Question

The refractive index of a material M1 changes by 0.014 and that of another material M2 changes by 0.024 as the colour of the light is changed from red to violet. Two thin prisms, one made of M1(A = 5.3°) and the other made of M2(A = 3.7°) are combined with their refracting angles oppositely directed. (a) Find the angular dispersion produced by the combination. (b) The prisms are now combined with their refracting angles similarly directed. Find the angular dispersion produced by the combination.

Answer 1

If μ'_v and μ'_r are the refractive indices of material M₁, then we have:
μ'_v – μ'_r = 0.014
If μ_v and μ_r are the refractive indices of material M₂, then we have:
μ_v – μ_r = 0.024
Now,
Angle of prism for M_1, A' = 5.3°
Angle of prism for M₂, A = 3.7°
(a) When the prisms are oppositely directed, angular dispersion $\left({\delta }_1\right)$ is given by
δ₁ = (μ_v – μ_r)A – (μ'_v – μ'_r)A'
On substituting the values, we get:
δ₁ = 0.024 × 3.7° – 0.014 × 5.3°
= 0.0146°
So, the angular dispersion is 0.0146°.

(b) When the prisms are similarly directed, angular dispersion$\left({\delta }_2\right)$ is given by
δ₂ = (μ_v – μ_r)A + (μ'_v – μ'_r)A'
On substituting the values, we get:
δ₂ = 0.024 × 3.7° + 0.014 × 5.3°
= 0.163°
So, the angular dispersion is 0.163°.