In the given figure, if observer sees the bottom of vessel at $8 \mathrm{cm}$, find the refractive index of the medium in which observer is present.

 

A man starting from point $P$ crosses a $4 \mathrm{km}$ wide lagoon and reaches point $Q$ in the shortest possible time by the path shown in the figure. If the person swims at a speed of $3 \mathrm{km} {\mathrm{h}}^{-1}$ and walks at a speed of $4 \mathrm{km} {\mathrm{h}}^{-1}$, then his time in minutes of journey is $\left({\mu }_{\mathrm{salt}water}=\frac{4}{3}\right)$:

In the given figure, the faces of the prism $ABCD$ made of glass with a refractive index $n$ form dihedral angles $\angle A=90°, \angle \mathrm{B}=75°, \angle \mathrm{C}=135° \& \angle \mathrm{D}=60°$ (The Abbe's prism ). A beam of light falls on the face $\mathrm{AB}$ & after total internal reflection from face, $BC$ escapes through face $AD$. Find the range of $n$ and angle of incidence $\mathrm{\alpha }$ of the beam onto face $AB$, if a beam that has passed through the prism in this manner is perpendicular to the incident beam.

A point source of light is placed at a distance h below the surface of a large deep lake. (a) Show that the fraction $f$ of the light energy that escapes directly from the water surface is independent of h and is given by $f=\frac{1}{2}-\frac{1}{2n}\sqrt{n^2-1}$, where $n$ is the index of refraction of water.

(Note: Absorption within the water and reflection at the surface except where it is total, have been neglected)

(b) Evaluate this ratio for $n=\frac{4}{3}$.

A glass prism with a refracting angle of $60°$ has a refractive index $1.52$for red and $1.6$ for violet light. A parallel beam of white light is incident on one face at an angle of incidence which gives minimum deviation for red light. Find:

(a) the angle of incidence

(b) angular width of the spectrum

(c) the length of the spectrum if it is focussed on a screen by lens of focal length $100 \mathrm{cm}$.

In the given figure, $O$ is a point object kept on the principal axis of a concave mirror $M$ of radius of curvature $20 \mathrm{cm}$. $P$ is a prism of angle $\alpha =1.8°$. Light falling on the prism (at a small angle of incidence) gets refracted through the prism and then falls on the mirror. The refractive index of the prism is $3/2$. Find the distance between the images formed by the concave mirror due to this light.

In the figure shown $L$ is a converging lens of focal length $10 \mathrm{cm}$ and $M\mathit{ }$is a concave mirror of radius of curvature $20 \mathrm{cm}$. A point object $O$ is placed in front of the lens at a distance $15 \mathrm{cm}$. $AB$ and $CD$ are optical axes of the lens and mirror respectively. Find the distance of the final image formed by this system from the optical center of the lens. The distance between $CD \& AB$ is $1 \mathrm{cm}$.