The locus of all such particles of the medium which are vibrating in the same phase is called a wavefront.
Question 2 Give the relation between path difference and wavelength for constructive interference between two waves. Solution
For constructive interference the path difference between two waves must be an integral multiple of \(\lambda\) i.e., \(p=n\lambda, \,\, n=0,1,2,3,......\)
Question 3 State two conditions to obtain sustained interference of light Solution
The conditions for obtaining sustained interference of light are
(a) the two light sources should be coherent
(b) the two light sources should be narrow and placed close to each other
Question 4 If the separation between the two slits is decreased in Young's double-slit experiment keeping the screen position fixed. What will happen to the fringe width? Solution
Fringe width \(\beta=\frac{D\lambda}{d}\)
As the separation \(d\) between two slits decrease, fringe width \(\beta\) decrease.
Question 5 Why are coherent sources necessary to produce a sustained interference pattern? Solution
Coherent sources have a constant phase difference. This ensures that the position of Maxima and minima do not change with time i.e., a sustained interference pattern is obtained.
Question 6 Sunglasses are made of Polaroid and not colored glasses. Why? Solution
Polaroid absorbs only that part of the light which produces a dazzling effect in the eye. But colored glasses absorb more light incident on them. This makes the image appears to be dim.
Question 7 Why does the intensity of a secondary maximum become less as compared to the central maximum? Solution
The intensity of light of a secondary maxima decreases with the order of maximum. This happens because, the intensity of central maximum is due to wavelets from all parts of the slit. First secondary maxima is formed due wavelets from one third parts of the slit and second secondary maxima is due to wavelets from one fifth part of the slit and so on. This is the reason why intensity of secondary maxima becomes less as compared to central maxima.
Question 8 Will ultrasonic big show any polarization? Give a reason for your answer. Solution
no ultrasonic waves are longitudinal in nature so they cannot be polarized.
Question 9 Why longitudinal waves cannot be polarized? Solution
In polarization vibrations perpendicular to the direction of propagation is restricted to just one direction this is possible in transverse waves that have such vibrations. in longitudinal waves, vibrations occur along the direction of propagation. So their polarization is not possible.
Answer Question 10 Is there any difference between the colors emerging from a prism and the colors of a soap film seen in sunlight? Solution
Yes. In the prism, colors are produced due to the dispersion of light. The colors of a soap film are due to the interference of light.
Short answers type questions
Question 1 How is a wavefront different from a ray? Draw the geometrical shape of the wavefront when
(a) light diverges from a point source and
(b) light emerges out of convex lens when a point source placed at its focus Solution
A wavefront is a surface obtained by joining all points vibrating in the same phase. A ray is a line drawn perpendicular to the wavefront in the direction of propagation of light waves.
(a) light diverges from a point source
(b) light emerges out of convex lens when a point source placed at its focus
Question 2 Two narrow slits are illuminated by a single monochromatic source. Name the pattern obtained on the screen. One of the slits is now completely covered what is the name of the pattern now obtained on the screen. Draw intensity pattern obtained in the two cases. Also, write two differences between the pattern obtained in the above two cases. Solution
With narrow slits an interference pattern is obtained.
When one slit is completely covered diffraction pattern is obtained.
Intensity distribution curve for interference
Intensity distribution curve for diffraction
Interference
Diffraction
Fringes are equal in width
Fringes are not equal in width
The pattern is formed due to superposition of two wavefronts
The pattern is formed due to superposition of secondary wavelets from different parts of the same wavefront
Maxima occur at \(\theta =\frac{\lambda}{a}\)
Minima occur at \(\theta =\frac{\lambda}{a}\)
Fringes are of equal intensity
Fringes are of decreasing intensity
Question 3 What two main changes in the diffraction pattern of the single slit will you observe when the monochromatic source of light is replaced by a source of white light. Solution
When monochromatic source is replaced by a source of white light the diffraction pattern shows following changes (i) In each diffraction order the, diffracted image of the slit gets dispersed into component colors of white light. As fringe width is proportional to wavelength so, the red fringe with higher wavelength is wider than the violet fringe with smaller wavelength. (ii) In higher-order spectra, the dispersion is more and it causes overlapping of different colors.
