Question

A system of coordinates is drawn in a medium whose refractive index varies as $\mu =\frac{2}{1+y^2}$, where $0\le y\le 1$ and $\mu =2$ for $y<0$ as shown in figure. 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. ${(\frac{2}{\sqrt{3}}-1)}^{\frac{1}{2}}$
• B. ${\left(\frac{2}{\sqrt{3}}\right)}^{\frac{1}{2}}$
• C. ${(\sqrt{3}-1)}^{\frac{1}{2}}$
• D. ${(\frac{4}{\sqrt{3}}-1)}^{\frac{1}{2}}$

Answer 1

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



For two glass slabs kept in contact like in the diagram,
${\mu }_1\sin {\theta }_1={\mu }_2\sin {\theta }_2={\mu }_3\sin {\theta }_3$
The medium in question can be considered as made up of a series of glass slabs on top of each other.



Therefore by applying the previous equation, we get
${\mu }_1\sin {\theta }_1={\mu }_n\sin {\theta }_n$
At $y=0,{\mu }_1=2$ and ${\theta }_1={60}^{\circ }$
At $y=H,{\mu }_n=\frac{2}{1+H^2}$ and ${\theta }_n={90}^{\circ }$
i.e $2\sin {60}^{\circ }=\frac{2}{1+H^2}\sin {90}^{\circ }$
i.e $H=\sqrt{\frac{2}{\sqrt{3}}-1}$