Question

In an experiment for determination of refractive index of glass of a prism by $i - \delta$ plot, it was found that a ray incident at angle ${35^ \circ }$ suffers a deviation of ${40^ \circ }$and that it emerges at angle ${79^ \circ }$. In that case, which of the following is closest to the maximum possible value of the refractive index?
A) $1.6$
B) $1.7$
C) $1.8$
D) $1.5$

Answer 1

Hint The first relation that we will be using in this question is : $\delta = i + e - A$
Where: $\delta$ is the angle of deviation.
$i$ is the angle of incidence.
$e$ is the angle of emergence.
$A$is the angle of prism.
We are given the values of $\delta$, $i$ and $e$. So, we get the value of $A$.
Value of refractive index in terms of $A$ is
${\mu _{\max }} = \dfrac{{\sin \left( {\dfrac{{A + \delta }}{2}} \right)}}{{\sin \left( {\dfrac{A}{2}} \right)}}$
We have the values of $A$ and $\delta$, we can easily get the value of $\mu$.

Complete step-by-step solution :
We are given
Angle of deviation, $\delta = {40^ \circ }$
Angle of incidence, $i = {35^ \circ }$
Angle of emergence, $e = {79^ \circ }$
Now, using the relation
$\delta = i + e - A$
And putting in the values, we get
${40^ \circ } = {35^ \circ } + {79^ \circ } - A \\\ A = {74^ \circ } \\\$
So, angle of prism, $A = {74^ \circ }$
Refractive index $\left( \mu \right)$ is given in terms of angle of prism $\left( A \right)$ and angle of deviation $\left( \delta \right)$ as:
${\mu _{\max }} = \dfrac{{\sin \left( {\dfrac{{A + \delta }}{2}} \right)}}{{\sin \left( {\dfrac{A}{2}} \right)}}$
Putting in the values of $A$ and $\delta$, we get
${\mu _{\max }} = \dfrac{{\sin \left( {\dfrac{{{{74}^ \circ } + {{40}^ \circ }}}{2}} \right)}}{{\sin \left( {\dfrac{{{{74}^ \circ }}}{2}} \right)}} = \dfrac{{\sin {{57}^ \circ }}}{{\sin {{37}^ \circ }}} \\\ = \dfrac{{0.8386}}{{0.6018}} = 1.4 \\\$
The actual formula for ${\mu _{\max }}$involves the value of ${\delta _{\min }}$ and not $\delta$. We have assumed that ${\delta _{\min }} = {40^ \circ }$ which may not be true. Hence, maximum possible value of $\mu$ will be closest to $1.5$

Note:- Since, we are to find the maximum possible values of $\mu$, we have used approximations such as substituting ${\delta _{\min }} = {40^ \circ }$. One should take care during calculations especially, while dealing with sine values.