᵣed and violet rays bend most toward the prism base.
Key Equations & 3D Simulation Parameters
λᵥiol nᵣed ⇒ δᵥiol > δᵣed
- Prism apex angle φ: 35 … 62 ° (default: 52 °)
- Incident beam angle α₁: 25 … 65 ° (default: 44 °)
- Base glass index n₀: 1.42 … 1.72 (default: 1.5)
- Cauchy dispersion coeff. B: 0.02 … 0.12 μm² (default: 0.06 μm²)