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Question
Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet rays are μr, μy and μv 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.
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Solution
For the crown glass, we have:-
Refractive index for red rays = μr
Refractive index for yellow rays = μy
Refractive index for violet rays = μv
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 δcy and δfy 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:-
δy = δcy − δfy
= 2δcy − δfy
=2(μcy + 1)A − (μfy − 1)A'
Angular dispersion produced by the combination is given by
δv − δr = [(μvc − 1)A − (μvf − 1)A' + (μvc − 1)A] − \[\left[ \left( \mu_{rc} - 1 \right)A - \left( \mu_{rf} - 1 \right)A' + \left( \mu_{rc} - 1 \right)A \right]\]
Here,
μvc = Refractive index for the violet colour of the crown glass
μvf = 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
δv − δr = 2(μvc −1)A − (μvf − 1)A'
(a) For zero angular dispersion, we have:
δt − δt = 0 = 2(μvc −1)A − (μvf − 1)A'
\[\Rightarrow \frac{A'}{A} = \frac{2( \mu_{vf} - 1)}{( \mu_{vc} - 1)}\]
\[ = \frac{2( \mu_r - \mu_r )}{( \mu_r - \mu)}\]
(b) For zero deviation in the yellow ray, δy = 0.
⇒ 2(μcy − 1)A = (μfy − 1)A
\[\Rightarrow \frac{A'}{A} = \frac{2( \mu_{cy} - 1)}{( \mu_{fy} - 1)}\]
\[ = \frac{2( \mu_y - 1)}{(\mu '_y - 1)}\]
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