How a Rainbow Forms

How a Rainbow Forms

A rainbow is created when sunlight enters spherical raindrops and undergoes a sequence of optical phenomena: refraction (bending at the air-water interface), dispersion (wavelength-dependent bending that separates colors), internal reflection at the back of the drop, and a second refraction upon exiting. Each color emerges at a slightly different angle, producing the characteristic arc.

The Five Steps

• 1. Incident Sunlight: White light from the sun (solar irradiance S₀ = 1361 W/m²) travels toward raindrops suspended in the atmosphere.
• 2. First Refraction & Dispersion: Light enters the denser water (n ≈ 1.33). Different wavelengths refract at different angles — violet bends more than red because its refractive index is higher.
• 3. Internal Reflection: Light strikes the inner surface at the back of the raindrop and reflects. The angle of incidence equals the angle of reflection for all colors.
• 4. Second Refraction & Angular Separation: Light exits the raindrop, refracting again. The cumulative effect separates colors by about 2°: red at 42° from the anti-solar direction, violet at 40°.
• 5. Observer's Perspective: Each raindrop contributes one color to the observer's eye based on its position. The collective effect of millions of drops forms the rainbow arc.

Key Quantities

• n_red = 1.330 — refractive index of water for red light (λ ≈ 700 nm).
• n_violet = 1.342 — refractive index of water for violet light (λ ≈ 400 nm).
• θ_red = 42° — angle of red light from the anti-solar point (rainbow angle).
• θ_violet = 40° — angle of violet light from the anti-solar point.
• S₀ = 1361 W/m² — solar constant, the intensity of sunlight at Earth's distance.

Sources

D. R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 84th Edition. CRC Press, 2003-2004.
J. D. Walker, "Multiple rainbows from single drops of water," American Journal of Physics, vol. 44, no. 5, pp. 421-433, 1976.
Kopp, G., & Lean, J. L. (2011). A new, lower value of total solar irradiance. Geophysical Research Letters, 38(1), L01706.
Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.). Cengage Learning.
Hecht, E. (2017). Optics (5th ed.). Pearson.