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Refracta

Fermat’s Principle / Snell’s Law Physics

\[\frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2}\]
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Why does light bend when it enters water? Refracta is a lifeguard who runs faster on sand than she swims, so her quickest route to a swimmer is not a straight line: it bends at the waterline. Fermat proposed that light follows the same least-time rule, which gives Snell’s law of refraction. This is a 3" pixel-art sticker from Pixelated Physics.

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3″ × 3″ kiss-cut sticker.

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The physics

Light takes the path between two points that takes the least time. That single rule gives Snell's law of refraction: light bends toward the slower medium, so that the sine of the angle divided by the speed is the same on both sides.

Refracta runs twice as fast on sand as she swims. The swimmer is out to sea and to one side.

The straight line is the shortest route but not the fastest, because too much of it is in the water. The fastest route runs further along the sand and bends at the waterline so that sin θ₁/v₁ = sin θ₂/v₂.

In the art she runs at 50° from the normal and swims at 22.5°, since sin 22.5° ≈ sin 50°/2. A brute-force search over every crossing point gives the same crossing as the one drawn.

Light does the same. Pierre de Fermat argued in 1662 that light takes the path of least time; with n = c/v the rule becomes Snell’s law, n₁ sin θ₁ = n₂ sin θ₂.

Run more. Swim less.

Equations

\[\frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2}\]
\[n_1\sin\theta_1 = n_2\sin\theta_2,\qquad n = c/v\]
\[\delta\!\int n\,ds = 0\]
SymbolMeaningUnit
\(\theta_1\) angle of the path from the normal on the sand (light: in the first medium) \(\mathrm{rad}\)
\(\theta_2\) angle of the path from the normal in the water (light: in the second medium) \(\mathrm{rad}\)
\(v_1, v_2\) speeds on each side \(\mathrm{m/s}\)
\(n_1, n_2\) refractive indices \(\mathrm{1}\)
\(c\) speed of light in vacuum \(\mathrm{m/s}\)
\(ds\) element of path length \(\mathrm{m}\)

Sources

  1. Feynman Lectures on Physics Vol. I Ch. 26: Optics: The Principle of Least Time
  2. Hecht, Optics, 5th ed., Ch. 4: The Propagation of Light

Source Fermat, P. de (1662). Letter to Marin Cureau de la Chambre, 1 January 1662 (No. CXII), on refraction by the principle of least time. In Œuvres de Fermat, Vol. II (P. Tannery & C. Henry, eds.), Paris: Gauthier-Villars, 1894, p. 457.

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