Light: Reflection, Refraction, Snell's Law and Prisms

Straight lines and a speed limit

Light travels in straight lines, which is why objects cast sharp shadows: the light that would pass through them is blocked. In a vacuum its speed is c = 299 792 458 m/s, a value so well fixed that since 1983 the metre has been defined from it. That is roughly 300 000 km every second. Light from the Moon reaches us in 384 400 km / c ≈ 1.28 s, and from the Sun, 149.6 million km away, in about 499 s, which is 8 min 19 s. When you look at the Sun, you see it as it was just over eight minutes ago.

The law of reflection

A mirror sends light back. Angles are measured from the normal, an imaginary line standing at right angles to the surface. The law of reflection says the angle of incidence equals the angle of reflection. Send a ray at 30° to the normal and it leaves at 30° on the other side, so the incoming and outgoing rays are 60° apart. That is why the image in a flat mirror appears as far behind the glass as you stand in front of it. Change the incidence angle in the sandbox and check that the two angles always stay equal.

Light slows down in matter

Each material has a refractive index n = c / v, the ratio of the speed of light in vacuum to its speed in the material. For water n = 1.33, so light moves at 299 792 458 / 1.33 ≈ 225 000 km/s. In glass (n = 1.5) it moves at about 200 000 km/s, and in diamond (n = 2.42) at only about 124 000 km/s. When a ray crosses a boundary at an angle, one side of the beam slows first, and the ray turns toward the normal in the slower material.

Snell's law

Willebrord Snell found the rule in the early 1600s: n₁·sin θ₁ = n₂·sin θ₂, where θ₁ and θ₂ are the angles from the normal in the first and the second material. A ray from air (n₁ = 1) hits water (n₂ = 1.33) at 45°: sin θ₂ = 0.7071 / 1.33 = 0.532, so θ₂ ≈ 32.1°. It bends toward the normal. Your eyes assume light always arrives straight, so a spoon in a glass looks broken, and a pool 2 m deep, seen from straight above, seems only 2 / 1.33 ≈ 1.5 m deep.

Trapped light

Go the other way, from glass into air, and the ray bends away from the normal. Make the angle larger and the refracted ray finally skims the surface. Beyond this critical angle the light cannot leave at all, and the surface acts as a perfect mirror: total internal reflection. The critical angle follows from sin θc = n₂ / n₁. For glass to air, sin θc = 1 / 1.5 = 0.667, so θc ≈ 41.8°. For water it is 48.8°, and for diamond 24.4°, which is partly why diamonds sparkle. An optical fibre is a thin glass thread in which light zigzags by total internal reflection, carrying phone and internet signals as flashes.

Newton's prism

In 1666 Isaac Newton darkened a room, let a thin beam of sunlight in through a hole and passed it through a glass prism. The beam spread into a band of colours, which Newton counted as seven: red, orange, yellow, green, blue, indigo and violet. He also showed that a second prism turns the colours back into white light, so white light is a mixture. Each colour bends by a slightly different amount: red (about 700 nm) bends least and violet (about 400 nm) bends most. The visible range spans roughly 380 to 750 nm. Raindrops make rainbows by the same effect.

Light, reflection and refraction