$A$,$B$ and $C$ are parallel-sided transparent media of refractive indices $n_{1}$,$n_{2}$ and $n_{3}$ respectively. They are arranged as shown in the figure. $A$ ray is incident at an angle $i$ on the surface of separation of $A$ and $B$. After refraction into medium $B$,the ray grazes the surface of separation of media $B$ and $C$. Then,$\sin i$ equals to

$A$ glass slab consists of thin uniform layers of progressively decreasing refractive indices $(RI)$ such that the $RI$ of any layer is $\mu - m \Delta \mu$. Here, $\mu$ and $\Delta \mu$ denote the $RI$ of the $0^{\text{th}}$ layer and the difference in $RI$ 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 $0^{\text{th}}$ layer enters the $1^{\text{st}}$ layer at an angle of incidence of $30^{\circ}$. After undergoing the $m^{\text{th}}$ refraction, the ray emerges parallel to the interface. If $\mu = 1.5$ and $\Delta \mu = 0.015$, the value of $m$ is:

$A$ light beam is incident on a denser medium whose refractive index is $1.414$ at an angle of incidence $45^o$. Find the ratio of the width of the refracted beam in the medium to the width of the incident beam in air.

$A$ concave mirror is placed on a horizontal surface and two thin uniform layers of different transparent liquids (which do not mix or interact) are formed on the reflecting surface. The refractive indices of the upper and lower liquids are $\mu_1$ and $\mu_2$ respectively. The bright point source at a height $d$ ($d$ is very large in comparison to the thickness of the film) above the mirror coincides with its own final image. The radius of curvature of the reflecting surface is

$A$ mark at the bottom of a liquid appears to rise by $0.1 \ m$. The depth of the liquid is $1 \ m$. The refractive index of the liquid is

Corpuscular theory of light predicts the speed of light to be