Dispersion and Rainbow Formation

Why This Topic Matters for CDS

Optics is one of the most dependable scoring areas in CDS / OTA General Science. Questions on dispersion, prisms and rainbows appear almost every alternate year, usually as single-fact MCQs that reward a clear conceptual picture rather than heavy calculation.

The examiner loves to test three things: the order of colours, the reason violet bends most, and the geometry of a rainbow (which colour is on top, the role of the Sun's position, primary vs secondary). Once you understand the single underlying idea — that refractive index depends on colour — every question in this chapter becomes easy.

This chapter also connects neatly to refraction, lenses and total internal reflection, so the effort you put in here pays off across the whole optics unit. A confident grasp of dispersion lets you reason about everyday observations the examiner may dress up as questions: the colours on a soap bubble, the sparkle of a diamond, the way a glass of water throws a tiny spectrum on a wall, and of course the rainbow. Treat the topic as a set of linked ideas rather than isolated facts and you will rarely lose a mark on it.

White Light and the Seven Colours

Sunlight appears white but is actually a mixture of seven colours. When separated, these colours appear in a fixed sequence remembered by the word VIBGYOR:

• V — Violet
• I — Indigo
• B — Blue
• G — Green
• Y — Yellow
• O — Orange
• R — Red

This band of colours obtained when white light is split is called the spectrum. Violet has the shortest wavelength and red the longest. Because colour is fundamentally a matter of wavelength, each colour behaves slightly differently when it enters glass or water.

What Exactly is Dispersion?

Dispersion is the splitting of white light into its constituent colours when it passes from one medium into another. It happens because the refractive index of a medium is not the same for all colours — it is slightly larger for violet and smaller for red.

Refraction is the bending of light as it changes speed on entering a new medium. Since each colour travels at a slightly different speed inside glass, each one bends by a slightly different angle. They enter together but leave fanned out.

Inside the medium, speed depends on colour too: v_red > v_violet. Red travels fastest in glass and is therefore bent the least. One vital fact: the frequency of light never changes on refraction — only its speed and wavelength change. That is why a colour stays the same colour as it crosses a boundary.

It is worth pausing on the cause of dispersion. When light enters glass or water it is slowed because the electrons in the medium respond to the wave, and this response is stronger for higher-frequency (shorter-wavelength) light. Violet, being higher in frequency, interacts more strongly and is slowed more, giving glass a larger refractive index for violet than for red. The variation of refractive index with wavelength is the heart of dispersion; without it, white light would pass straight through and emerge unchanged, and there would be no spectrum and no rainbow at all.

Dispersion Through a Prism

A glass prism is the classic tool to demonstrate dispersion, first shown clearly by Isaac Newton. A prism has two refracting surfaces inclined at an angle (the angle of the prism, often 60°). Because the two surfaces are not parallel, the emerging rays do not become parallel again — instead the colours separate and stay separated.

When a narrow beam of white light strikes one face of the prism:

1. The light refracts on entering the denser glass and bends towards the base.
2. Each colour bends by a different amount — violet most, red least.
3. On leaving the second face the light bends again, increasing the separation.
4. A band of seven colours, the spectrum, appears on a screen with red at the top and violet at the bottom of the deviated beam.

The angle by which a ray is turned from its original path is the angle of deviation. Violet, bending most, has the largest deviation; red the smallest. The difference between them is the angular dispersion.

Angle of Deviation: The Working Relation

For a thin prism the deviation produced for any colour is given by a neat relation. If A is the prism angle and µ the refractive index for that colour, the deviation δ is:

This shows clearly why a larger refractive index (violet) gives a larger deviation, and why a prism with a larger angle A spreads the colours further apart. For CDS you rarely need to compute a numerical answer, but understanding this relation lets you predict how changing the prism or the medium changes the spectrum.

Worked Example: Deviation of Two Colours

Let us see how the formula plays out with simple numbers.

So violet is deviated more than red (3.192° vs 3.084°), exactly as expected, and the colours are spread over a small angle of about 0.108°. The wider this gap, the broader and clearer the visible spectrum.

How a Rainbow Forms

A rainbow is nature's giant dispersion experiment. It appears when sunlight falls on millions of tiny water droplets suspended in the air after rain. Each droplet acts like a miniature prism plus mirror.

Three optical events happen inside a single droplet:

1. Refraction and dispersion — sunlight enters the droplet and splits into its colours as it bends.
2. Internal reflection — the light strikes the back of the droplet and is reflected by total internal reflection.
3. Refraction again — on leaving the droplet the colours bend once more and emerge further separated.

Each droplet sends out only one colour towards your eye depending on its position, so the overall arch is built from countless droplets working together.

The shape is an arc of a circle because only droplets lying at a fixed angle from the line joining the Sun, your eye and your shadow can send light back to you. That angle is roughly 42° for the primary bow. Droplets higher in the sky deliver red to your eye, while droplets a little lower deliver violet, which is why the colours appear in a smooth band from top to bottom. From an aeroplane, where droplets exist all around, a rainbow can even appear as a complete circle rather than an arch.

Primary and Secondary Rainbows

You can sometimes see two rainbows at once. They differ in brightness, colour order and the number of internal reflections.

Primary rainbow

Formed by one internal reflection inside each droplet. It is the brighter of the two. Here red is on the outer (upper) edge and violet on the inner (lower) edge. The red light emerges at about 42° and violet at about 40° from the antisolar point.

Secondary rainbow

Formed by two internal reflections inside each droplet. It is fainter because some light is lost at each reflection, and its colours are reversed — violet on the outer edge and red on the inner edge. It appears at a larger angle (around 50–53°).

Common Mistakes to Avoid

Previous-Year Style Question

A Related Idea: Why the Sky is Blue

Examiners sometimes pair dispersion with scattering, so know the difference. Dispersion is about refraction splitting colours by bending. Scattering is the redirection of light by tiny particles in the atmosphere.

Shorter wavelengths (blue, violet) are scattered far more than longer ones (red). This Rayleigh scattering is why the daytime sky looks blue and why the setting Sun looks red — at sunset light travels through more atmosphere, blue is scattered away, and mostly red reaches the eye.

A handy way to keep the two ideas apart: dispersion depends on the angle of bending being different for each colour, so it needs a refracting surface such as a prism or a raindrop. Scattering depends on the size of the particles and the wavelength of light, and it happens throughout the open atmosphere without any single refracting surface. Both phenomena involve colour, but the mechanism is completely different, and CDS examiners often place them side by side in the same question to test whether you can tell them apart.

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