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.