Question

A prism of refractive index $\sqrt{2}$ and refracting angle A produces minimum deviation ${\delta }_m$ of a ray on one face at an angle of incidence ${45}^{\circ }$, The values of A and ${\delta }_m$ are respectively.

• A. ${45}^{\circ }$, ${45}^{\circ }$
• B. ${45}^{\circ }$, ${60}^{\circ }$
• C. ${60}^{\circ }$, ${30}^{\circ }$
• D. ${60}^{\circ }$, ${45}^{\circ }$

Answer 1

The correct option is C ${60}^{\circ }$, ${30}^{\circ }$

$\mu =\frac{\sin i}{\sin r}$

Again for minimum deviation

$\mu =\frac{\sin \left(\frac{A+{\delta }_{min}}{2}\right)}{\sin \frac{A}{2}}$

$\therefore \mu =\frac{\sin i}{\sin \frac{A}{2}}$

$\Rightarrow \sin \frac{A}{2}=\frac{\sin i}{\mu }$

$\Rightarrow \sin \frac{A}{2}=\frac{\sin {45}^{\circ }}{\sqrt{2}}=\frac{1}{2}$

$\Rightarrow \frac{A}{2}={30}^{\circ }$

$\Rightarrow A={60}^{\circ }$

Again $i+e=A+{\delta }_{min}$

For minimum deviation $i=e$

$\therefore 2i=A+{\delta }_{min}$

$\Rightarrow {\delta }_{min}=2i-A$

$\Rightarrow {\delta }_{min}=2\times {45}^{\circ }-{60}^{\circ }={30}^{\circ }$

Hence the correct answer is option (C).