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Questions:

Case Study Questions

5 x 4 = 20

Interference is based on the superposition principle. According to this principle, at a particular point in the medium, the resultant displacement produced by a number of waves is the vector sum of the displacements produced by each of the waves. If two sodium lamps illuminate two pinholes S₁ and S₂. The intensities will add up and no interference fringes will be observed on the screen. Here the source undergoes abrupt phase change in times of the order of 10^-10 seconds.

(i) Two coherent sources of intensity 10 W/m²and 25 W/m² interfere to form fringes. Find the ratio of maximum intensity to minimum intensity

(a) 15.54

(b) 16.78

(c) 19.72

(d) 18.39

(ii) Which of the following does not show interference?

(a) Soap bubble

(b) Excessively thin film

(c) A thick film

(d) Wedge shaped film

(iii) In a Young's double-slit experiment, the slit separation is doubled. To maintain the same fringe spacing on the screen, the screen-to-slit distance D must be changed to

(a) 2D

(b) 4D

(c) D/2

(d) D/4

(iv) 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

(v) The resultant amplitude of a vibrating particle by the superposition of the two waves \(y_{1}=a \sin \left[\omega t+\frac{\pi}{3}\right] \text { and } y_{2}=a \sin \omega t \text { is }\)

(a) a

(b)\(\sqrt{2}\) a

(c) 2a

(d) \(\sqrt{3}\) a

The phenomenon of bending of light around the sharp corners and the spreading of light within the geometrical shadow of the opaque obstacles is called diffraction of light. The light thus deviates from its linear path. The deviation becomes much more pronounced, when the dimensions of the aperture or the obstacle are comparable to the wavelength of light.

(i) Light seems to propagate in rectilinear path because

(a) its spread is very large

(b) its wavelength is very small

(c) reflected from the upper surface of atmosphere

(c) it is not absorbed by atmosphere

(ii) In diffraction from a single slit the angular width of the central maxima does not depends on

(a) \(\lambda\) of light used	(b) width of slit
(c) distance of slits from the screen	(d) ratio of \(\lambda\) and slit width

(iii) For a diffraction from a single slit, the intensity of the central point is

(a) infinite

(b) finite and same magnitude as the surrounding maxima

(c) finite but much larger than the surrounding maxima

(d) finite and substantially smaller than the surrounding maxima

(iv) .Resolving power of telescope increases when

(a) wavelength of light decreases	(b) wavelength of light increases
(c) focal length of eye-piece increases	(d) focal length of eye-piece decreases

(v) In a single diffraction pattern observed on a screen placed at D metre distance from the slit of width d metre, the ratio of the width of the ce,ntral maxima to the width of other secondary maxima is

(a) 2: 1

(b) 1: 2

(c) 1: 1

(d) 3: 1

Huygen's principle is the basis of wave theory of light. Each point on a wavefront acts as a fresh source of new disturbance, called secondary waves or wavelets. The secondary wavelets spread out in all directions with the speed light in the given medium.An initially parallel cylindrical beam travels in a medium of refractive index \(\mu(I)=\mu_{0}+\mu_{2} I\), where \(\mu_{0} \text { and } \mu_{2}\)are positive constants and I is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius.

(i) The initial shape of the wavefront of the beam is

(a) planar	(b) convex
(c) concave	(d) convex near the axis and concave near the periphery

(ii) According to Huygens Principle, the surface of constant phase is

(a) called an optical ray	(b) called a wave
(c) called a wavefront	(d) always linear in shape

(iii) As the beam enters the medium, it will

(a) travel as a cylindrical beam	(b) diverge
(c) converge	(d) diverge near tile axis and converge near the periphery.

(iv) Two plane wavefronts of ligbt, one incident on a thin convex lens and another on the refracting face of a thin prism. After refraction at them, the emerging wavefronts respectively become

(a) plane wavefront and plane wavefront	(b) plane wavefront and spherical wavefront
(c) spherical wavefront and plane wavefront	(d) spherical wavefront and spherical wavefront

(v) Which of the following phenomena support the wave theory of light?
1. Scattering
2. Interference
3. Diffraction
4. Velocity of light in a denser medium is less than the velocity of light in the rarer medium

(a) 1,2,3

(b) 1,2,4

(c) 2,3,4

(d) 1,3,4

Wavefront is a locus of points which vibratic in same phase. A ray of light is perpendicular to the wavefront. According to Huygens principle, each point of the wavefront is the source of a secondary disturbance and the wavelets connecting from these points spread out in all directions with the speed of wave. The figure shows a surface XY separating two transparent media, medium-1 and medium-2. The lines ab and cd represent wavefronts of a light wave travelling in medium- 1 and incident on XY. The lines ef and gh represent wavefronts of the light wave in medium -2 after refraction.

