Question

A glass slab consists of thin uniform layers of progressively increasing refractive indices such that the R.I. of any layer is $\mu +m\Delta \mu$. Here $\mu$ and $\Delta \mu$ denote the RI of $0^{\text{th }}$ layer and the difference in R.I. between any two consecutive layers, respectively. The integer $m=0,1,2,3\dots$. denotes the number of the successive layers. A ray of light from the upper most layer enters the consecutive layer at an angle of incidence of $30^{\circ }$. After undergoing the $m^{\text{th }}$ refraction, the ray emerges parallel to the interface. If $\mu =1.26$ and $\Delta \mu =0.012$, the value of $m$ is________.

Answer 1

Answer: 105

By Snell's law,
$\frac{{\mu }_2}{{\mu }_1}=\frac{\sin i}{\sin r}$
$\therefore {\mu }_1\sin i={\mu }_2\sin r$
$\therefore (\mu +m\Delta \mu )\sin 30^{\circ }=1.26\sin 90^{\circ },$
$\therefore (1.26+m\times 0.012)\times \frac{1}{2}=1.26\times 1$
$\therefore \frac{1.26}{2}+m\times 0.006=1.26$
$\therefore m=\frac{1.26}{2\times 0.006}=105$
$\frac{{\mu }_2}{{\mu }_1}=\frac{\sin i}{\sin r}$
$\therefore {\mu }_1\sin i={\mu }_2\sin r$
$\therefore (\mu +m\Delta \mu )\sin 30^{\circ }=1.26\sin 90^{\circ },$
$\therefore (1.26+m\times 0.012)\times \frac{1}{2}=1.26\times 1$
$\therefore \frac{1.26}{2}+m\times 0.006=1.26$
$\therefore m=\frac{1.26}{2\times 0.006}=105$