Question

A ray enters a glass slab at the origin at an angle of incidence $\theta$ from air where refractive index varies along the slab as $\mu ={\mu }_0-k{y}^2$, where $y$ is the distance measured along the normal and ${\mu }_0$ and $k$ are positive contains. If the glass slab is sufficiently thick, the maximum value of $y$ coordinate of the refracted ray is

• A. $\frac{{\mu }_0-\sin \theta }{k}$
• B. $\frac{{\mu }_0+\sin \theta }{k}$
• C. $\sqrt{\frac{{\mu }_0-\sin \theta }{k}}$
• D. $\sqrt{\frac{{\mu }_0+\sin \theta }{k}}$

Answer 1

The correct option is C $\sqrt{\frac{{\mu }_0-\sin \theta }{k}}$

$\begin{array}{c}\text{let at depth h, T.I.R}\\ \text{takes place.}\\ \text{Just before}T,I,R\\ r={90}^{\circ }\end{array}$

$\begin{array}{cc}{\mu }_1\sin i& ={\mu }_2\sin r\\ {\mu }_1=1,& {\mu }_2={\mu }_0-k{h}^2,i=\theta ,r=90^{\circ }\\ 1\times \sin \theta & ={\mu }_0-k{h}^2\times \sin {90}^{\circ }\\ \Rightarrow & \sin \theta ={\mu }_0-k{h}^2\\ \Rightarrow & h=\sqrt{\frac{{\mu }_0-\sin \theta }{k}}\end{array}$