(i) Light travels as a

(a) parallel beam in each medium	(b) convergent beam in each medium
(c) divergent beam in each medium	(d) divergent beam in one medium and convergent beam in the other medium.

(ii) The phases of the light wave at c, d, e and f are \(\phi_{c}, \phi_{d}, \phi_{e}\) and \( \phi_{f}\) respectively. It is given that \(\phi_{c} \neq \phi_{f}\)

\(\text { (a) } \phi_{c} \text { cannot be equal to } \phi_{d}\)	\(\text { (b) } \phi_{d} \text { can be equal to } \phi_{e}\)
\(\text { (c) }\left(\phi_{d}-\phi_{f}\right) \text { is equal to }\left(\phi_{c}-\phi_{e}\right)\)	\(\text { (d) }\left(\phi_{d}-\phi_{c}\right) \text { is not equal to }\left(\phi_{f}-\phi_{e}\right)\)

(iii) Wavefront is the locus of all points, where the particles of the medium vibrate with the same

(a) phase

(b) amplitude

(c) frequency

(d) period

(iv) A point source that emits waves uniformly in all directions, produces wavefronts that are

(a) spherical

(b) elliptical

(c) cylindrical

(d) planar

(v) What are the types of wavefronts ?

(a) Spherical

(b) Cylindrical

(c) Plane

(d) All of these

Consider the situation shown in figure. The two slits S₁ and S₂ placed symmetrically around the central line are illuminated by monochromatic light of wavelength \(\lambda\). The separation between the slits is d. The light transmitted. by the slits falls on a screen S₀ place at a distance D from the slits. The slits S₃ is at the central line and the slit S₄ is at a distance from S₃. Another screen S_c is placed a further distance D away from S_c.

(i) Find the path difference if \(z=\frac{\lambda D}{2 d}\)

\((a) \lambda\)

\((b) \lambda / 2\)

\((c) 3 / 2 \lambda\)

\((d) 2 \lambda\)

(ii) Find the ratio of the maximum to minimum intensity observed on \(S_{c} \text { if } z=\frac{\lambda D}{d}\)

(a) 4

(b) 2

(c) \(\infty\)

(d) 1

(iii) Two coherent point sources S₁ and S₂ are separated by a small distance d as shown in figure. The fringes obtained on the screen will be

(a) Concentric Circles

(b) points

(c) Straight lines

(d) semi-circles

(iv) In the case of light waves from two coherent sources S₁ and S₂. there will be constructive interference at an arbitrary point P, if the path difference S₁P - S₂P is

\(\text { (a) }\left(n+\frac{1}{2}\right) \lambda \)

\( \text { (b) } n \lambda\)

\( \text { (c) }\left(n-\frac{1}{2}\right) \lambda\)

\( \text { (d) } \frac{\lambda}{2}\)

(v) Two monochromatic light waves of amplitudes 3A and 2A interfering at a point have a phase difference of 60°. The intensity at that point will be proportional to

(a) 5A²

(b) 13A²

(c) 7A²

(d) 19A²

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Answers Key:

Case Study Questions

5 x 4 = 20

(i) (c) : Given \(I_{1}=10 \mathrm{~W} / \mathrm{m}^{2} \text { and } I_{2}=25 \mathrm{~W} / \mathrm{m}^{2}\)
\(\frac{I_{1}}{I_{2}}=\frac{a_{1}^{2}}{a_{2}^{2}}=\frac{10}{25} \Rightarrow \frac{a_{1}}{a_{2}}=\frac{3.16}{5} \text { or } a_{1}=\frac{3.16}{5} a_{2}=0.6324 a_{2}\)
\(\frac{I_{\max }}{I_{\min }}=\frac{\left(a_{1}+a_{2}\right)^{2}}{\left(a_{1}-a_{2}\right)^{2}}=\frac{\left[0.6324 a_{2}+a_{2}\right]^{2}}{\left[0.6324 a_{2}-a_{2}\right]^{2}}=19.724\)
(ii) (b): In an excessively thin film, the thickness of the film is negligible. Thus the path difference between the reflected rays becomes \(\lambda\)/2 which produces a minima.
(iii) (a): Since, \(\beta=\frac{\lambda D}{d} \text { for } d=2 d\)
\(\beta^{\prime}=\frac{\lambda D^{\prime}}{2 d}=\beta(\text { Gives })\)
\(\therefore \quad D_{1}=2 D\)
(iv) (b): The condition for possible interference maxima on the screen is, dsin \(\theta\) = n\(\lambda\)
where d is slit separation and Ais the wavelength.
\(\text { As } d=2 \lambda \text { (given) } \quad \therefore 2 \lambda \sin \theta=n \lambda \text { or } 2 \sin \theta=n\)
For number of interference maxima to be maximum,sin\(\theta\) = 1 :. n = 2
The interference maxima will be formed when n = 0, ± 1, ± 2
Hence the maximum number of possible maxima is 5.
(v) (d): \(y_{1}=a \sin \left(\omega t+\frac{\pi}{3}\right) \text { and } y_{2}=a \sin \omega t\)
\(A=\sqrt{a_{1}^{2}+a_{2}^{2}+2 a_{1} a_{2} \cos \phi}, \text { where } \phi=\frac{\pi}{3}\)
\(=\sqrt{a^{2}+a^{2}+2 a a \cos \frac{\pi}{3}}=\sqrt{3} a\)

