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 $m=0,1,2,3 \ldots$. 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

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}, $
$\therefore (1.26+ m \times 0.012) \times \frac{1}{2}=1.26 \times 1$
$\therefore \frac{1.26}{2}+ m \times 0.006=1.26$
$\therefore m =\frac{1.26}{2 \times 0.006}=105$