Dispersion of light — prism, rainbow formation, chromatic aberration

Question

What causes dispersion of light through a prism, how does a rainbow form, and what is chromatic aberration in lenses?

Solution — Step by Step

Why dispersion happens

Dispersion is the splitting of white light into its constituent colours because the refractive index of a medium depends on the wavelength of light.

$n = n(\lambda)$

Violet light ( $\lambda \approx 400$ nm) has a higher refractive index than red light ( $\lambda \approx 700$ nm) in glass. So violet bends more than red.

This wavelength-dependence of $n$ is called Cauchy's relation:

$n = A + \frac{B}{\lambda^2}$

where $A$ and $B$ are material constants.

Dispersion through a prism

When white light enters a prism:

All colours refract at the first surface (violet bends most)
They travel through the prism at slightly different angles
At the second surface, they refract again, increasing the angular separation

The angle between the red and violet emergent rays is the angular dispersion:

$\delta_v - \delta_r = (n_v - n_r) \times A$

where $A$ is the angle of the prism (for small angles).

The dispersive power of the material:

$\omega = \frac{n_v - n_r}{n_y - 1}$

where $n_y$ is the refractive index for yellow (mean) light.

Rainbow formation

A rainbow is nature's dispersion experiment, using water droplets as prisms.

Primary rainbow:

Light enters a raindrop and refracts (disperses slightly)
It reflects off the back of the drop (TIR-like reflection)
It refracts again while exiting
Red appears on top (42 degrees from antisolar point), violet at bottom (40 degrees)

Secondary rainbow:

Two internal reflections instead of one
Colours are reversed (violet on top, red at bottom)
Fainter than primary (extra reflection means energy loss)
Appears at about 51-53 degrees

Chromatic aberration in lenses

A single convex lens acts like a stack of tiny prisms. Different colours focus at different points:

Violet focuses closer to the lens (shorter focal length)
Red focuses farther from the lens (longer focal length)

This creates a blurred, colour-fringed image called chromatic aberration.

Fix: Use an achromatic doublet — a convex lens of crown glass combined with a concave lens of flint glass. The dispersions cancel while the net converging power remains.

Condition for achromatism:

$\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0$

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Why This Works

Dispersion occurs because the electromagnetic wave interacts differently with the atoms of the medium depending on its frequency. Higher frequency (violet) light interacts more strongly, experiencing a greater "slowing down" (higher $n$ ) as it passes through the medium. This frequency-dependent refractive index is the root cause of all dispersion phenomena.

Alternative Method

For prism problems, if the prism angle $A$ and minimum deviation $\delta_m$ are given, use:

$n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

This avoids dealing with individual surface refractions. For each colour, use its own $\delta_m$ to find the corresponding $n$ .

Common Mistake

⚠️ Common Mistake

Students often say "red light has a higher refractive index because it has more energy." This is doubly wrong. First, violet has more energy (higher frequency), not red. Second, higher energy (violet) corresponds to higher refractive index, not lower. Red has the lowest refractive index and bends the least. The mnemonic VIBGYOR gives the order from most deviated (V) to least deviated (R). JEE Main and CBSE boards both test this ordering.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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