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## Equation: The Law of Rainbows

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• 12:30 pm  |
• Wired September 2011

Illustration: Mark Weaver

When sunlight hits a raindrop, some photons glance off the surface. But others penetrate, bounce off the opposite side of the droplet, and shoot back out the front. That internally reflected light is what makes rainbows, and their distribution of colors is determined by Snell’s law, which describes how light refracts as its speed changes. It works like this: Light moves slower in water than in air, and the sudden deceleration alters its direction of travel. Short wavelengths like violet bend more than long ones like red, causing that beam of white sunlight to fan out into a spectrum of colors. The result: Each color hits the back of the raindrop at a different spot and bounces out at a slightly different angle. Now picture a sky full of raindrops. Each one reflects the full spectrum, but because of the varying angles, we see different colors from drops at different heights. Imagine a line extending from your head to its shadow on the ground. The red light that you see is from drops that are 42.4 degrees above that line. The violet band, at the bottom of the rainbow, is from drops that are a little lower—40.7 degrees.

A measure of the angle of incidence
(1)—the angle at which a ray of sunlight strikes a raindrop.

A measure of the angle of refraction
(2)—the ray’s angle of travel after entering the drop. The difference in the two angles is the amount the light bends.

The velocity of light through air. This is very close to c, the speed of light in a vacuum, and is almost identical for all colors.

The velocity of light through water—about 25 percent slower than v1. In water, violet light travels about 1 percent slower than red light.

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