What is dipersion? Obtain the equation for dispersive power of a medium.

Explanation

Dispersion. Dispersion is splitting of white light into its costituent colours.
Dispersive power:
Consider beam of white light passes through a prism: It gets dispersed into its constituent colours. Let `delta_(v), delta_(R)` are the angle of deviation for violet and red light. Let `n_(v) and n_(R)` are the refractive indices for the violet and red light respectively.

Angle of deviation for different colours
The refractive index of the material of a prism is given by the equation
`n=(sin(A+D)/(2))/(sin((A)/(2)))`
Here A is the angle of the prism and D is the angle of minimum deviation. If the angle of prism is small of the order of `10^(@)`, the prism is said to be a small angle prism. When rays of light pass through such prisms. the angle of deviation also becomes small. If A be the angle of a small angle prism and `delta` the angle of deviation then the prism formula becomes.
`n=(sin(A+delta)/(2))/(sin((A)/(2)))`
For small angles of A and `delta_(m).`
`sin((A+delta)/(2))=((A+delta)/(2))`
`sin((A)/(2))=((A)/(2))`
`thereforen=(((A+delta)/(2)))/(((A)/(2)))=(A+delta)/(A)=1+(delta)/(A)`
Further simplifying, `(delta)/(A) = n - 1`
`delta = (n - 1) A`
When white light enters the prism, the deviation is defferent colours. Thus, the refractive index is also different colours.
For Violet light, `delta_(v) = (n_(v) - 1)A`
For Red light, `delta_(R) = (n_(R) - 1)A)`
As, angle of deviation for violet colours `delta_(v)` is greater than angle of deviation for red colour `delta_(R)`, the refractive index for violet colour `n_(v)` is greater than the refractive index for red colours `n_(R)`. Subtracting `delta_(v) "from" delta_(R)` we get,
`delta_(v)-delta_(R)=(n_(v)-n_(R))A`
The term `(delta_(v) - delta_(R))` is the angular separation between the two extreme colours (violet and red) in the spectrum is called the angular dispersion. Clearly, the angular dispersion produced by a prism depends upon.
(i) Angle of the prism
(ii) Nature of the material of the prism.
If we take `delta` is the angle of deviation for any middly ray (green or yellow) and n the corresponding refractive index. Then,
`delta = (n - 1)A`
Dispersive power `(omega)` is the ability of the material of the prism to cause disperison. It is defined as the ratio of the angular dispersion for the extreme colours to the deviation for any mean colour. Dispersive power `(omega)`,
`omega=("angular dispersios")/("mean deviation")=(delta_(v) - delta_(R))/(delta)`
Substituting for `(delta_(v) - n_(R)) and (delta)`,
`omega = ((n_(v) - n_(R))/((n - 1)))`
Dispersive power is a dimensionless quality. It has no unit. Dispersive power is always positive. The dispersive power of a prism depends only on the nature of material of the prism and it is independent of the prims.