Interference Problems

This section provides 100 problems to test your understanding of interference of light, including calculations of resultant amplitude, fringe spacing, path difference, thin film thickness, and intensity distribution, as well as applications like interferometry and anti-reflective coatings. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering wave optics, a key topic for JEE/NEET success.

Numerical Problems

1. Two coherent waves with amplitudes 4 V/m and 3 V/m interfere with a phase difference $\varphi =0$. Calculate the resultant amplitude.

   • (a) 6.9 V/m
   • (b) 7.0 V/m
   • (c) 7.1 V/m
   • (d) 7.2 V/m

2. Two coherent waves with equal amplitudes of 5 V/m interfere with a phase difference $\varphi =\pi$. Calculate the resultant amplitude.

   • (a) 0.0 V/m
   • (b) 0.1 V/m
   • (c) 0.2 V/m
   • (d) 0.3 V/m

3. Two coherent waves with amplitudes 2 V/m and 3 V/m interfere with a phase difference $\varphi =\pi /2$. Calculate the resultant amplitude.

   • (a) 3.5 V/m
   • (b) 3.6 V/m
   • (c) 3.7 V/m
   • (d) 3.8 V/m

4. In a Young's double-slit experiment, $d=0.2\text{mm}$, $\lambda =600\text{nm}$, and $D=1.5\text{m}$. Calculate the fringe spacing $\beta$.

   • (a) 4.4 mm
   • (b) 4.5 mm
   • (c) 4.6 mm
   • (d) 4.7 mm

5. In a Young's double-slit experiment, $d=0.5\text{mm}$, $\lambda =500\text{nm}$, and $D=2\text{m}$. Calculate the position of the third bright fringe ($y_3$).

   • (a) 5.9 mm
   • (b) 6.0 mm
   • (c) 6.1 mm
   • (d) 6.2 mm

6. In a Young's double-slit experiment, $d=0.1\text{mm}$, $\lambda =400\text{nm}$, and $y_2=8\text{mm}$. Calculate the screen distance $D$.

   • (a) 0.99 m
   • (b) 1.00 m
   • (c) 1.01 m
   • (d) 1.02 m

7. A thin film with $n=1.33$ and $t=300\text{nm}$ is viewed at normal incidence. Find the wavelength $\lambda$ for constructive interference with $m=1$.

   • (a) 531 nm
   • (b) 532 nm
   • (c) 533 nm
   • (d) 534 nm

8. An anti-reflective coating has $n=1.38$, designed for $\lambda =550\text{nm}$. Calculate the thickness $t$.

   • (a) 99.5 nm
   • (b) 99.6 nm
   • (c) 99.7 nm
   • (d) 99.8 nm

9. Two coherent waves with amplitudes 6 V/m interfere with $\varphi =2\pi /3$. Calculate the resultant amplitude.

   • (a) 5.9 V/m
   • (b) 6.0 V/m
   • (c) 6.1 V/m
   • (d) 6.2 V/m

10. In a Young's double-slit experiment, $d=0.4\text{mm}$, $\lambda =600\text{nm}$, $D=1\text{m}$. Calculate $\beta$.

    • (a) 1.4 mm
    • (b) 1.5 mm
    • (c) 1.6 mm
    • (d) 1.7 mm

11. A thin film with $n=1.5$, $t=200\text{nm}$ at normal incidence has $\lambda =600\text{nm}$. Find $m$ for constructive interference.

    • (a) 0
    • (b) 1
    • (c) 2
    • (d) 3

12. Two coherent waves with amplitudes 3 V/m and 4 V/m interfere with $\varphi =\pi /3$. Calculate the resultant amplitude.

    • (a) 6.0 V/m
    • (b) 6.1 V/m
    • (c) 6.2 V/m
    • (d) 6.3 V/m

13. In a Young's double-slit experiment, $d=0.3\text{mm}$, $\lambda =500\text{nm}$, $D=1.5\text{m}$. Find the position of the first dark fringe.

