Question

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

Answer 1

Dispersion. Dispersion is splitting of white light into its constituent colours. 

Dispersive Power:

Consider a beam of white light passes through a prism; It gets dispersed into its constituent colours. Let δ_v , δ_R are the angles of deviation for violet and red light. Let n_v and n_R are the refractive indices for the violet and red light respectively.

The refractive index of the material of a prism is given by the equation

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 5 the angle of deviation then the prism formula becomes.

n = \( {\frac{sin {\frac{A+δ}{2}}}{sin {\frac{A}{2}}}}\)……… (1)

For small angles of A and δ_m

\(sin(\frac{A + δ}{2})\) ≈ \(\frac{A+δ}{2}\)……. (2) 

sin(\(\frac{A}{2}\))≈ (\(\frac{A}{2}\))…… (3)

∴ n = \(\frac{\frac{A + δ}{2}}{\frac{A}{2}}\) = \(\frac{A+δ}{A}\) = I + \(\frac{δ}{A}\)

Further simplifying, δ/A = n – 1

δ = (n – 1) A ……. (4)

When white light enters the prism, the deviation is different for different colours. Thus, the refractive index is also different for different colours.

For Violet light, δ_v = (n_v – 1)A …(5) 

For Red light, δ_R = (n_R – 1) …(6)

As, angle of deviation for violet colour δ_v is greater than angle of deviation for red colour δ_R , the refractive index for violet colour n_v is greater than the refractive index for red colour n_R . Subtracting δ_v from δ_R we get,

δ_v – δ_R = (n_v – n_R )A ….. (7)

The term (δ_v – δ_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.

1. Angle of the prism 

2. Nature of the material of the prism.

If we take 8 is the angle of deviation for any middly ray (green or yellow) and n the corresponding refractive index. Then,

8 = (n – 1) A … (8)

Dispersive power (ω) is the ability of the material of the prism to cause dispersion. It is defined as the ratio of the angular dispersion for the extreme colours to the deviation for any mean colour. Dispersive power (ω),

Substituting for (δ_v -δ_R )and (δ),

ω = \(\frac{n_v -n_R}{n-1}\) ……. (10)

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 angle of the prism.