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 $0^{th}$ layer and the difference in $RI$ between any two consecutive layers, respectively. The integer $m = 0, 1, 2, 3.....$ denotes the numbers of the successive layers. A ray of light from the 0th layer enters the $1^{st}$ layer at an angle of incidence of $30^\circ$. 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

Answer: (D)

50

Answer 2

By Snell's law,
$\mu \sin i =$ constant
$\mu \sin i (\mu-m \Delta \mu) \sin r$
where $\mu=1.5, i=30^{\circ}, r=90^{\circ}, \Delta \mu=0.015$
$1.5 \sin 30^{\circ} =(1.5-m \times 0.15) \sin 90^{\circ}$
$\frac{3}{2} \times \frac{1}{2} =(1.5-m \times 0.015) \times 1$
$\frac{3}{4} =\frac{3}{2}-\frac{15\, m}{1000}$
$15 \,m =\left(\frac{3}{2}-\frac{3}{4}\right) \times 1000$
$15 \,m =\frac{6-3}{4} \times 1000$
$\Rightarrow m=\frac{3}{4} \times \frac{1000}{15}$
$m =\frac{3000}{60}$
$\Rightarrow m=50$