Question

A light ray is incident on an inclined glass-air interface as shown. The largest angle $\varphi$ for which the light ray is totally reflected at the surface $\text{AC}$ is -

• A. ${\cos }^{-1}\left(\frac{2}{3}\right)$
• B. ${\cos }^{-1}\left(\frac{1}{3}\right)$
• C. ${\sin }^{-1}\left(\frac{2}{3}\right)$
• D. ${\sin }^{-1}\left(\frac{1}{3}\right)$

Answer 1

The correct option is A ${\cos }^{-1}\left(\frac{2}{3}\right)$


For $\text{TIR}$ at surface $\text{AC}$,
$i\ge {\theta }_c$

In critical case,
$i={\theta }_c$

But, $i={90}^{\circ }-\varphi$ [from geometry]

So, largest value of angle $\varphi$ for $\text{TIR}$ at surface $\text{AC}$ is for $i={\theta }_c$

So, from Snell's law at surface $\text{AC}$, for $i={\theta }_c$

${\mu }_1\sin i={\mu }_2\sin r$

$\Rightarrow 1.5\sin ({90}^{\circ }-{\varphi }_{max})=1\times \sin {90}^{\circ }$

$\Rightarrow \cos {\varphi }_{max}=\frac{2}{3}$

$\Rightarrow {\varphi }_{max}={\cos }^{-1}\left(\frac{2}{3}\right)$
Hence, option $\left(a\right)$ is correct.

Why this question ? This question is for testing the concept of total internal reflection and Snell's law