$A$ narrow parallel beam of light is incident paraxially on a solid transparent sphere of radius $r$ kept in air. What should be the refractive index if the beam is to be focused at the farther surface of the sphere?

An image is formed at a distance of $100 \ cm$ from the glass surface when light from a point source in air falls on a spherical glass surface with a refractive index of $1.5$. The distance of the light source from the glass surface is $100 \ cm$. The radius of curvature is (in $cm$)

Derive the relation between object distance $(u)$,image distance $(v)$,refractive indices of the media ($n_1$ and $n_2$),and the radius of curvature $(R)$ for a spherical refracting surface.

$A$ laser beam falls on a crystal ball of radius $R$ as shown in the figure. What is its refractive index?

$A$ luminous point object $O$ is placed at a distance $2R$ from the spherical boundary separating two transparent media of refractive indices $n_1$ and $n_2$ as shown,where $R$ is the radius of curvature of the spherical surface. If $n_1 = \frac{4}{3}$,$n_2 = \frac{3}{2}$ and $R = 10 \text{ cm}$,the image is obtained at a distance from $P$ equal to:

$A$ fish is at the centre of a spherical water-filled $(\mu = 4/3)$ fish bowl. $A$ child stands in air at a distance $2R$ ($R$ is the radius of curvature of the sphere) from the centre of the bowl. At what distance from the centre would the child's nose appear to the fish situated at the centre (in $R$)?