Question

In the diagram shown, a light ray is incident on the lower medium boundary at an angle of ${45}^0$ with the normal. Which of the following statements is/are true?

This question has multiple correct options
$A.$ If ${\displaystyle \frac{{\mu }_2}{\sqrt{2}}}$, then angle of deviation is ${45}^0$
$B.$ If ${\displaystyle \frac{{\mu }_2}{\sqrt{2}}}$, then angle of deviation is ${90}^0$
$C.$ If ${\displaystyle \frac{{\mu }_2}{\sqrt{2}}}$, then angle of deviation is ${135}^0$
$D.$ If ${\displaystyle \frac{{\mu }_2}{\sqrt{2}}}$, then angle of deviation is $0^0$

Answer 1

HINT- In this question we will use the concept of total internal reflection. Total internal reflection (TIR) is the optical phenomenon in which the surface of the water in a fish tank (for example) when viewed from below the water level, reflects the underwater scene like a mirror, with no loss of brightness. In general, TIR occurs when waves in one medium reach the boundary with another medium at a sufficiently slanting angle, provided that the second medium is transparent to the waves and allows them to travel faster than in the first medium.

Step-By-Step answer:
Now coming back to question, Critical angle is given by:
$${\theta }_c={\sin }^{-1}({\displaystyle \frac{{\mu }_2}{{\mu }_1}})={\sin }^{-1}({\displaystyle \frac{{\mu }_2}{2}})$$
If $\mu <\sqrt{2}$, then total internal reflection takes place and the ray reflects back, the angle of deviation is ${45}^0+{45}^0={90}^0$
If $\mu =\sqrt{2}$, then the ray passes tangentially at the first medium. Then, the angle of deviation is ${45}^0$
If $\mu >\sqrt{2}$, then the ray enters the second medium and the angle of refraction at third interface is given by:
${\mu }_1\sin {\theta }_1={\mu }_2\sin {\theta }_2={\mu }_3\sin {\theta }_3$
For 1st and 3rd medium,
$2\sin 45=\sqrt{2}\sin {\theta }_3$
$\sin {\theta }_3=1$
The angle of refraction is ${90}^0$
The deviation is $90-45={45}^0$
Therefore, option $A.$ and $B.$ are correct options for the above question.

NOTE- CRITICAL ANGLE
The angle of incidence beyond which rays of light passing through a denser medium to the surface of a less dense medium are no longer refracted but totally reflected.
So the critical angle is the angle of incidence that provides an angle of refraction of 90-degrees. For the crown glass-water boundary, the critical angle is 61.0 degrees. The actual value of the critical angle is dependent upon the combination of materials present on each side of the boundary.