Question

A ray of light falls on the plane surface of a half cylinder at an angle ${45}^{\circ }$ in the plane perpendicular to the axis (see figure). Refractive index of glass is $\sqrt{2}$ and radius of cylinder is $\sqrt{3}\text{m}$. The value of $\text{OP}$ in critical case is

• A. $\sqrt{2}\text{m}$
• B. $\sqrt{5}\text{m}$
• C. $\sqrt{3}\text{m}$
• D. $\sqrt{6}\text{m}$

Answer 1

The correct option is A $\sqrt{2}\text{m}$


At surface $\text{AB}$, from Snell's law,

$1\times \sin {45}^{\circ }=\sqrt{2}\sin r$

$\Rightarrow \sin r=\frac{1}{2}$

$\Rightarrow r={30}^{\circ }..........\left(1\right)$

In critical case, at $\text{C}$

$\sin {\theta }_c=\frac{{\mu }_r}{{\mu }_d}$

$\Rightarrow \sin {\theta }_c=\frac{1}{\sqrt{2}}.......\left(2\right)$

In $\Delta OPC$,

$\frac{OP}{\sin {\theta }_c}=\frac{OC}{\sin ({90}^{\circ }+r)}$ [Sine Law]

$\Rightarrow OP=\frac{OC\sin {\theta }_c}{\cos r}$

From eqs. (1) and (2)

$\Rightarrow OP=\frac{\sqrt{3}\times \frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2}}$

$\Rightarrow OP=\sqrt{2}\text{m}$

Hence, option $\left(a\right)$ is the correct answer.

Why this question ? It tests your geometrical understanding, and its application in refraction of light.