Question

Consider a light ray travelling in air is incident into a medium of refractive index $\sqrt{2 n}$. The incident angle is twice that of refracting angle. Then, the angle of incidence will be :

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

Answer 1

Answer: (D)

$i=2 r$
$\sin i \times n_{1}=\sin r \times n_{2}$
$\sin i \times 1=\sin \frac{i}{2} \times \sqrt{2 n}$
$\frac{\sin i}{\sin \frac{i}{2}}=\sqrt{2 n}$
$\frac{2 \sin \frac{i}{2} \cos \frac{i}{2}}{\sin \frac{i}{2}}=\sqrt{2 n}$
$\cos \frac{i}{2}=\sqrt{\frac{n}{2}}$
$\frac{i}{2}=\cos ^{-1}\left(\sqrt{\frac{n}{2}}\right)$
$i=2 \cos ^{-1}\left(\sqrt{\frac{n}{2}}\right)$