Embibe Experts Solutions for Chapter: Ray Optics, Exercise 4: Exercise - 4

A pole of length $2.00 \mathrm{m}$ stands half dipped in a swimming pool with water level $1 \mathrm{m}$ higher than the bed (bottom). The refractive index of water is $4/3$ and sunlight is coming at an angle of $37°$ with the vertical. Find the length of the shadow of the pole on the bed. Use${\sin }^{–1}(0.45)=26.8°, \tan (26.8°)=0.5.$

A fly $F$ is sitting on a glass slab $S, 45\mathrm{cm}$ thick and of refractive index $3/2$. The slab covers the top of a container $C$ containing water $(\mathrm{R}.\mathrm{I}.= 4/3)$upto a height of $20 \mathrm{cm}$. The bottom of the container is closed by a concave mirror $M$ of radius of curvature $40 \mathrm{cm}$. Locate the final image formed by all refractions and reflection assuming paraxial rays.

A glass porthole is made at the bottom of a ship for observing sea life. The hole diameter $D$ is much larger than the thickness of the glass. Determine the area $S$ of the field of vision at the sea bottom for the porthole if the refractive index of water is ${\mathit{\mu }}_{\mathrm{w}}$ and the sea depth is $h$.

The figure below depicts a concave mirror with centre of curvature $C$ focus $F$, and a horizontally drawn $OFC$ as the optic axis. The radius of curvature is $R(OC=R)$ and $OF=\frac{R}{2}$. A ray of light $QP$, parallel to the optical axis and at a perpendicular distance $w(w\le \frac{R}{2})$ from it, is incident on the mirror at $P$. It is reflected to the point $B$ on the optical axis, such that $BF=k$. Here $k$ is a measure of lateral aberration.

(a) Express $k$ in terms of $\left\{w,R\right\}$.

(b) Sketch $k \mathrm{vs} w \text{for }w\in [0, \frac{R}{2}]$

(c) Consider points $P_1 , P_2 ......P_n$ on the concave mirror which are increasingly further away from the optic centre $O$ and approximately equidistant from each other(see figure below). Rays parallel to the optic axis are incident at $P_1 , P_2 ,.......P_n$ and reflected to points on the optic axis. Consider the points where these rays reflected from $P_n , P_{n–1} , .....P_2$ intersect the rays reflected from $P_{n–1} , P_{n–2} , ..... P_1,$ respectively. Qualitatively sketch the locus of these points in figure below for a mirror (shown with solid line) with radius of curvature $2 \mathrm{cm}$.

A symmetrical converging convex lens of focal length $10 \mathrm{cm}$ & diverging concave symmetrical lens of focal length $- 20 \mathrm{cm}$ are cut from the middle and perpendicularly and symmetrically to their principal axis. The parts thus obtained are arranged as shown in the figure. Find the focal length $(\mathrm{in} \mathrm{cm})$ of this arrangement.

In the given figure, a stationary observer $O$ looking at a fish $F$ (in water of, $\mu =\frac{4}{3}$) through a converging lens of focal length $90 \mathrm{cm}$. The lens is allowed to fall freely from a the height $62.0 \mathrm{cm}$ with its axis vertical. The fish and the observer are on the principal axis of the lens. The fish moves up with constant velocity $100 \mathrm{cm} {\mathrm{s}}^{-1}$. Initially, it was at a depth of $44.0 \mathrm{cm}$. The velocity with which the fish appears to move to the observer at $t=0.2 \mathrm{s}$ is, $\left(x+\frac{3}{4}\right) \mathrm{m} {\mathrm{s}}^{-1}$. Find the value of $x. (g\mathit{=}10 \mathrm{m} {\mathrm{s}}^{-2} )$

A glass rod of the refractive index $1.50$of rectangular cross-section, $\left\{\mathrm{d}\times l\right\}$ is bent into a $“\mathrm{U}”$ shape see figure (A). The cross-sectional view of this rod is shown in figure (B).

The bent portion of the rod is semi-circular with inner and outer radii $R$and $R+d$, respectively. A parallel monochromatic beam of light is incident normally on face $ABCD$.

(a) Consider two monochromatic rays $r_1$ and $r_2$ in figure(B). State whether the following statements are True or False.

(b) Consider the ray $r_1$ whose point of incidence is very close to the edge $\mathrm{BC}$. Assume it undergoes total internal reflection at $p_1$. In cross-sectional view below, draw the trajectory of this reflected ray beyond the next glass-air boundary that it encounters.

(c) Obtain the minimum value of the ratio $\frac{R}{d}$ for which any light ray entering the glass normally through the face $ABCD$ undergoes at least one total internal reflection.

(d) A glass rod with the above computed minimum ratio of $\frac{R}{d}$, is fully immersed water of refractive index $1.33$. What fraction of light flux entering the glass through the plane surface $ABCD$ undergoes at least one total internal reflection?

The following figure shows the section $ABC$ of an equilateral triangular prism. A ray of light enters the prism along$\mathit{ }LM$ and emerges along $QD$. If the refractive index of the material of the prism is $1.6$, angle $LMN$ is