    • (a) 1.24 mm
    • (b) 1.25 mm
    • (c) 1.26 mm
    • (d) 1.27 mm

14. An anti-reflective coating with $n=1.4$ is designed for $\lambda =480\text{nm}$. Calculate $t$.

    • (a) 85.6 nm
    • (b) 85.7 nm
    • (c) 85.8 nm
    • (d) 85.9 nm

15. A thin film with $n=1.6$, $t=250\text{nm}$, $\lambda =400\text{nm}$ at normal incidence. Check interference type for $m=0$.

    • (a) Constructive
    • (b) Destructive
    • (c) No interference
    • (d) Partial interference

16. Two coherent waves with amplitudes 2 V/m interfere with $\varphi =\pi /4$. Calculate the resultant amplitude.

    • (a) 3.7 V/m
    • (b) 3.8 V/m
    • (c) 3.9 V/m
    • (d) 4.0 V/m

17. In a Young's double-slit experiment, $\beta =2\text{mm}$, $d=0.25\text{mm}$, $D=1\text{m}$. Calculate $\lambda$.

    • (a) 499 nm
    • (b) 500 nm
    • (c) 501 nm
    • (d) 502 nm

18. A thin film with $n=1.33$, $t=500\text{nm}$, $\lambda =665\text{nm}$ at normal incidence. Find $m$ for destructive interference.

    • (a) 0
    • (b) 1
    • (c) 2
    • (d) 3

19. Two coherent waves with amplitudes 5 V/m and 5 V/m interfere with $\varphi =\pi /6$. Calculate the intensity ratio $I/I_{\text{max}}$.

    • (a) 0.92
    • (b) 0.93
    • (c) 0.94
    • (d) 0.95

20. In a Young's double-slit experiment, $d=0.15\text{mm}$, $\lambda =450\text{nm}$, $D=1.2\text{m}$. Calculate $\beta$.

    • (a) 3.5 mm
    • (b) 3.6 mm
    • (c) 3.7 mm
    • (d) 3.8 mm

21. A thin film with $n=1.5$, $t=300\text{nm}$, $\lambda =450\text{nm}$ at normal incidence. Find $m$ for constructive interference.

    • (a) 1
    • (b) 2
    • (c) 3
    • (d) 4

22. Two coherent waves with amplitudes 4 V/m and 2 V/m interfere with $\varphi =0$. Calculate the resultant amplitude.

    • (a) 5.9 V/m
    • (b) 6.0 V/m
    • (c) 6.1 V/m
    • (d) 6.2 V/m

23. In a Young's double-slit experiment, $d=0.1\text{mm}$, $\lambda =500\text{nm}$, $D=2\text{m}$. Find $y_4$.

    • (a) 19.9 mm
    • (b) 20.0 mm
    • (c) 20.1 mm
    • (d) 20.2 mm

24. An anti-reflective coating with $n=1.35$ is designed for $\lambda =540\text{nm}$. Calculate $t$.

    • (a) 99.9 nm
    • (b) 100.0 nm
    • (c) 100.1 nm
    • (d) 100.2 nm

25. A thin film with $n=1.4$, $t=400\text{nm}$, $\lambda =560\text{nm}$ at normal incidence. Find $m$ for destructive interference.

    • (a) 1
    • (b) 2
    • (c) 3
    • (d) 4

26. Two coherent waves with amplitudes 3 V/m interfere with $\varphi =\pi /2$. Calculate the resultant amplitude.

    • (a) 4.2 V/m
    • (b) 4.3 V/m
    • (c) 4.4 V/m
    • (d) 4.5 V/m

27. In a Young's double-slit experiment, $d=0.2\text{mm}$, $\lambda =700\text{nm}$, $D=1\text{m}$. Calculate $\beta$.

    • (a) 3.4 mm
    • (b) 3.5 mm
    • (c) 3.6 mm
    • (d) 3.7 mm

28. A thin film with $n=1.5$, $t=150\text{nm}$, $\lambda =450\text{nm}$ at normal incidence. Check interference type for $m=0$.

    • (a) Constructive
    • (b) Destructive
    • (c) No interference
    • (d) Partial interference

29. Two coherent waves with amplitudes 6 V/m and 8 V/m interfere with $\varphi =\pi$. Calculate the resultant amplitude.

    • (a) 1.9 V/m
    • (b) 2.0 V/m
    • (c) 2.1 V/m
    • (d) 2.2 V/m

30. In a Young's double-slit experiment, $d=0.25\text{mm}$, $\lambda =600\text{nm}$, $D=1.5\text{m}$. Find the position of the second dark fringe.

