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

Two identical monochromatic light sources $A\&B$ intensity ${10}^{-15} \mathrm{W} {\mathrm{m}}^2$ produce wavelength of light $4000\sqrt{3} \stackrel{°}{\mathrm{A}}$. A glass of thickness $3 \mathrm{mm}$ is placed in the path of the ray as shown in figure. The glass has a variable refractive index $n=1+\sqrt{x}$ where $x$ (in mm) is distance of plate from left to right. Calculate total intensity at focal point $F$ of the lens.

Two slits $S_1$ and $S_2$ on the $x$-axis and symmetric with respect to y-axis are illuminated by a parallel monochromatic light beam of wavelength $\lambda$. The distance between the slits is $d(>>\lambda )$. Point $M$ is the mid point of the line $S_1{ S}_2$ and this point is considered as the origin. The slits are in horizontal plane. The interference patten is observed on a horizontal plate (acting as screen) of mass $M$, which is attached to one end of a vertical spring of spring constant $K$. The other end of the spring is fixed to ground. At $t=0$ the plate is at a distance $D(>>d)$ below the plane of slits and the spring is in its natural length. The plate is left from rest from its initial position. Find the $x$ and $y$ co-ordinates of the $n^{th }$  maxima on the plate as a function of time. Assume that spring is light and plate always remains horizontal.

In a YDSE a parallel beam of light of wavelength $6000\stackrel{°}{\mathrm{A}}$ is incident on slits at angle of incidence $30°$. $A$ and $B$ are two thin transparent films each of refractive index $1.5$. Thickness of $A$ is $20.4 \mathrm{\mu m}$. Light coming through $A$ and $B$ have intensities $I$ and $4I$ respectively on the screen. Intensity at point $O$ which is symmetric relative to the slits is $3I$. The central maxima is above $O$.
(i) What is the maximum thickness of $B$ to do so.  Assuming thickness of $B$ to be that found in part (i) answer the following parts.

(ii) Find fringe width, maximum intensity and minimum intensity on screen.

(iii) Distance of the nearest minima from $O$.

(iv) Intensity at $5 \mathrm{cm}$ on either side of $O$.

A screen is at a distance $D=80 \mathrm{cm}$ from a diaphragm having two narrow slits $S_1$ and $S_2$ which are $d=2 \mathrm{mm}$ apart. Slit $S_1$ is covered by a transparent sheet of thickness $t_1=2.5 \mathrm{\mu m}$ and $S_2$ by another sheet of thickness $t_2=1.25 \mathrm{\mu m}$ as shown in figure. Both sheets are made of same material having refractive index $\mu =1.40$.

Water is filled in space between the diaphragm and screen. A monochromatic light beam of wavelength $\mathrm{\lambda }=5000 \stackrel{°}{\mathrm{A}}$ is incident normally on the diaphragm. Assuming intensity of beam to be uniform and slits of equal width, calculate ratio of intensity at C to maximum intensity of interference pattern obtained on the screen, where $C$ is foot of perpendicular bisector of $S_1{S}_2$. (Refractive index of water, ${\mathrm{\mu }}_{\mathrm{w}}=\frac{4}{3}$ )

Two plane mirrors, a source $S$ of light, emitting monochromatic rays of wavelength $\lambda$ and a screen are arranged as shown in figure. If angle $\theta$ is very small, calculate fringe width of the interference pattern formed by reflected rays.

Two parallel beams of light $P$ and $Q$ (separation $d$) containing radiations of wavelengths $4000\stackrel{°}{\mathrm{A}}$ and $5000\stackrel{°}{\mathrm{A}}$ (which are mutually coherent in each wavelength separately) are incident normally on a prism as shown in figure The refractive index of the prism as a function of wavelength is given by the relation, $\mu \left(\lambda \right)=1.20+\frac{b}{{\lambda }^2}$, where $\lambda$ is in $\stackrel{°}{\mathrm{A}}$ and $b$ is a positive constant.

The value of $b$ is such that the condition for total reflection at the face $AC$ is just satisfied for one wavelength and is not satisfied for the other. A convergent lens is used to bring these transmitted beams into focus. If the intensities of the upper and the lower beams immediately after transmission from the face $AC$, are $4I$ and $I$ respectively, find the resultant intensity at the focus.

A narrow monochromatic beam of light of intensity $I$ is incident on a glass plate as shown in figure. Another identical glass plate is kept close to the first one & parallel to it. Each glass plate reflects $25\%$ of the light incident on it & transmits the remaining. Find the ratio of the minimum & the maximum intensities in the interference pattern formed by the two beams obtained after one reflection at each plate.

Monochromatic radiation of wavelength ${\lambda }_1=3000\stackrel{.}{A}$ f falls on a photocell operating in saturating mode. The corresponding spectral sensitivity of photocell is $J=4.8\times {10}^{-3} \frac{A}{W}$. When another monochromatic radiation of wavelength ${\lambda }_2=1650\stackrel{.}{A}$ and power $P=5\times {10}^{-3} W$ is incident, it is found that maximum velocity of photoelectrons increases $n=2$ times. Assuming efficiency of photoelectron generation per incident photon to be same for both the cases, calculate

(i) threshold wavelength for the cell.

(ii) saturation current in second case.