7.1 Dispersive power

1.1 Refractive index of light
1.2 Types of medium

2.1 Refraction from a spherical surface

3.1 Shift due to a glass slab
3.2 Lateral Magnification

4.1 Sign convention
4.2 Some important terms
4.3 Ray tracing
4.4 Image formed by covex lens
4.5 Image formed by concave lens

5.1 Thin Lens Formula
5.2 Magnification and Power of lens
5.3 Combination of lenses
5.4 Displacement method to find focal length.
5.5 Silvering of lens

6.1 Critical angle
6.2 Application of total internal reflection

7.1 Dispersive power

8.1 Rainbow

9.1 Spectrometer
9.2 Simple microscope
9.3 Compound microscope
9.4 Astronomical telescope (Refracting type)
9.5 Terrestrial telescope
9.6 Galileo's terrestrial telescope
9.7 Reflecting type telescope

The dispersive power of a material is defined as the ratio of angular dispersion to the average deviation when a white beam of light is passed through it.

It is denoted by $\omega $.

Mathematically it is given by,

$$\begin{equation} \begin{aligned} \omega = \frac{{{\text{angular dispersion}}}}{{{\text{average deviation}}}} \\ \omega = \frac{{\left( {{\mu _V} - {\mu _R}} \right)A}}{{\left( {\mu - 1} \right)A}} \\ \omega = \frac{{\left( {{\mu _V} - {\mu _R}} \right)}}{{\left( {{\mu _y} - 1} \right)}} \\\end{aligned} \end{equation} $$

Two cases are possible.

1. Dispersion without average deviation

2. Average deviation without dispersion

Suppose we combine two prisms of refracting angles $A$ & $A'$ and dispersive power $\omega $ and $\omega '$ respectively in such a way that their refracting angle are reversed with respect to each other as shown in the figure.

The deviation produced by the prism is given by,

$$\begin{equation} \begin{aligned} {\delta _1} = + \left( {\mu - 1} \right)A \\ {\delta _2} = - \left( {\mu - 1} \right)A \\\end{aligned} \end{equation} $$

Note: $-ve$ sign shows that the deviation are in opposite direction.

Net deviation is given by,

$$\begin{equation} \begin{aligned} \delta = {\delta _1} + {\delta _2} \\ \delta = \left( {\mu - 1} \right)A - \left( {\mu ' - 1} \right)A'\quad ...(i) \\\end{aligned} \end{equation} $$

Similarly, average deviation produced by combination if white light is passed,

$${\delta _y} = \left( {{\mu _y} - 1} \right)A - \left( {\mu {'_y} - 1} \right)A'\quad ...(ii)$$

$${\delta _y} = 0$$

or

$$\begin{equation} \begin{aligned} \left( {{\mu _y} - 1} \right)A - \left( {\mu {'_y} - 1} \right)A' = 0 \\ \left( {\frac{{{\mu _y} - 1}}{{\mu {'_y} - 1}}} \right) = \frac{A}{{A'}} \\ A' = \left( {\frac{{{\mu _y} - 1}}{{\mu {'_y} - 1}}} \right)A\quad ...(iii) \\\end{aligned} \end{equation} $$

Dispersion is given by,

$$\begin{equation} \begin{aligned} {\delta _V} - {\delta _R} = \left[ {\left( {{\mu _V} - 1} \right)A - \left( {\mu {'_V} - 1} \right)A'} \right] - \left[ {\left( {{\mu _R} - 1} \right)A - \left( {{\mu _R} - 1} \right)A'} \right] \\ {\delta _V} - {\delta _R} = \left( {{\mu _V} - {\mu _R}} \right)A - \left( {{\mu _V} - {\mu _R}} \right)A'\quad ...(iv) \\\end{aligned} \end{equation} $$

From equation $(iii)$ and $(iv)$ we get,

$$\begin{equation} \begin{aligned} {\delta _V} - {\delta _R} = \left( {{\mu _V} - {\mu _R}} \right)A - \left( {{\mu _y} - 1} \right)\left( {\frac{{\mu {'_V} - \mu {'_R}}}{{\mu {'_y} - 1}}} \right)A \\ {\delta _V} - {\delta _R} = \left( {{\mu _y} - 1} \right)A\left[ {\left( {\frac{{{\mu _V} - {\mu _R}}}{{{\mu _y} - 1}}} \right) - \left( {\frac{{\mu {'_V} - \mu {'_R}}}{{\mu {'_y} - 1}}} \right)} \right] \\\end{aligned} \end{equation} $$

$${\delta _V} - {\delta _R} = \left( {{\mu _y} - 1} \right)A\left( {\omega - \omega '} \right)$$

Dispersion is given by,

$${\delta _V} - {\delta _R} = \left( {{\mu _V} - {\mu _R}} \right)A - \left( {\mu {'_V} - \mu {'_R}} \right)A$$

For without dispersion,

$$\begin{equation} \begin{aligned} {\delta _V} - {\delta _R} = 0 \\ \left( {{\mu _V} - {\mu _R}} \right)A - \left( {\mu {'_V} - \mu {'_R}} \right)A' = 0 \\\end{aligned} \end{equation} $$

or

$$A' = \left( {\frac{{{\mu _V} - {\mu _R}}}{{\mu {'_V} - \mu {'_R}}}} \right)A\quad ...(v)$$

From equation $(ii)$ and $(v)$ we get,

$${\delta _y} = \left( {{\mu _y} - 1} \right)A\left[ {1 - \frac{{\left( {\mu {'_y} - 1} \right)\left( {{\mu _V} - {\mu _R}} \right)}}{{\left( {\mu {'_V} - \mu {'_R}} \right)\left( {{\mu _y} - 1} \right)}}} \right]$$