Question

The critical angle of a prism is $30^\circ $. The velocity of light in the medium is
(A) $1.5 \times {10^8}\,m/s$
(B) $3 \times {10^8}\,m/s$
(C) $4.5 \times {10^8}\,m/s$
(D) None of these

Answer 1

Hint
As the value of the critical angle is given, firstly we will find the value of the refractive index of the medium. And further using this value of the refractive index, we will be able to calculate the velocity of the light in the medium.
Formulae used: In this solution we will be using the following formula,
$\Rightarrow n = \dfrac{1}{{\sin {\theta _C}}}$
Where $n$ is the refractive index and ${\theta _C}$ is the critical angle of the prism
$\Rightarrow n = \dfrac{c}{v}$
where $c$ is the speed of light in vacuum and $v$ is the speed of light in any other medium.

Complete step by step answer
The formula for calculating the refractive index of a medium, for a given critical angle of a prism is given by, $n = \dfrac{1}{{\sin {\theta _C}}}$
Now we can substitute the given value of the critical angle from the question to find the value of the refractive index of the medium. We are given ${\theta _C} = 30^\circ $. Therefore,
$\Rightarrow n = \dfrac{1}{{\sin 30^\circ }}$
Now the value of $\sin 30^\circ $ is $\dfrac{1}{2}$
So substituting the value,
$\Rightarrow n = \dfrac{1}{{\dfrac{1}{2}}}$
Upon further calculation, we obtain the value of the refractive index of the medium as $n = 2$.
So far, we have obtained the value of the refractive index. Now, we need to proceed with the calculation in order to find the value of the velocity of the light in the medium, whose refractive index we have just calculated.
The equation that relates the refractive index, the velocity of light in air and the velocity of light in a medium is given as follows.
$\Rightarrow n = \dfrac{c}{v}$
So, now will substitute the obtained value of the refractive index and the constant value of the velocity of the light in air is given by $c = 3 \times {10^8}m/s$. Hence we can find the velocity of light in a medium as,
$\Rightarrow 2 = \dfrac{{3 \times {{10}^8}}}{v}$
To calculate the velocity, we can rearrange the equation as,
$\Rightarrow v = \dfrac{{3 \times {{10}^8}}}{2}$
Upon further calculation, we obtain the value of the velocity of light in a medium as,
$\Rightarrow v = 1.5 \times {10^8}\,{m \mathord{\left/
 {\vphantom {m s}} \right.} s}$
$\therefore $ The velocity of light in the medium is $1.5 \times {10^8}\,{m \mathord{\left/
 {\vphantom {m s}} \right.} s}$.
Thus, option (A) is correct.

Note
The velocity of light varies in different mediums with different refractive indices. Due to this property of light, when white light is passed through a prism, it breaks up into its constituent colours.