$A$ 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

$A$ beaker containing liquid is placed on a table,underneath a microscope which can be moved along a vertical scale. The microscope is focused,through the liquid,onto a mark on the table when the reading on the scale is $a$. It is next focused on the upper surface of the liquid and the reading is $b$. More liquid is added and the observations are repeated; the corresponding readings are $c$ and $d$. The refractive index of the liquid is

If a wave gets refracted into a denser medium,then which of the following is true?

If the angle of incidence is twice the angle of refraction in a medium of refractive index $\mu$,then the angle of incidence is

$A$ monochromatic light is travelling in a medium of refractive index $n=1.6$. It enters a stack of glass layers from the bottom side at an angle $\theta=30^{\circ}$. The interfaces of the glass layers are parallel to each other. The refractive indices of different glass layers are monotonically decreasing as $n_m=n-m \Delta n$,where $n_m$ is the refractive index of the $m^{\text{th}}$ slab and $\Delta n=0.1$ (see the figure). The ray is refracted out parallel to the interface between the $(m-1)^{\text{th}}$ and $m^{\text{th}}$ slabs from the right side of the stack. What is the value of $m$?

$A$ beam of light has a small wavelength spread $\delta \lambda$ about a central wavelength $\lambda$. The beam travels in vacuum until it enters a glass plate at an angle $\theta$ relative to the normal to the plate,as shown in the figure. The index of refraction of the glass is given by $n(\lambda)$. The angular spread $\delta \theta'$ of the refracted beam is given by