    • (a) 3.5 mm
    • (b) 3.6 mm
    • (c) 3.7 mm
    • (d) 3.8 mm

31. A spacecraft sensor uses a thin film with $n=1.4$, $t=250\text{nm}$ at normal incidence for $\lambda =700\text{nm}$. Find $m$ for constructive interference.

    • (a) 0
    • (b) 1
    • (c) 2
    • (d) 3

32. An anti-reflective coating with $n=1.3$ is designed for $\lambda =520\text{nm}$. Calculate $t$.

    • (a) 99.9 nm
    • (b) 100.0 nm
    • (c) 100.1 nm
    • (d) 100.2 nm

33. In a Young's double-slit experiment, $\beta =3\text{mm}$, $d=0.15\text{mm}$, $\lambda =600\text{nm}$. Calculate $D$.

    • (a) 0.74 m
    • (b) 0.75 m
    • (c) 0.76 m
    • (d) 0.77 m

34. Two coherent waves with amplitudes 5 V/m interfere with $\varphi =\pi /3$. Calculate the intensity ratio $I/I_{\text{max}}$.

    • (a) 0.74
    • (b) 0.75
    • (c) 0.76
    • (d) 0.77

35. A thin film with $n=1.6$, $t=200\text{nm}$, $\lambda =640\text{nm}$ at normal incidence. Find $m$ for destructive interference.

    • (a) 0
    • (b) 1
    • (c) 2
    • (d) 3

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Conceptual Problems

36. What does the principle of superposition state for wave interference?

• (a) Waves cancel each other out completely
• (b) Resultant displacement is the sum of individual displacements
• (c) Waves must have different frequencies
• (d) Waves must be incoherent

37. What condition is necessary for sustained interference of light?

• (a) Different wavelengths
• (b) Coherence of the sources
• (c) Incoherent sources
• (d) Different amplitudes

38. What is the unit of fringe spacing $\beta$ in SI units?

• (a) Meter
• (b) Radian
• (c) Watt
• (d) Hertz

39. What happens at a point of destructive interference in Young's double-slit experiment?

• (a) Maximum intensity
• (b) Minimum intensity
• (c) No interference
• (d) Partial interference

40. What does a thin film interference pattern depend on?

• (a) Thickness of the film only
• (b) Refractive index and thickness of the film
• (c) Wavelength of light only
• (d) Angle of incidence only

41. What is the unit of phase difference $\varphi$?

• (a) Radian
• (b) Meter
• (c) Hertz
• (d) Watt

42. What does a path difference of $\lambda$ indicate in interference?

• (a) Destructive interference
• (b) Constructive interference
• (c) No interference
• (d) Partial interference

43. What happens to the fringe spacing $\beta$ if the slit separation $d$ increases?

• (a) Increases
• (b) Decreases
• (c) Remains the same
• (d) Becomes zero

44. What does an anti-reflective coating achieve?

• (a) Increases reflection
• (b) Reduces reflection by destructive interference
• (c) Increases intensity
• (d) Reduces wavelength

45. What is the dimension of path difference $\delta$?

• (a) $[\text{L}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

46. What does a zero resultant amplitude indicate?

• (a) Constructive interference
• (b) Destructive interference
• (c) No interference
• (d) Partial interference

47. What is the significance of $\frac{\lambda D}{d}$?

• (a) Path difference
• (b) Fringe spacing in Young's experiment
• (c) Intensity of light
• (d) Phase difference

48. What does a phase difference of $\pi$ result in?

• (a) Constructive interference
• (b) Destructive interference
• (c) No interference
• (d) Partial interference

49. What does holography rely on?

• (a) Diffraction
• (b) Interference of coherent light
• (c) Refraction
• (d) Reflection

50. How does interference assist in spacecraft sensor systems?

• (a) Increases intensity
• (b) Enables precise detection through patterns
• (c) Reduces wavelength
• (d) Increases path difference

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Derivation Problems

51. Derive the resultant amplitude for two interfering waves $E_0=\sqrt{E_{10}^2+E_{20}^2+2E_{10}{E}_{20}\cos \varphi }$.

