# Dispersion of Light

Splitting of light into constituent colors is called dispersion. Dispersion occurs due to wavelength dependence of the refractive index. The refractive index of materials varies according to Cauchy's formula \begin{align} \mu=\mu_0+\frac{A}{\lambda^2}, \end{align} where $\mu_0$ and $A > 0$ are constants.

Let a light beam is incident on a prism of small angle $A$. If the angle of incidence is small then the mean deviation $\delta_y$ (yellow color) and angular dispersion $\theta$ are given by \begin{align} \delta_y &= (\mu_y-1)A, \\ \theta & =(\mu_v-\mu_r)A, \end{align} where $\mu_y$ and $\mu_v$ are refractive indices for the yellow and violet colors, respectively.

The dispersive power of a prism is defined as \begin{align} \omega &=\frac{\mu_v-\mu_r}{\mu_y-1} \\ &\approx \frac{\theta}{\delta_y}\quad\text{(if $A$ and $i$ small)} \end{align}

In dispersion without deviation, a light beam is dispersed without deviating from its path. This is achieved by using two prisms ($A,\mu$) and ($A^\prime,\mu^\prime$) such that \begin{align} (\mu_y-1)A+(\mu_y^\prime-1)A^\prime=0. \end{align}

The condition for deviation without dispersion is \begin{align} (\mu_v-\mu_r)A=(\mu_v^\prime-\mu_r^\prime)A^\prime \end{align}

## Demo of dispersion by a prism

Refractive index of a material varies with wavelength. The relation is approximately given as $$\mu=\mu_0+\frac{A}{\lambda^2}$$.

Make a narrow slit on a stiff piece of paper and make it stand vertically. Allow sunlight or torchlight to fall on the slit, to create a narrow beam of light. Let this beam fall on a rectangular face of a prism placed near a wall. Light will pass through the prism and fall on the wall. Rotate the prism till you see a band of colours on the wall.

Dispersion of light using a Water and Plane Mirror: Get a plane mirror of suitable size. Place it in a bowl (Katori) as shown in figure. Fill the bowl with water. Put this arrangement near a window such that direct sunlight falls on the mirror. Adjust the position of the mirror such that the reflected light from the mirror falls on a wall. If the wall is not white, fix a sheet of white paper on it. Reflected light will be seen to have many colours. How can you explain this? The mirror and water form a prism. This split the light into multiple colours. Splitting of light into its colours is known as dispersion of light. Rainbow is a natural phenomenon showing dispersion.

## Problems from IIT JEE

Problem (IIT JEE 2008): Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is 60 degree). In the position of minimum deviation, the angle of refraction will be,

1. 30 degree for both the colours.
2. greater for the violet colour.
3. greater for the red colour.
4. equal but not 30 degree for both the colours.

Solution: At the angle of minimum deviation ($\delta_m$), angle of incidence is equal to the angle of emergence, the angle of refraction ($r$) is equal to half of the prism angle ($A$) and ray inside the prism is parallel to the prism base. Further, refractive index is given by, \begin{align} \mu=\frac{\sin\frac{A+\delta_m}{2}}{\sin\frac{A}{2}}, \end{align} and the angle of incidence by, \begin{align} i=\frac{A+\delta_m}{2}. \end{align} From above equations, $\delta_m$ and $i$ depend on $\mu$ (colours). However, for the given prism, $r=A/2={30}$ deg is independent of $\mu$. Readers are encouraged to find $\delta_m$ and $i$ for red and violet colours if $\mu_\text{red}=1.514$ and $\mu_\text{violet}=1.523$.