Question

A crown glass prism of angle $ {6.20^ \circ } $ is to be combined with a flint glass prism in such a way that the mean ray passes undeviated. The angle of the flint glass prism needed if the refractive indices of crown glass and flint glass for yellow light are $ 1.517 $ and $ 1.620 $ respectively is
(A) $ {2.6^ \circ } $
(B) $ {5.17^ \circ } $
(C) $ {51.7^ \circ } $
(D) $ {26^ \circ } $

Answer 1

Hint : For the ray to pass undeviated, the deviation produced by the 2 prisms should be equal. The net deviation is thus equal to zero. So by finding the deviations and equating it to zero we will get the answer.

Formula Used: The formulae used in the solution are given here.
The deviation produced by the crown prism is $ \delta = \left( {\mu - 1} \right)A $ where $ A $ is the angle of crown glass and $ \mu $ is the refractive index of crown glass.
 $ \left( {{\mu _1} - 1} \right){A_1} = \left( {{\mu _2} - 1} \right){A_2} $ where, $ {\mu _1} $ and $ {\mu _2} $ are the refractive indices of crown glass and flint glass, $ {A_1} $ is the angle of crown glass and $ {A_2} $ is the angle of flint glass.

Complete answer
The deviation produced by the crown prism is $ \delta = \left( {{\mu _1} - 1} \right){A_1} $ where $ {A_1} $ is the angle of crown glass and $ {\mu _1} $ is the refractive index of crown glass.
The deviation produced by the flint glass is $ {\delta _{{\text{flint}}}} = \left( {{{{\mu }}_2}{\text{ - 1}}} \right){A_2} $ where $ {A_2} $ is the angle of flint glass and $ {{{\mu }}_2} $ is the refractive index of flint glass.
The prisms are placed with respect to each other. The deviations are also in the opposite direction. Thus, the net deviation is
 $ D = \delta - {\delta _{{\text{flint}}}} $ .
Substituting the values of $ \delta $ and $ {\delta _{{\text{flint}}}} $ in the above equation,
 $ \delta - {\delta _{{\text{flint}}}} = \left( {{\mu _1} - 1} \right){A_1} - \left( {{{{\mu }}_2}{\text{ - 1}}} \right){A_2} $
The net dispersion is zero. Thus, we get,
 $ \left( {{\mu _1} - 1} \right){A_1} = \left( {{\mu _2} - 1} \right){A_2} $ where, $ {\mu _1} $ and $ {\mu _2} $ are the refractive indices of crown glass and flint glass, $ {A_1} $ is the angle of crown glass and $ {A_2} $ is the angle of flint glass.
Substituting the values for the variables, $ {\mu _1} = 1.517 $ , $ {\mu _2} = 1.620 $ and $ {A_1} = {6.20^ \circ } $ , we get,
 $ \left( {1.517 - 1} \right)6.20 = \left( {1.620 - 1} \right){A_2} $
Simplifying the equation,
 $ {A_2} = \dfrac{{3.2054}}{{0.620}} $
 $ \Rightarrow {A_2} = {5.17^ \circ } $
The angle of the flint glass prism is $ {5.17^ \circ } $ .
The correct answer is Option B.

Note:
It is given that, for yellow light,the refractive index of crown glass is $ 1.517 $ and the refractive index of flint glass is $ 1.620 $ .
The angular dispersion produced by the crown prism is $ {\delta _v} - {\delta _r} = \left( {{\mu _{_v}} - {\mu _r}} \right)A $ where $ {\mu _{_v}} $ is the refractive index of violet light and $ {\mu _r} $ is the refractive index of red light.