$A$ paraxial beam is incident on a glass $(n = 1.5)$ hemisphere of radius $R = 6\, cm$ in air as shown. The distance of the point of convergence $F$ from the plane surface of the hemisphere is......$cm$.

The time required for the light to pass through a glass slab (refractive index $= 1.5$) of thickness $4 \,mm$ is ($c = 3 \times 10^{8} \,m/s$, speed of light in free space).

$A$ ball is dropped from a height of $20 \,m$ above the surface of water in a lake. The refractive index of water is $4/3$. $A$ fish inside the lake, in the line of fall of the ball, is looking at the ball. At an instant, when the ball is $12.8 \,m$ above the water surface, the fish sees the speed of the ball as (Take $g = 10 \,m/s^2$): (in $\,m/s$)

The figure shows a transparent slab of length $1 \, m$ placed in air,whose refractive index in the $x$-direction varies as $\mu = 1 + x^2$ for $0 \leq x \leq 1$. The optical path length of ray $R$ will be:

Light travels through a glass plate of thickness $t$ and having refractive index $\mu$. If $c$ is the velocity of light in vacuum, the time taken by the light to travel through this thickness of glass is:

$A$ microscope is focused on a mark on a piece of paper and then a slab of glass of thickness $3\, cm$ and refractive index $1.5$ is placed over the mark. How should the microscope be moved to get the mark in focus again?