Illustrated Physics
physics, drawn
Concept

Rainbow — where it appears

An arc of colour seen opposite the Sun, formed by sunlight refracted into water drops and reflected once inside them. Its angle of about 42 degrees from the antisolar point is where the deflection of the rays has a minimum, and the colours separate because that minimum depends slightly on wavelength.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Rays inside a drop never reach the critical angle. Rays of light entering a drop of water (index 1.333) from the left at four heights — 0.3, 0.6, 0.85, 0.97 of the radius — followed through six internal reflections, each chord drawn with a width proportional to the light it carries. Every chord inside a sphere meets the surface at the same angle at both ends, so each ray strikes the inside of the drop again and again at its refraction angle: 13.0°, 26.8°, 39.6°, 46.7°, all below the critical angle of 48.6°. None is ever totally reflected; at each contact most of the light leaves, and what is left inside dwindles — the light that, leaving after one and two reflections, makes the primary and secondary rainbows.

The drop that cannot hold the light it lets in

Light inside water meeting the surface at more than 48.6° is totally reflected, and every raindrop is made of water. Yet no ray of sunlight that enters a raindrop is ever totally reflected inside it. A sphere's geometry forbids it: a ray refracted on the way in meets the inside of the surface at its refraction angle, and keeps meeting it at that same angle every time it bounces, and that angle is always smaller than the critical one. The light leaks out a little at every bounce, which is why there are rainbows at all. Light born inside the drop is another matter, and a sphere can keep it for ever.

optics · Total internal reflection The rings the full solution finds. The light scattered back by water drops of about 10 µm radius, at 550 nm, from the exact Mie series with around 134 terms for each drop, averaged over a ten per cent spread of sizes so that the fine ripple each single drop has is washed out. Instead of the smooth, weak background of the ray picture, the intensity rises sharply toward 180° through a series of rings about one and a half degrees apart, each brighter than the last, to a first ring 1.5° from straight back. These are what an observer sees round the shadow of an aircraft on a cloud, or round a shadow on a mist at a mountain top: the glory. It is present in the same mathematics that gives the rainbow, and it takes the whole series, with its surface-wave terms, to find.

The rings no ray can reach

Look down from an aircraft at a cloud, and round the aircraft's shadow there may be coloured rings, centred exactly on the point opposite the Sun. Mountain climbers have seen them round their own shadows for centuries, and every drop in that cloud is a sphere of water. Yet ray optics says that a drop of water cannot send light straight back. Its best ray falls fourteen degrees short. The glory lives entirely in those fourteen degrees.

optics · Scattering

Named alongside it

The objects these essays reach for when they reach for this one.

BackscatterCloud dropletCritical angleEscape coneFresnel equationsGeometric opticsGloryMie scatteringRefractionSnell's lawSurface waveTotal internal reflection

All concepts