Question

A ray of light is incident on the surface of a transparent medium at an angle of $45^\circ $ and is refracted in the medium at an angle$30^\circ $. What will be the velocity of light in the transparent medium?
A) $1.96 \times {10^8}\dfrac{m}{s}$
B) $2.12 \times {10^8}\dfrac{m}{s}$
C) $2.65 \times {10^8}\dfrac{m}{s}$
D) $1.25 \times {10^8}\dfrac{m}{s}$

Answer 1

Hint: The speed of light is different in different mediums and due to which light bends as it enters from one medium to another. The refractive index is the ratio of speed of light in vacuum to the speed of light in another medium.
Formula used: The formula of Snell’s law is given by, $\dfrac{{\operatorname{sin} i}}{{\operatorname{sin} r}} = \dfrac{c}{v}$ where i is incident angle, r is the refracted angle, c is the speed of light in vacuum and v is the speed of light in the medium.

Complete step by step answer:
As it is given that the refracted angle is given as $45^\circ $ and $30^\circ $ with the vertical also so applying the Snell’s law.
Therefore,
According to Snell’s law,
$\dfrac{{\operatorname{sin} i}}{{\operatorname{sin} r}} = \dfrac{c}{v}$, where $i$ is the incident angle, $r$ is refracted angle, $c$ is the speed of light in vacuum and v is the speed of light in the medium.
So here we need to find the speed of light in the medium and therefore rearrange the terms.
$\dfrac{{\operatorname{sin} i}}{{\operatorname{sin} r}} = \dfrac{c}{v}$
$\Rightarrow v = c \cdot \left( {\dfrac{{\operatorname{sin} r}}{{\operatorname{sin} i}}} \right)$
So here the incident angle is $45^\circ $ and the refracted angle is $30^\circ $ also the speed of light in vacuum is $c = 3 \times {10^8}\dfrac{m}{s}$.
Replace the values of speed of light, incident angle, and refracted angle.
$ \Rightarrow v = c \cdot \left( {\dfrac{{\operatorname{sin} r}}{{\operatorname{sin} i}}} \right)$
On substituting the corresponding values,
$ \Rightarrow v = \left( {3 \times {{10}^8}} \right) \cdot \left( {\dfrac{{\operatorname{sin} 30^\circ }}{{\operatorname{sin} 45^\circ }}} \right)$
On simplification,
$ \Rightarrow v = \left( {3 \times {{10}^8}} \right) \cdot \left[ {\dfrac{{\left( {\dfrac{1}{2}} \right)}}{{\left( {\dfrac{1}{{\sqrt 2 }}} \right)}}} \right]$
$ \Rightarrow v = \left( {3 \times {{10}^8}} \right) \cdot \dfrac{1}{{\sqrt 2 }}$
On further simplification,
$ \Rightarrow v = \left( {3 \times {{10}^8}} \right) \cdot \dfrac{1}{{\sqrt 2 }}$
$ \Rightarrow v = 2 \cdot 12 \times {10^8}\dfrac{m}{s}$

Therefore, the speed of light in the medium is equal to $v = 2.12 \times {10^8}\dfrac{m}{s}$. So the correct option for this problem is potion B.

Note:
Snell's law should be understood and remembered by the students as it can help in solving these types of problems. Also the Snell’s law relation involving the refractive index should be well remembered by students as sometimes problems involve refractive index instead of the speed of light in different mediums.