52. Derive the fringe spacing in Young's double-slit experiment $\beta =\frac{\lambda D}{d}$.

53. Derive the intensity distribution in Young's double-slit experiment $I=4I_0{\cos }^2\left(\frac{\pi d\sin \theta }{\lambda }\right)$.

54. Derive the condition for constructive interference in a thin film $2nt\cos \theta =(m+\frac{1}{2})\lambda$.

55. Derive the thickness of an anti-reflective coating $t=\frac{\lambda }{4n}$.

56. Derive the path difference in Young's double-slit experiment $\delta =d\sin \theta$.

57. Derive the intensity ratio $I/I_{\text{max}}$ for two interfering waves.

58. Derive the condition for destructive interference in a thin film $2nt\cos \theta =m\lambda$.

59. Derive the position of the $m$-th bright fringe in Young's experiment $y_m=\frac{m\lambda D}{d}$.

60. Derive the phase difference due to path difference $\varphi =\frac{2\pi \delta }{\lambda }$.

61. Derive the wavelength for constructive interference in a thin film given $n$ and $t$.

62. Derive the fringe spacing if the wavelength $\lambda$ changes in Young's experiment.

63. Derive the resultant intensity for two waves with a phase difference of $\pi /2$.

64. Derive the thickness of a thin film for destructive interference given $\lambda$.

65. Derive the intensity distribution for constructive interference in Young's experiment.

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NEET-style Conceptual Problems

66. What is the unit of wavelength $\lambda$ in SI units?

• (a) Meter
• (b) Radian
• (c) Hertz
• (d) Watt

67. What does a phase difference of $2\pi$ indicate in interference?

• (a) Destructive interference
• (b) Constructive interference
• (c) No interference
• (d) Partial interference

68. What is the relationship between fringe spacing $\beta$ and wavelength $\lambda$ in Young's experiment?

• (a) $\beta \propto \frac{1}{\lambda }$
• (b) $\beta \propto \lambda$
• (c) $\beta$ is independent of $\lambda$
• (d) $\beta \propto {\lambda }^2$

69. What happens to the interference pattern in a thin film if the thickness increases?

• (a) Pattern disappears
• (b) Pattern shifts with new wavelengths
• (c) Pattern remains the same
• (d) Pattern becomes brighter

70. What is the dimension of intensity $I$?

• (a) $[\text{M}{\text{T}}^{-3}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

71. What does the coherence of light sources ensure in interference?

• (a) Changing phase difference
• (b) Constant phase difference
• (c) No phase difference
• (d) Random phase difference

72. What is the role of interference in holography?

• (a) Increases wavelength
• (b) Records 3D images through patterns
• (c) Reduces intensity
• (d) Increases path difference

73. What happens to the intensity at a point of constructive interference?

• (a) Becomes zero
• (b) Becomes maximum
• (c) Remains the same
• (d) Becomes minimum

74. Why does a soap film show colorful patterns?

• (a) Due to diffraction
• (b) Due to interference of different wavelengths
• (c) Due to refraction
• (d) Due to reflection only

75. What is the unit of refractive index $n$?

• (a) Dimensionless
• (b) Meter
• (c) Hertz
• (d) Watt

76. What does a path difference of $\lambda /2$ indicate?

• (a) Constructive interference
• (b) Destructive interference
• (c) No interference
• (d) Partial interference

77. Which type of light source is used in Young's double-slit experiment?

• (a) Incoherent
• (b) Coherent
• (c) Polychromatic
• (d) Static

78. What is the orientation of interference fringes in Young's experiment?

• (a) Circular
• (b) Linear
• (c) Random
• (d) No fringes

79. What does a pseudo-force do in a non-inertial frame for interference calculations?

• (a) Affects perceived path difference
• (b) Affects intensity
• (c) Creates interference
• (d) Reduces phase difference

