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# 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/s2) $\sqrt 3 \times {10^8}$m/s3) $\dfrac{1}{{\sqrt 3 }} \times {10^8}$m/s4) $3\sqrt 3 \times {10^8}$m/s

Last updated date: 20th Sep 2024
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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.

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.