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, $\hat{r} = \frac{4}{5}\hat{i} -\frac{3}{5}\hat{j}$