Question

The speed of light in air is $3 \times {10^8}$m/s. If the same light undergoes minimum deviation by 60 degree in a an equilateral glass prism, its speed in prism will be
1) $3 \times {10^8}$m/s
2) $\sqrt 3 \times {10^8}$m/s
3) $\dfrac{1}{{\sqrt 3 }} \times {10^8}$m/s
4) $3\sqrt 3 \times {10^8}$m/s

Answer 1

Hint: Use the formula of minimum deviation and find out the value of $\mu \;$. Then use the relation $\mu \; = \dfrac{c}{v}$ to find out the value of V.

Complete step-by-step answer:
Formula required:
\[\mu \; = \dfrac{{\sin (\dfrac{{A + \partial }}{2})}}{{\sin (\dfrac{A}{2})}}\] to find out the refractive index.
$\mu \; = \dfrac{c}{v}$ to find out the value of v.
Given that the angle of prism =${60^ \circ }$
Also given that the angle of minimum deviation = ${60^ \circ }$
Speed of light = $3 \times {10^8}$m/s
To find out the refractive index of the medium we use the reqd, formula :
\[\mu \; = \dfrac{{\sin (\dfrac{{A + \partial }}{2})}}{{\sin (\dfrac{A}{2})}}\]
Putting the values in the above eqn, we have:
\[ \Rightarrow \mu \; = \dfrac{{\sin (\dfrac{{60 + 60}}{2})}}{{\sin (\dfrac{{60}}{2})}}\]
Further simplifying we get :
\[
   \Rightarrow \mu \; = \dfrac{{\sin (60)}}{{\sin (30)}} \\
   \Rightarrow \mu \; = \dfrac{{\dfrac{{\sqrt 3 }}{2}}}{{\dfrac{1}{2}}} \\
   \Rightarrow \mu \; = \sqrt 3 \\
 \]
Hence the refractive index of the prism = \[\mu \; = \sqrt 3 \]
Now finding out the velocity or speed of light in the given medium by the mentioned formula we have,
$\mu \; = \dfrac{c}{v}$
 $
   \Rightarrow \sqrt 3 = \dfrac{{3 \times {{10}^8}}}{v} \\
   \Rightarrow v = \sqrt 3 \times {10^8}m{s^{ - 1}}. \\
 $
Hence the refractive index of the prism = \[\mu \; = \sqrt 3 \] and the velocity is $\sqrt 3 \times {10^8}$m/s.

Hence the correct answer is option 2) $\sqrt 3 \times {10^8}$m/s.

Note:
If the prism is thin (prism angles up to ${5^ \circ }$) then we can use the direct formula to find refractive index by using the formula ${\partial _m} = A(\mu \; - 1)$.
This formula is a special case of prism which is derived from the formula \[\mu \; = \dfrac{{\sin (\dfrac{{A + \partial }}{2})}}{{\sin (\dfrac{A}{2})}}\]. Since the prism angle is very small, hence just substitute sin A = A and solve the equation and find out the value of the refractive index.