Question

A monochromatic light is traveling in a medium of refractive index $n=1.6$. It enters a stack of glass layer from the bottom side at an angle $\theta ={30}^{\circ }.$ The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layer are monotonically decreasing as $n_m=n-m\Delta n$, where $n_m$ is the refractive index of the $m^{th}$ slab and $\Delta n=0.1$ (see the figure). The ray is refracted out parallel to the interface between the $(m-1{)}^{th}$ and $m^{th}$ slabs from the right side of the stack. What is the value of $m$ is

Answer 1

Applying Snell's law,
$n\sin \theta =n_m\sin {90}^{\circ }$
$n\sin {30}^{\circ }=n_m\sin {90}^{\circ }$

$n\times \frac{1}{2}=(n-m\Delta n)\left(1\right)$

$\frac{1.6}{2}=[1.6-m(0.1\left)\right]$

$m\left(0.1\right)=\frac{1.6}{2}$

$m=8$