Question

A parallel beam of light is incident from air at an angle $\alpha$ on the side PQ of a right-angled triangular prism of refractive index $n=\sqrt{2}$. Light undergoes total internal reflection in the prism at the face $PR$ when $\alpha$ has minimum value of ${45}^{\circ }$. The angle $\theta$ of the prism is

• A. ${15}^{\circ }$
• B. ${22.5}^{\circ }$
• C. ${30}^{\circ }$
• D. ${45}^{\circ }$

Answer 1

The correct option is A ${15}^{\circ }$


Applying Snell's law at surface PQ,
$n_1\times sin\alpha =n_2\times sin\beta$
$\therefore 1\times \sin \alpha =n\times \sin \beta$
at $\alpha =45°,n=\sqrt{2}$
$\sin \beta =\frac{1}{2}\therefore \beta =30°$
for total internal reflection (TIR),
$\gamma =90-(180-(90+\theta +\beta ))$

$\therefore$ Applying Snell's law at surface PR
$\therefore \sqrt{2}\sin \gamma =1$
$\therefore \sin (\theta +\beta )=\frac{1}{\sqrt{2}}$
$\therefore \theta +\beta =45°$
$\therefore \theta +30°=45°$
$\therefore \theta =15°$
Thus option A matches correctly.