The length of the optical path of two media in contact of lengths $d_1$ and $d_2$ with refractive indices $\mu_1$ and $\mu_2$ respectively,is

$A$ small coin is fixed at the centre of the base of an empty cylindrical steel container having radius $R=1 \,m$ and height $d=4 \,m$. At time $t=0$,the container starts getting filled with water at a flow rate of $Q=0.1 \,m^3/s$ without disturbing the coin. Find the approximate time $t$ in seconds when the coin will first be seen by the observer $O$ from a height of $H=5.75 \,m$ above and $L=1.5 \,m$ radially away from the coin as shown in the figure. (Take refractive index of water,$n=1.33$ or $4/3$)

$A$ vessel of height $2d$ is half-filled with a liquid of refractive index $\sqrt{2}$ and the other half with a liquid of refractive index $n$ (the given liquids are immiscible). Then the apparent depth of the inner surface of the bottom of the vessel (neglecting the thickness of the bottom of the vessel) will be

$A$ light ray is incident on a glass slab at an angle of incidence of $60^o$. If the angle between the reflected ray and the refracted ray is $90^o$,what is the refractive index of the glass slab?

$A$ light beam is travelling from Region $I$ to Region $IV$ (Refer Figure). The refractive indices in Regions $I$,$II$,$III$,and $IV$ are $n_0$,$\frac{n_0}{2}$,$\frac{n_0}{6}$,and $\frac{n_0}{8}$,respectively. The angle of incidence $\theta$ for which the beam just misses entering Region $IV$ is:

The refractive indices of water and glass with respect to air are $1.2$ and $1.5$ respectively. The refractive index of glass with respect to water is