Question

A glass slab consists of thin uniform layers of progressively decreasing refractive indices $RI$ (see figure) such that the $RI$ of any layer is $\mu -m\Delta \mu$. Here, $\mu$ and $\Delta \mu$ denote the $RI$ of 0th layer and the difference in $RI$ between any two consecutive layers, respectively. The integer $m=0,1,2,3,\dots$ denotes the numbers of the successive layers.
A ray of light from the 0th layer enters the 1st layer at an angle of incidence of ${30}^0$. After undergoing the $m$th refraction, the ray emerges parallel to the interface. If $\mu =1.5$ and $\Delta \mu =0.015$, the value of $m$ is:

• A. 20
• B. 30
• C. 40
• D. 50

Answer 1

The correct option is D 50

Given : $\mu =1.5$ $\Delta \mu =0.015$

Refractive index of $mth$ layer ${\mu }_m=\mu -m\Delta \mu$

Thus refractive index of $0th$ layer i.e $m=0$, ${\mu }_o=\mu$

Also the angle of incidence in $0th$ layer $i={30}^o$

Using Snell' law of refraction for $0th$ and $mth$ layer ${\mu }_o\times sini={\mu }_m\times sinr$

$\mu \times sin{30}^o=(\mu -m\Delta \mu )\times sin{90}^o$

$1.5\times 0.5=[1.5-m(0.015\left)\right]\times 1$

$⟹$ $m=50$