Question

A system of coordinates is drawn in a medium whose refractive index varies as $\mu =\frac{2}{1+y^2}$, where $0\le y\le 1$. A ray of light is incident at origin at an angle ${60}^{\circ }$ with $y$ -axis as shown in the figure. At point P ray becomes parallel to $x-axis$. The value of $H$ is

• A. ${\{\left(\frac{2}{\sqrt{3}}\right)-1\}}^{\frac{1}{2}}$
• B. ${\left\{\frac{2}{\sqrt{3}}\right\}}^{\frac{1}{2}}$
• C. ${\{\left(\sqrt{3}\right)-1\}}^{\frac{1}{2}}$
• D. ${\{(\sqrt{3}+1)-1\}}^{\frac{1}{2}}$

Answer 1

The correct option is A ${\{\left(\frac{2}{\sqrt{3}}\right)-1\}}^{\frac{1}{2}}$

Applying Snell’s law between the points $O$ and $P$, we have
$2\times \sin \left({60}^{\circ }\right)=\sin \left({90}^{\circ }\right)\times \frac{2}{(1+H^2)}$, $2\times \frac{\sqrt{3}}{2}=1\times \frac{2}{(1+H^2)}$
$(1+H^2)=\frac{2}{\sqrt{3}}$, $H=\sqrt{(\frac{2}{\sqrt{3}}-1)}$
$\therefore \left(a\right)$ is the correct answer.