Rainbow prism lab

Definitions

ROYGBIV

Mnemonic for the visible spectrum: Red (640–750 nm), Orange (590–640 nm), Yellow (570–590 nm), Green (495–570 nm), Blue (450–495 nm), Indigo (420–450 nm), Violet (380–420 nm). Newton's choice to include indigo as a separate band was motivated by numerology (seven colors matching musical octaves); modern spectroscopy shows it blends continuously with blue and violet.

Refraction

Bending of light when it crosses a boundary between media of different refractive indices.

Refractive index n

Ratio of speed of light in vacuum (c) to speed in medium (v): n = c/v. Dimensionless. Glass ≈ 1.5, water ≈ 1.33.

Snell's law

n₁ sin(θ₁) = n₂ sin(θ₂), where θ is angle to the normal. Holds at any interface between isotropic media.

Dispersion

Wavelength dependence of refractive index. Shorter wavelengths (blue, violet) refract more than longer ones (red, orange).

Cauchy equation

Empirical formula for refractive index: n(λ) = A + B/λ² + C/λ⁴, where λ is in micrometers. Coefficients depend on material; for borosilicate crown glass: A ≈ 1.45, B ≈ 0.003, C ≈ 0.00001 (Hecht 2017).

Critical angle θ_c

Angle at which total internal reflection begins: sin(θ_c) = n₂/n₁. Light is trapped inside the denser medium.

Rainbow angle

The rainbow appears at ~42° from the antisolar point (for primary rainbow, one internal reflection in water droplets). This is not the deviation angle of a prism, but a property of spherical geometry and Descartes' minimum-deviation ray for n ≈ 1.33 (water).

Physics

When white light enters a prism, each wavelength refracts by a slightly different amount. This is dispersion: the refractive index of glass depends on wavelength. The Cauchy equation gives a good fit:

n(λ) = 1.45 + 0.003/λ² + 10⁻⁵/λ⁴

where λ is in micrometers. As wavelength decreases (toward violet, ~380 nm), n increases slightly. As it increases (toward red, ~750 nm), n decreases. The difference is tiny—about 1% for glass—but enough to separate white light into a spectrum.

Snell's law governs each wavelength:

n(air) sin(θ₁) = n(λ) sin(θ₂)

When light hits the first prism surface (air to glass), it bends toward the normal. At the second surface (glass to air), it bends away. The total deviation (deflection) is maximized around 42° for a symmetric ray path—this is why rainbows appear at a fixed angle above the antisolar point.

A prism's deviation angle depends on its shape, incident angle, and refractive index. In a water droplet (spherical, not a prism), light refracts in, reflects internally once, and refracts out. This geometry produces a minimum-deviation angle of ~42° for the primary rainbow and ~51° for the secondary rainbow (two internal reflections, colors reversed). The 42° angle is not universal—it depends on the droplet's refractive index (n ≈ 1.33 for water).

Sources

• Born, M., & Wolf, E. (1999). Principles of Optics, 7th ed. Cambridge University Press. (Classical treatment of wave optics, diffraction, polarization.)
• Hecht, E. (2017). Optics, 5th ed. Pearson. (Cauchy equation coefficients for borosilicate crown glass; comprehensive treatment of refraction, dispersion, rainbows, prisms.)
• Descartes, R. (1637). La Dioptrique. (Original explanation of rainbow via refraction and internal reflection in spherical water drops.)
• Airy, G. B. (1838). On the intensity of light in the neighbourhood of a caustic. Transactions of the Cambridge Philosophical Society, 6, 379–402. (Quantum correction to Descartes' rainbow angle; Airy integral.)