Question 4 When a sheet of transparent plastic is placed between two crossed polarizers, no light is transmitted. When the sheet is stretched in one direction, some light passes through the crossed polarizers. What is happening? Solution
The transparent plastic sheet is not a Polaroid. So when two Polaroids are placed with crossed axes, no light is transmitted, whether the plastic sheet it is placed between them or not. But when the sheet is a stretched the polymer molecules in it make it a polaroid with its own polaroid Axis which may make some angle with the axis of two polaroids it becomes a case of three polaroids with the middle polaroid having its Axis between the axes of two fixed polaroids that is why some light is transmitted in this case.
Question 5 Show that when a Ray of light is incident on the surface of a transparent medium at the polarizing angle, the reflected and transmitted rays are perpendicular to each other. Solution
From Snell's law,
\[\frac{\sin i_P}{\sin r_P}=\mu \]
From Brewster's Law,
\[\(\tan i_p=\frac{\sin i_P}{\cos i_P}=\mu
\\
\frac{\sin i_P}{\sin r_P}=\frac{\sin i_P}{\cos i_P}
\\
or, \sin r_P=\cos i_P
\\
\rightarrow \,\,\sin r_P=\sin \left( 90^{\circ}-i_P \right)
\\
Therefore, r_P=\left( 90^{\circ}-i_P \right)
\\
\rightarrow \,\,i_P+r_P=90^{\circ}\) \]
Hence, the reflected and transmitted rays are perpendicular to each other.
Question 6 Define critical angle and polarizing angle. What is the relation between the two angles. Solution
Critical Angle \(i_C\) The angle of incidence in the denser medium, at which the angle of refraction in the rarer medium is \(90^{\circ}\)is called the critical medium of the denser medium. Polarizing Angle \(i_P\) It is the angle of incidence, at which when unpolarized light is incident on a transparent medium, the reflected light becomes completely plane polarized.
The relation between two angles is \(\tan i_p=\frac{1}{\sin i_C}\)
Multiple Choice Questions
Question 1
To demonstrate the phenomenon of interference, we require two sources, which emit radiation of
(a) nearly the same frequency
(b) the same frequency
(c) different wavelength
(d) the same frequency and have a definite phase relationship Solution
(d) To observe interference, the two sources must emit radiation of the same frequency and have a definite phase relationship.
Question 2
The maximum number of possible interference maxima for slit separation equal to twice the wavelength in Young’s double-slit experiment is
(a) infinite
(b) five
(c) three
(d) zero Solution
(b) The condition for possible interference maxima on screen is
\(d=\sin \theta = n\lambda\)
Given that, \(d=2\lambda\)
Therefore,
\(2\lambda \sin \theta = n\lambda\)
Or, \(2\sin \theta =n\)
For a number of interference maxima to be maximum,
\(\sin \theta = 1\)
Therefore, \(n=2\)
The interference maxima would be formed when
\(n=0, \pm 1, \pm 2\)
Hence the maximum number of possible interference maxima =5
Question 3
When an unpolarized light intensity \(I_0\) is incident on a polarizing sheet, the intensity of the light which does not get transmitted is
(a) $\frac{I_0}{2}$
(b) $\frac{I_0}{4}$
(c) zero
(d) \(I_0\) Solution
(a)
Intensity transmitted by the polarizing sheet,
$I=I_0\cos ^2\theta $
The average value of \(\cos ^2\theta \) over one cycle (when \(\theta\) varies from \(0\) to \(2\pi\)) is \(\frac{1}{2}\).
$I=I_0\times \frac{1}{2}=\frac{I_0}{2}$
The intensity of light which does not get transmitted is
$I'=I_0-\frac{I_0}{2}$
Question 4
Select the right option in the following
(a) Christian Huygens, a contemporary of Newton's established the wave theory of light by assuming that light waves are transverse.
(b) Maxwell provided the theoretical evidence that light is a transverse wave.
