Question

Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet rays are μ_r, μ_y and μ_v respectively and those for the flint glass are μ'_r, μ'_y and μ'_v respectively. Find the ratio A'/A for which (a) there is no net angular dispersion, and (b) there is no net deviation in the yellow ray.
Figure

Answer 1

For the crown glass, we have:
Refractive index for red rays = μ_r
Refractive index for yellow rays = μ_y
Refractive index for violet rays = μ_v
For the flint glass, we have:
Refractive index for red rays = μ'_r
Refractive index for yellow rays = μ'_y
Refractive index for violet rays = μ'_v
Let δ_cy and δ_fy be the angles of deviation produced by the crown and flint prisms for the yellow light.
Total deviation produced by the prism combination for yellow rays:
δ_y = δ_cy − δ_fy
= 2δ_cy − δ_fy
=2(μ_cy + 1)A − (μ_fy − 1)A'
Angular dispersion produced by the combination is given by
δ_v − δ_r = [(μ_vc − 1)A − (μ_vf − 1)A' + (μ_vc − 1)A − $\left[\left({\mu }_{rc}-1\right)A-\left({\mu }_{rf}-1\right)A\text{'}+\left({\mu }_{rc}-1\right)A\right]$
Here,
μ_{vc =} Refractive index for the violet colour of the crown glass
μ_vf = Refractive index for the violet colour of the flint glass
${\mu }_{rc}$ = Refractive index for the red colour of the crown glass
${\mu }_{rf}$ = Refractive index for the red colour of the flint glass
On solving, we get:
δ_v − δ_r = 2(μ_vc −1)A − (μ_vf − 1)A'
(a) For zero angular dispersion, we have:
δ_t − δ_t = 0 = 2(μ_vc −1)A − (μ_vf − 1)A'
$$\begin{gathered} \Rightarrow \frac{A\text{'}}{A}=\frac{2({\mu }_{\mathrm{vf}}-1)}{({\mu }_{vc}-1)} \\ =\frac{2({\mu }_r-{\mu }_r)}{({\mu }_r-\mu )} \end{gathered}$$
(b) For zero deviation in the yellow ray, δ_y = 0.
⇒ 2(μ_cy − 1)A = (μ_fy − 1)A
$$\begin{gathered} \Rightarrow \frac{A\text{'}}{A}=\frac{2({\mu }_{\mathrm{cy}}-1)}{({\mu }_{\mathrm{fy}}-1)} \\ =\frac{2({\mu }_y-1)}{(\mu {\text{'}}_y-1)} \end{gathered}$$