A ray $R_1$ is incident on the plane surface of the glass slab (kept in air) of refractive index $\sqrt{2}$ at an angle of incidence equal to the critical angle for this air-glass system. The refracted ray $R_2$ undergoes partial reflection and refraction at the other surface. The angle between the reflected ray $R_3$ and the refracted ray $R_4$ at that surface is

 

One face of a rectangular glass slab of the thickness $6 \mathrm{cm}$ is silvered. A point object is placed at $8 \mathrm{cm}$ in front of the unsilvered face and its image is formed $10 \mathrm{cm}$ behind the silvered face. If the refractive index of glass is $\frac{x}{10}$ then find the value of $x$.

An air bubble is formed inside water. The refractive index of water is $\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$. At what distance from the air bubble should a point object be placed, so as to form a real image at the same distance from the bubble?    

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

A ray of light falls on a transparent sphere with centre at $C$ as shown in figure. The ray emerges from the sphere parallel to line $AB$. Find the refractive index of the sphere.

In the given figure, if object is placed at a distance of $9 \mathrm{cm}$ from $O_1$, then find the image distance $\left(\mathrm{in} \mathrm{cm} \right)$ from $O_2$•

A point object is placed at a distance of $3R$ from point $P$ on glass hemisphere of radius $R$. If refractive index of hemisphere is $1.5$, then position of its image from $C$ is 

The observer $O$ sees the distance $AB$ as infinitely large. If the refractive index of the liquid is ${\mu }_1$ and that of glass is ${\mu }_2$, then $\frac{{\mu }_1}{{\mu }_2}$ is 

In the figure shown below, an air lens of radii of curvature $10 \mathrm{cm}$ $\left(R_1=R_2=10 \mathrm{cm}\right)$ is cut in a cylinder of glass $\left(\mu =1.5\right)$. The focal length and the nature of the lens is