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

(a) at the farther surface of the sphere,

(b) at the centre of the sphere.

A quarter cylinder of radius $R$ and refractive index $1.5$ is placed on a table. A point object $P$ is kept at a distance of $mR$ from it. Find the value of $m$ for which a ray from $P$ will emerge parallel to the table as shown in the figure.

The focal length of lens shown in the figure. Solve for three cases $n_{\mathrm{s}}=1.5, n_{\mathrm{s}}=2.0, n_{\mathrm{s}}=2.5.$

A thin lens, made of a material of refractive index ${\mu }_2$ has a medium of refractive index ${\mu }_1$ on one side and a medium of refractive index ${\mu }_3$ on the other side. The lens is biconvex and the two radii of curvature have equal magnitudes, $R$. A beam of light travelling parallel to the principal axis is incident on the lens. Where will the image be formed, if the beam is incident from

(a) the medium ${\mu }_1$and

(b) from the medium ${\mu }_3$?

Two glasses with refractive indices of $1.5 \mathrm{and} 1.7$ are used to make two identical double-convex lenses.

(i) Find the ratio of their focal lengths.

(ii) How will each of these lenses act on a ray parallel to its optical axis if the lenses are submerged.