Question

Obtain the equation for dispersive power of a medium

Answer 1

1. Consider a beam of white light passes through a prism; It gets dispersed into its constituent colours.
2. 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.
   
   Angle of deviation for different colours
3. 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.
   n = `(sin ("A + δ"/2))/(sin ("A"/2))`
   For small angles of A and δ_m,
   `sin ("A + δ"/2) ≈ ("A + δ"/2)`
   `sin ("A"/2) ≈ ("A"/2)`
   ∴ n = `(("A + δ"/2))/(("A"/2))`
   `= ("A + δ")/"A" = 1 + δ/"A"`
   Further simplification, 
   `δ/"A" = "n - 1"`
   δ = (n – 1) A
   Whn 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 =(n_v – 1) A
   For Red light, δ_R =(n_R – 1) A
4. As, angle of deviation for violet colour δ_v is greater than the angle of deviation for red colour, δ_v 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
5. The term (δ_v – δ_R) is the angular separation between the two extreme colours (violet and red) in the spectrum is called 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.
6. It is defined as the ratio of the angular dispersion for the extreme colours to the deviation for any mean colour.
   Dispersive power (ω),
   ω = `"angular dispersion"/"mean deviation" = (δ_"v" - δ_"R")/δ`
   Substituting for (δ_v – δ_R) and (δ),
   ω = `("n"_"v" - "n"_"R")/("n - 1")`