80. What is the dimension of $\frac{\lambda D}{d}$?

• (a) $[\text{L}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

81. What is the role of interference in spacecraft optical systems?

• (a) Increases intensity
• (b) Enables precise measurements through patterns
• (c) Reduces wavelength
• (d) Increases path difference

82. What happens to the fringe spacing if the screen distance $D$ decreases?

• (a) Increases
• (b) Decreases
• (c) Remains the same
• (d) Becomes zero

83. Why does an anti-reflective coating reduce reflection?

• (a) Due to constructive interference
• (b) Due to destructive interference
• (c) Due to diffraction
• (d) Due to refraction

84. What is the significance of $2nt\cos \theta$?

• (a) Intensity in Young's experiment
• (b) Path difference in thin film interference
• (c) Fringe spacing
• (d) Phase difference

85. What is the unit of thickness $t$ in thin film interference?

• (a) Meter
• (b) Radian
• (c) Hertz
• (d) Watt

86. What does a maximum intensity in Young's experiment indicate?

• (a) Destructive interference
• (b) Constructive interference
• (c) No interference
• (d) Partial interference

87. What is the physical significance of ${\cos }^2\left(\frac{\pi d\sin \theta }{\lambda }\right)$?

• (a) Path difference
• (b) Intensity variation in Young's experiment
• (c) Fringe spacing
• (d) Wavelength

88. Why does the interference pattern in Young's experiment require small slit separation $d$?

• (a) To increase intensity
• (b) To ensure measurable fringe spacing
• (c) To reduce wavelength
• (d) To increase path difference

89. What is the dimension of $\frac{2\pi }{\lambda }$?

• (a) $[{\text{L}}^{-1}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

90. How does structural coloration in biology use interference?

• (a) Increases intensity
• (b) Produces colors through thin film interference
• (c) Reduces wavelength
• (d) Increases path difference

91. What is the role of slit separation $d$ in Young's experiment?

• (a) Determines the wavelength
• (b) Determines the fringe spacing
• (c) Determines the intensity
• (d) Determines the phase difference

92. What does a high intensity ratio $I/I_{\text{max}}$ indicate?

• (a) Destructive interference
• (b) Constructive interference
• (c) No interference
• (d) Partial interference

93. What is the physical significance of $\frac{\lambda }{4n}$?

• (a) Path difference
• (b) Thickness for anti-reflective coating
• (c) Fringe spacing
• (d) Intensity

94. What is the dimension of $\cos \varphi$?

• (a) Dimensionless
• (b) $[\text{L}{\text{T}}^{-1}]$
• (c) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (d) $[\text{L}{\text{T}}^{-2}]$

95. Why does the interference pattern in Young's experiment depend on $\lambda$?

• (a) Due to $\beta =\frac{\lambda D}{d}$
• (b) Due to intensity
• (c) Due to phase difference
• (d) Due to coherence

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NEET-style Numerical Problems

96. Two coherent waves with amplitudes 3 V/m and 3 V/m interfere with $\varphi =0$. Calculate the resultant amplitude.

• (a) 5.9 V/m
• (b) 6.0 V/m
• (c) 6.1 V/m
• (d) 6.2 V/m

97. In a Young's double-slit experiment, $d=0.2\text{mm}$, $\lambda =500\text{nm}$, $D=1\text{m}$. Calculate $\beta$.

• (a) 2.4 mm
• (b) 2.5 mm
• (c) 2.6 mm
• (d) 2.7 mm

98. A thin film with $n=1.5$, $t=200\text{nm}$, $\lambda =600\text{nm}$ at normal incidence. Find $m$ for constructive interference.

• (a) 0
• (b) 1
• (c) 2
• (d) 3

99. An anti-reflective coating with $n=1.4$ is designed for $\lambda =560\text{nm}$. Calculate $t$.

• (a) 99.9 nm
• (b) 100.0 nm
• (c) 100.1 nm
• (d) 100.2 nm

100. Two coherent waves with amplitudes 4 V/m and 4 V/m interfere with $\varphi =\pi /2$. Calculate the resultant amplitude.
     - (a) 5.6 V/m
     - (b) 5.7 V/m
     - (c) 5.8 V/m
     - (d) 5.9 V/m

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