Question

A ray of light of pure single colour is incident on the face of a prism having angle of the prism 30° at an angle of incidence 45°. The refracted ray does not change its direction as it crosses the other face and emerges out of the prism. The refractive index of the material of the prism is
A. $\dfrac{2}{{\sqrt 3 }}$
B. 2
C. $\sqrt 2 $
D. $\sqrt 3 $

Answer 1

Hint: Use ${r_2}$= 0° since the refracted ray emerges from the prism without changing its direction. Also remember to use formulav$A = {r_1} + {r_2}$ and find the correct option.
Formula used: ${\mu _{21}} = \dfrac{{\sin {i_1}}}{{\sin {r_1}}}$and $A = {r_1} + {r_2}$

Complete Step-by-Step solution:

According to the given information A = 30° and ${i_1} = 45^\circ $
Since according to the given information the refracted ray does not change its direction
Also we know that for a prism of angle A, ${r_1}$ and ${r_2}$ are the refraction angle
Therefore $A = {r_1} + {r_2}$ (equation 1)
Since the refracted ray emerges out of the prism without change in direction .i.e. ${r_2}$= 0°
Substituting the values in the equation 1
$ \Rightarrow $\[{30^ \circ } = {r_1} + {0^ \circ }\]
$ \Rightarrow $\[{r_1} = {30^ \circ }\]
Therefore the relation ${\mu _{21}} = \dfrac{{\sin {i_1}}}{{\sin {r_1}}}$$ \Rightarrow $$\dfrac{{\sin {{45}^ \circ }}}{{\sin {{30}^ \circ }}}$
So, ${\mu _{21}} = \sqrt 2 $
Hence C is the correct option.

Note: Refraction is the bending of light when going from one medium to another, and as a light ray passes through a glass prism, light refraction happens both when it reaches the prism and when it leaves the prism. Furthermore, because the refracting surfaces are not parallel, the evolving radius and incident rays are not parallel to each other. In this case, on passing through the prism, the beam of light deviates.