(c) Thomas Young experimentally proved the wave behavior of light and Huygens assumption.
(d) All the statements given above correctly answer the question whet is light. Solution
(b) Maxwell proved theoretically that light has a transverse wave nature.
Question 5
In a double-slit experiment, the distance between slits is increased ten times whereas their distance from the screen is halved. What is the fringe width?
(a) It remains the same
(b) it becomes 1/10
(c) it becomes 1/20
(d) it becomes 1/90 Solution
Question 6 The colors seen in the reflected white light from a thin oil film are due to
(a) diffraction
(b) interference
(c) polarization
(d) dispersion Solution
(b) the colors of a thin film are due to interference.
Question 7 In diffraction from a single slit, the angular width of the central maxima does not depend on
(a) \(\lambda\) of light used
(b) width of slit
(c) the distance of slit from screen
(d) ratio of \(\lambda\) and slit width. Solution
(c) Angular width of central maximum in a diffraction pattern $=\frac{2\lambda}{d}$
It does not depend on the distance of the slit from the screen. Question 8 Which of the following can measure the position of particle most accurately?
(a) polarized light
(b) light with high wavelength
(c) light with low wavelength
(d) none of the above. Solution
(c)
Question 9
We can obtain polarized light with the help of which of the following instrument?
(a) Nicol Prism
(b) bi prism
(c) Polarimeter
(d) none of the above Solution
(a) A Nicole prism is used to obtain polarized light.
Question 10 If in Young's double-slit experiment of light interference is performed in water, which of the following is correct?
(a) fringe width will decrease
(b) fringe width will increase
(c) there will be no fringe
(d) fringe width will remain unchanged Solution
(a) The wavelength of light in water, \(\left( \lambda'=\frac{\lambda}{\mu} \right) \) is less than that in air. When the apparatus is immersed in water, the fringe width \(\left( \beta \propto \lambda' \right) \) will decrease.
Question 11 Interference occurs in which of the following waves
(a) longitudinal
(b) transverse
(c) electromagnetic
(d) all of these Solution
(d) interference occurs in all types of waves
Question 12 A monochromatic beam of light is used for the formation of fringes on the screen by illuminating the two sits in Young's double-slit experiment. When a thin film of mica is interposed in the path of one of the interfering beams, then
(a) the fringe width increases
(b) the fringe width decreases
(c) the fringe width remains same but the pattern shifts
(d) the fringe pattern disappears Solution
(c) When a thin film of mica is inserted in the path of one beam, the entire fringe pattern shifts towards the side in which the film is inserted.
Question 13 When exposed to sunlight, thin films of oil on water often exhibit brilliant colors due to the phenomenon of
(a) interference
(b) diffraction
(c) dispersion
(d) polarization Solution
(a)
Question 14 The ratio of intensity of two waves is given by 4:1. Then ratio of amplitudes of two waves is
(a) 2:1
(b) 1:2
(c) 4:1
(d) 1:4 Solution
Question 15 In a Fresnel bi-prism experiment, the two positions of lense gives separation between the slits as 16 cm and 9 cm respectively. What is the actual distance of separation.
(a) 12.5 cm
(b) 12 cm
(c) 13 cm
(d) 14 cm Solution
(b) In Fresnel bi-prism experiment, the actual distance of separation between two slits,
$d=\sqrt{d_1d_2}=\sqrt{16\times 9}=1212 cm.$
Question 16 Which of the following phenomenon is not common to sound and light waves?
(a) interference
(b) diffraction
(c) coherence
(d) polarization Solution
(d) Sound waves are longitudinal waves and they can not be polarized. Light waves can be polarized because they are transverse waves.
Question 17 Which of the following statements is true?
(a) both light and sound waves can travel in vacuum
(b) both light and sound waves in air are transverse
(c) the sound waves in air are longitudinal, while light waves are transverse
(d) both light and sound waves in air are longitudinal Solution
(c) Sound waves are longitudinal and they can not travel through vacuum. Light waves are transverse and can travel through vacuum.
