Question

A ray of light travelling in air is incident at angle of ${30}^{\circ }$ on one surface of a slab in which refractive index varies with $y$. The light travels along the curve $y=4x^2$ ($y$ and $x$ are in metre) in the slab. If the refractive index of the slab at $y=\frac{1}{2}m$ in the slab is $\frac{15}{x_o}$, then $x_o$ is

Answer 1

By snell's law at incidence and at $y$
$1\times sin{30}^{\circ }=nsini$
where $n$ is Refractive index at $y$ and $i$ is angle of incidence at $y$.

Now from slope
$tan(90-i)=\frac{dy}{dx}=8x=4\sqrt{y}$
$coti=4\sqrt{y}=4\sqrt{\frac{1}{2}}=2\sqrt{2}$
$\Rightarrow sini=\frac{1}{3}\therefore n=\frac{sin{30}^o}{sini}=\frac{\frac{1}{2}}{\frac{1}{3}}=1.5$