Dispersion, rainbows and scattering

Dispersion by a prism

Narrated video. Transcript:

This topic is beyond the current NCERT syllabus, but it explains some beautiful sights and is useful for competitive exams. Let us start with a beam of sunlight and a glass prism.

A narrow beam of white light enters the prism. What comes out is a fan of colours: violet, indigo, blue, green, yellow, orange and red. VIBGYOR. This splitting is called dispersion.

Red is bent the least, and violet is bent the most. So the refractive index of glass is not one number: it is larger for violet light than for red light.

For crown glass, n is about 1.514 for red light and about 1.532 for violet light. Shorter wavelengths travel more slowly in glass, so they bend more.

For a thin prism, the deviation is δ = (n − 1)A. With A = 6°, red is deviated by 0.514 × 6° = 3.08°, and violet by 0.532 × 6° = 3.19°.

The difference, δ_v − δ_r = (n_v − n_r)A = 0.018 × 6° ≈ 0.11°, is the angular dispersion: the angle over which the colours spread.

Are the colours made by the prism? Newton tested this. He placed a second, identical prism upside down after the first. The colours recombined into white light.

So white light is itself a mixture of colours. The prism only separates them, because each colour has a different refractive index.

In a vacuum, all colours travel at the same speed c. Dispersion happens only in a medium, where the speed of light depends on its wavelength.

Rainbows, blue sky and red sunsets

Narrated video. Transcript:

After a monsoon shower, with the sun behind you, you may see a rainbow. Each raindrop acts like a tiny prism and mirror together.

Sunlight enters a drop and refracts, splitting into colours. It reflects once off the back of the drop, by internal reflection, and refracts again as it leaves.

The light comes out at a definite angle to the incoming sunlight: about 40° for violet and 42° for red. Drops at 42° from the point opposite the sun send you red; drops at 40° send you violet.

So the primary rainbow is an arc with red on the outside and violet on the inside.

Light that reflects twice inside the drop forms a fainter secondary rainbow, higher up, at about 50° to 53°. Its colours are reversed: red inside, violet outside.

Now, why is the sky blue? Sunlight hits air molecules, which are much smaller than the wavelength of light. They scatter the light in all directions.

This scattering is much stronger for short wavelengths. Its intensity goes as 1/λ⁴. Light at the blue–violet end, 400 nanometres, is scattered (700/400)⁴, about 9 times more than red light at 700 nanometres.

So scattered blue light reaches your eyes from every part of the sky. Violet is scattered even more, but sunlight has less violet and our eyes are less sensitive to it.

At sunrise and sunset, sunlight travels a long path through the atmosphere. Most of the blue is scattered away, and the red and orange that remain colour the sun and the clouds.

Clouds are white because their water droplets are much larger than the wavelength. Large particles scatter all colours about equally, so the scattered light stays white.

Try it: split white light

Interactive simulation. Things to try:

• White light enters from the left and leaves as a fan of colours. Which colour is bent most?
• Raise the spread n_violet − n_red: the fan widens. Set it to 0: no dispersion at all.
• On the graph, the violet curve lies above the red one: violet is always deviated more.
• Make i small until violet is totally reflected at the second face while red still gets out.

Which colour bends most?

White light passes through a glass prism. Which colour is deviated the most, and why?

1. Red, because it has the longest wavelength
2. Violet, because glass has the largest refractive index for violet light
3. All colours equally; the prism only colours the light
4. Green, because it is in the middle of the spectrum

Blue sky

Scattering by air molecules varies as 1/λ⁴. How many times more strongly is light of wavelength 400 nm scattered than light of 800 nm?

1. 2 times
2. 4 times
3. 16 times
4. 8 times

Summary

• Thin prism: δ = (n − 1)A
• Angular dispersion: δ_v − δ_r = (n_v − n_r)A
• Scattering by small molecules: intensity ∝ 1/λ⁴

• Dispersion: n depends on wavelength (larger for violet), so a prism spreads white light into VIBGYOR.
• A rainbow comes from refraction, dispersion and internal reflection in raindrops: primary at about 40°–42° (red outside), secondary at about 50°–53° (colours reversed).
• Air molecules scatter blue light much more than red: the sky is blue, and the setting sun is red.
• Large droplets scatter all colours equally, so clouds look white.