Question 18 Laser beam is coherent because it contains
(a) waves of several wavelength
(b) incoherent waves of single wavelength
(c) coherent waves of several wavelengths
(d) coherent waves of single wavelength Solution
(d) A laser beam is coherent because it contains coherent waves of single wavelength.
Question 19 Resolving power of a telescope can be increased by increasing
(a) the wavelength
(b) the diameter of objective
(c) the diameter of eyepiece
(d) the focal length of eyepiece Solution
(b) Resolving power of telescope $=\frac{D}{1.22\lambda}$
R.P. of telescope can be increased by increasing diameter \(D\) of the objective.
Question 20 A diffraction pattern is obtained by using a beam of red light. What will happen if red light is replaced by blue light?
(a) bands disappear
(b) bands become broader and farther apart
(c) no change will take place
(d) diffraction bands become narrower and crowded together. Solution
(d) Directions of various minima in diffraction pattern are given by
\(\theta _n=\frac{n\lambda}{d}\)
We know that wavelength of blue light is less then that of red light. So diffraction bands become narrower and crowded together.
Long answer type questions
Question 1
(a) Use Huygens's geometrical construction to show how a plane wavefront at \(t=0\) propagates and produces a wavefront at a later time.
(b) Verify using Huygens's principal Snell's law of refraction of a plane wave propagating from a denser to a rarer medium.
(c) Illustrate with the help of diagrams the action of a convex lens and concave mirror on a plane wavefront incident on it Question 2
(i) In a Young's double-slit experiment, deduce the conditions for constructive and destructive interference. Write the expression for the distance between two consecutive bright or dark fringes.
(ii) What changes in the interference pattern do you observe if the two slits \(S_1\) and \(S_2\) are taken as point sources?
(iii) Plot a graph of the intensity distribution vs. path difference in this experiment. Compare this with the intensity distribution of fringes due to diffraction at a single slit. What important difference do you observe? Question 3
(i) State Huygens's principle. Using this principle explain, how a diffraction pattern is obtained on a screen due to a narrow slit on which and narrow beam coming from a monochromatic source of light is incident normally.
(ii) Show that the angular width of the first diffraction fringe is half of that of the central fringe.
(iii) If a monochromatic source of light is replaced by white light what changes would you observe in the differential pattern. Question 4
(i) How does one demonstrate using a suitable diagram that unpolarized light, when passed through a Polaroid, get polarized.
(ii) A beam of unpolarized light is incident on a glass-air interface. Show using a suitable ray diagram that light reflected from the interface is totally polarized when \(\mu =\tan i_B\) where \(\mu\) is the refractive index of glass with respect to air and \(i_B\) is the Brewster's angle.
Numerical Type Questions
Question 1 Two identical coherent waves each of intensity \(I\), are producing an interference pattern. Write the value of resultant intensity at a point of (i) constructive interference and (ii) destructive interference. Solution
\(I_R=I_1+I_2+2\sqrt{I_1I_2}\cos \phi \\I_1=I_2=I\)
(i) For constructive interference
\(\cos \phi =1, I_{max}=4I\)
(ii) For destructive interference
\(\cos \phi =-1, I_{\min}=0\)
Question 2 If the angle between the planes of the polarizer and analyzer is \(60^{\circ}\), by what factor does the intensity of transmitted light change when passing through the analyzer? Solution
\[I=I_0\cos ^2\theta \\
\frac{I}{I_0}=\cos ^260^{\circ}=\left( \frac{1}{2} \right) ^2\\
\frac{I}{I_0}=\frac{1}{4}\]
i.e., it reduces to one-fourth
Question 3 The ratio of intensity of maxima and minima in an interference pattern is 100:64. Calculate the ratio of intensities of the coherent sources producing the pattern. Solution
The ratio of intensity of maxima and minima is
\[\frac{I_{\max}}{I_{\min}}=\frac{\left( \sqrt{I_1}+\sqrt{I_2} \right) ^2}{\left( \sqrt{I_1}-\sqrt{I_2} \right) ^2}=\frac{\left( A_1+A_2 \right) ^2}{\left( A_1-A_2 \right) ^2}\\
\frac{I_{\max}}{I_{\min}}=\frac{\left( A_1+A_2 \right) ^2}{\left( A_1-A_2 \right) ^2}=\frac{100}{64}\\
\Rightarrow \frac{\left( A_1+A_2 \right)}{\left( A_1-A_2 \right)}=\frac{10}{8}\\
\Rightarrow 8A_1+8A_2=10A_1-10A_2\\
\Rightarrow \frac{A_1}{A_2}=\frac{9}{1}\]
As,
\(\frac{I_1}{I_2}=\left( \frac{A_1}{A_2} \right) ^2\\\frac{I_1}{I_2}=\frac{81}{1}\)
Question 4 Draw a diffraction pattern due to single slit illuminated by monochromatic source of light.