(i) (b): The wavelength of visible light is very small, that is hardly shows diffraction, so it seems to propagate in rectilinear path,
(ii) (c): Angular width of central maxima, 2\(\theta\) = 2\(\lambda\)/e. Thus, \(\theta\) does not depend on screen i.e., distance between the slit and the screen.
(iii) (c) : The intensity distribution of single slit diffraction pattern is shown in the figure. From the graph it is dear that the intensity of the central point is finite but much larger than the surrounding maxima.

(iv) (a): Resolving power of telescope \(=\frac{a}{1.22 \lambda}\)
\(\therefore\) It increases when wavelength of light decreases and/or objective lens of greater diameter is used.
(v) (a): Width of central maxima = 2\(\lambda\) DIe
width of other secondary maxima = \(\lambda\)DI e
\(\therefore\)Width of central maxima: width of other secondary maxima
= 2 : 1

(i) (a): As the beam is initially parallel, the shape of wavefront is planar.
(ii) (c): According to Huygens Principle, the surface of constant phase is called a wavefront.
(iii) (c)
(iv) (c): After refraction, the emerging wavefronts respectively become spherical. wavefront and plane wavefront as shown in figures (a) and (b).

(v) (c)

(i) (a): Since the path difference between two waveform is equal, light traves as parallel beam in each medium.
(ii) (c): Since all points on the wavefront are in the same phase,
\(\phi_{d}=\phi_{c} \text { and } \phi_{f}=\phi_{e} \)
\(\therefore \phi_{d}-\phi_{f}=\phi_{c}-\phi_{e^{-}}\)
(iii) (a): Wavefront is the locus of all points, where the particles of the medium vibrate with the same phase
(iv) (a)
(v) (d)

(i) (b): \(\text { As } z=\frac{\lambda D}{2 d}\)
\(\text { At } S_{4}: \frac{\Delta x}{d}=\frac{z}{D}\)
\(\Rightarrow \Delta x=\frac{\lambda D}{2 d} \frac{d}{d}=\frac{\lambda}{2}\)
(ii) (c): \(z=\frac{\lambda D}{d}\)
\(\Delta x \text { at } S_{4}: \Delta x=\frac{\lambda D}{d} \frac{d}{d}=\lambda\)
Hence, maxima at S₄ as well as S₃'
Resultant intensity at S₄ I = 4I₀
\(\therefore \quad \frac{I_{\max }}{I_{\min }}=\frac{\left[\left(4 I_{0}\right)^{1 / 2}+4\left(4 I_{0}\right)^{1 / 2}\right]^{2}}{\left[\left(4 I_{0}\right)^{1 / 2}-\left(4 I_{0}\right)^{1 / 2}\right]^{2}}=\infty\)
(iii) (a): When the screen is placed perpendicular to the line joining the 'sources, the fringes will be concentric circles.
(iv) (b): Constructive interference occurs when the path difference (S₁P - S₂P) is an integral multiple of \(\lambda\).or S₁P - S₂P = n\(\lambda\), where n = 0,1,2,3, .....
(v) (d): Here, A₁ = 3A, A₂ = 2A and \(\varphi\)= 60° The resultant amplitude at a point is
\(R =\sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos \phi} \)
\(=\sqrt{(3 A)^{2}+(2 A)^{2}+2 \times 3 A \times 2 A \times \cos 60^{\circ}} \)
\(=\sqrt{9 A^{2}+4 A^{2}+6 A^{2}}=A \sqrt{19}\)
As, Intensity \(\infty\) (Amplituder)²Therefore, intensity at the same point is
\(I \propto 19 A^{2}\)