Question

A light ray incident upon the 1st surface of a prism at an angle of incidence $90^\circ$ passes through the other surface grazingly, given $\mu = \sqrt 2$. The angle of prism is?

(A) $45^\circ$
(B) $90^\circ$
(C) ${\sin ^{ - 1}}\left( {\dfrac{{\sqrt 2 }}{3}} \right)$
(D) Information insufficient.

Answer 1

Here, you are given a prism whose angle is not known to you and you are asked to find that angle. It is given that the angle of incidence at the first surface is $90^\circ$, where the light will refract and will transmit through the prism and then will reach the second surface and then will again refract, it is given that it emerges grazingly. Here, angles and refractive index are involved, so figure out what should be used in order to find the angle of the prism. Also, if needed, try using the relation between angle of refraction, angle of incidence and the angle of prism for events occurring inside the prism.

Complete step by step answer:
So, whenever we are given refraction of light, the first thing that should come in our mind is Snell’s law. Snell’s law gives you the relation between the angle of incidence, angle of refraction and the refractive indexes of both the media. If the angle of incidence is $i$, angle of refraction is $r$ and the light goes from medium ${\mu _1}$ to medium ${\mu _2}$, then Snell’s law is mathematically given as ${\mu _1}\sin i = {\mu _2}\sin r$.
Let us come back to our question. For refraction at first surface, we have,
$\left( 1 \right)\left( {\sin 90^\circ } \right) = {\mu _2}\sin r \\\ \Rightarrow 1 = \mu \sin r \\\ \Rightarrow 1 = \sqrt 2 \sin r \\\ \Rightarrow\sin r = \dfrac{1}{{\sqrt 2 }} \to r = 45^\circ \\\$
Now, let the angle of incidence at the second surface be $r'$, the angle of refraction is $90^\circ$ (grazingly) and the light is going from glass to air, again we apply Snell’s law, we get,
$\left( {\sqrt 2 } \right)\left( {\sin r'} \right) = \left( 1 \right)\left( {\sin 90^\circ } \right) \\\ \Rightarrow\sqrt 2 \sin r' = 1 \\\ \Rightarrow\sin r' = \dfrac{1}{{\sqrt 2 }} \to r' = 45^\circ \\\$
Now, we will use the relation between angle of refraction, angle of incidence and the angle of prism for events occurring inside the prism.
We have $r + r' = A$,
$r + r' = A \\\ \Rightarrow A = 45^\circ + 45^\circ \\\ \therefore A= 90^\circ \\\$
Therefore, the angle of the prism is $90^\circ$ and option B is correct.

Note: We have used two formulas, one is the Snell’s law and the other is the relation between the angles, so keep this in mind. These can be useful while solving problems related to prism. Also, you can see that the angle of the prism came out to be $90^\circ$ which according to physics is impossible. But note that you also have the angle of incidence as $90^\circ$ which is also impossible because the ray would never incident on the surface. The answer which we got is just based on mathematics and is only supported by mathematics, it is practically impossible.