Question

At what angle with the normal should a light beam fall on the surface of a material with refractive index $\sqrt{3}$, if the angle of refraction was half the angle of incidence?

• A. ${45}^{\circ }$
• B. ${60}^{\circ }$
• C. ${30}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is B ${60}^{\circ }$

As we know,

$\mu =\frac{\sin i}{\sin r}[\because r=\frac{i}{2}]$

$\mu =\frac{\sin i}{\sin \left(\frac{i}{2}\right)}=\frac{\sin \left(2\frac{i}{2}\right)}{\sin \left(\frac{i}{2}\right)}$

$\mu =\frac{2\sin \frac{i}{2}\cos \frac{i}{2}}{\sin \frac{i}{2}}$

$\Rightarrow \sqrt{3}=2\cos \frac{i}{2}[\because \mu =\sqrt{3}]$

$\Rightarrow \frac{\sqrt{3}}{2}=\cos \left(\frac{i}{2}\right)$

$\therefore \frac{i}{2}={30}^{\circ }\Rightarrow i={60}^{\circ }$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(B\right)$ is the correct answer.