Light of wavelength 500 nm, falls from a distant source of slit 0.50 mm wide. Find the distance between the two dark bands, on either side of the central bright band of the diffraction pattern observed on a screen placed 2m from the slit. Solution
Given that,
\(\lambda =500\times 10^{-9}m\\d=0.50\times 10^{-3}m\\D=2m\)
Position of first dark band
\(\sin \theta _1=\frac{\lambda}{d}\,\,or, \tan \theta _1=\frac{x}{D}\)
For small angle \(\sin \theta _1\approx \theta _1, \tan \theta _1\approx \theta _1\)
i.e., \(x=\frac{\lambda D}{d}\)
Separation between two dark bands
\(2x=\frac{2\lambda D}{d}=\frac{2\times 500\times 10^{-9}\times 2}{0.50\times 10^{-3}}=0.0044\times 10^{-3}m\)
Question 5 Laser light of wavelength 630 nm incident on a pair of slits produces an interference pattern in which the bright fringes are separated by 8.1 mm. A second light produces an interference pattern in which the fringes are separated by 7.2 mm. Calculate the wavelength of the second light. Solution
Question 6 Find the ratio of intensities of two points \(P\) and \(Q\) on a screen in Young’s double slit experiment when waves from sources \(S_1\) and \(S_2\) have phase difference of (i) \(0^0\) and (ii) \(\frac{\pi}{2}\) respectively. Solution
When rays are out of phase at \(0^0\), we get constructive interference and intensity obtained would be maximum.
\(I_1=I_P+I_Q+2\sqrt{I_PI_Q}\cos \phi \)
Here we consider \(I_P=I_Q=I\)
So, \(I_1=I+I+2I=4I\)
When they are out of phase by \(\frac{\pi}{2}\)
\(I_2=I+I+0=2I\)
\(\therefore \,\,I_1: I_2=4:2=2:1\)
Question 6 In Young’s Double Slit experiment using monochromatic light of wavelength \(\lambda\), the intensity of light at a point on screen where path difference is \(\lambda\); is \(K\) units. What is the intensity of light at a point where path difference is \(\frac{\lambda}{3}\)? Solution
Intensity of light at any point on the screen is given by
\(I_R=I_1+I_2+2\sqrt{I_1I_2}\cos \phi \)
(i) For a path difference of \(\lambda\) there is a phase difference of \(2\pi\)
As sources are coherent and are taken out of the same source in the experiment so we have
\[I_1=I_2=I\\
I_R=2I+2I\cos 2\pi \\
I_R=4I \tag{1}\]
Or, \(4I=K\,\,units\)
(ii) When path difference is \(\frac{\lambda}{3}\) phase difference is \(\frac{2\pi}{3}\).
\[I_{R}^{'}=2I+2I\cos \frac{2\pi}{3}=2I-I=I \tag{2}\]
From equation (1) and (2) we have
\(I_{R}^{'}=\frac{K}{4}\,\,units\)
Question 7
Interference pattern is obtained with two coherent light sources of intensity ratio n. Show that in the interference pattern
$\frac {(I_{max}-I_{min})}{(I_{max}+I_{min})}=2 \sqrt {\frac {n}{n+1}}$
