Question

What should be the value of angle $\theta$ so that light entering normally through the surface $AC$ of a prism with refractive index $(n=\frac{3}{2})$, does not cross the $2^{\text{nd}}$ refracting surface $AB$?

• A. $\theta ={\cos }^{-1}\frac{1}{3}$
• B. $\theta \le {\cos }^{-1}\frac{2}{3}$
• C. $\theta \ge {\cos }^{-1}\frac{1}{3}$
• D. $\theta ={45}^{\circ }$

Answer 1

The correct option is B $\theta \le {\cos }^{-1}\frac{2}{3}$

The light ray is incident normally at surface $AC$, hence it will pass without deviation. Let it is incident at angle $i$ at the second surface.

$\Rightarrow i={90}^{\circ }-\theta$

For the light ray to not pass through surface $AB$ it should meet the condition required for $\text{T.I.R}$

$\Rightarrow i\ge {\theta }_c$

$\Rightarrow {90}^{\circ }-\theta \ge {\theta }_c$

$\sin ({90}^{\circ }-\theta )\ge \sin {\theta }_c$

$\cos \theta \ge \sin {\theta }_c....\left(i\right)$

From definition of critical angle,

$\sin {\theta }_c=\frac{1}{n}=\frac{2}{3}....\left(ii\right)$

From eq.$\left(i\right)$ and $\left(ii\right)$,

$\Rightarrow \cos \theta \ge \frac{2}{3}$

Since $\cos \theta$ is a decreasing function in the interval $0^{\circ }\le \theta \le {90}^{\circ }$

$\Rightarrow \theta \le {\cos }^{-1}\frac{2}{3}$