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}

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.

**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,

- 30 degree for both the colours.
- greater for the violet colour.
- greater for the red colour.
- 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 $\mu_\text{violet}=1.523$.