Question

A ray of light moving along the vector $(\hat{i}-2\hat{j})$ undergoes refraction at an interface of two media, which is $x-z$ plane. The refractive index for $y>0$ is $2$ while for $y<0$ it is $\frac{\sqrt{5}}{2}$. The unit vector along the refracted ray is

• A. $\frac{-3\hat{i}-5\hat{j}}{\sqrt{34}}$
• B. $\frac{-\left(4\hat{i}-5\hat{j}\right)}{5}$
• C. $\frac{-3\hat{i}-4\hat{j}}{5}$
• D. $\frac{4\hat{i}-3\hat{j}}{5}$

Answer 1

Answer: (D)

$-\left(\hat{i}-2\hat{j}\right)\cdot \hat{j}=\left|\hat{i}-2\hat{j}\right|\left|\hat{j}\right|cos\theta$
$\Rightarrow cos\theta =\frac{2}{\sqrt{5}}$
$\therefore sin\theta =\frac{1}{\sqrt{5}}$
Using Snells law at the interface
$2\times \frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{2}sin\beta$
$\Rightarrow sin\beta =\frac{4}{5}$
$\therefore cos\beta =\frac{3}{5}$
So, $\hat{r}=\frac{4}{5}\hat{i}-\frac{3}{5}\hat{j}$