chapter (37)

Chapter 37

Interference of Light Waves

Multiple Choice

1. A laser beam (λ = 694 nm) is incident on two slits 0.100 mm apart. Approximately how far apart (in m) will the bright interference fringes be on the screen 5.00 m from the double slits?

a. 3.47 × 10–3 b. 3.47 × 10–2 c. 3.47 × 10–4 d. 3.47 × 10–6 e. 3.47 × 10–5

2. Estimate the distance (in cm) between the central bright region and the third dark fringe on a screen 5.00 m from two double slits 0.500 mm apart illuminated by 500-nm light.

a. 3.47 b. 2.15 c. 1.75 d. 1.50 e. 1.25

3. Light is incident on a double-slit. The fourth bright band has an angular distance of 7.0° from the central maximum. What is the distance between the slits (in μm)? (Assume the frequency of the light is 5.4 × 1014 Hz.)

a. 27 b. 21 c. 24 d. 18 e. 14

4. For small angle approximations

a. the angle must be 10° or less b. the angle must be 10 radians or less c. the angle must be 1° or less d. the angle must be 1 radian or less e. the angle must be 45° or less

275

276 CHAPTER 37

5. Two slits separated by 0.10 mm are illuminated with green light (λ = 540 nm). Calculate the distance (in cm) from the central bright-region to the fifth bright band if the screen is 1.0 m away.

a. 2.3 b. 2.5 c. 2.7 d. 2.1 e. 2.0

6. Two slits separated by 0.050 mm are illuminated with green light (λ = 540 nm). How many bands of bright lines are there between the central maximum and the 12-cm position? (The distance between the double slits and the screen is 1.0 m).

a. 1111 b. 111 c. 11 d. 1 e. 11111

7. Two slits are illuminated with green light (λ = 540 nm). The slits are 0.05 mm apart and the distance to the screen is 1.5 m. At what distance (in mm) from the central maximum on the screen is the average intensity 50% of the intensity of the central maximum?

a. 1 b. 3 c. 2 d. 4 e. 0.4

8. Two slits are illuminated with red light (λ = 650 nm). The slits are 0.25 mm apart and the distance to the screen is 1.25 m. What fraction of the maximum intensity on the screen is the intensity measured at a distance 3.0 mm from the central maximum?

a. 0.94 b. 0.92 c. 0.96 d. 0.98 e. 0.99

9. In a double-slit experiment, the distance between the slits is 0.2 mm, and the distance to the screen is 150 cm. What wavelength (in nm) is needed to have the intensity at a point 1 mm from the central maximum on the screen be 80% of the maximum intensity?

a. 900 b. 700 c. 500 d. 300 e. 600

Interference of Light Waves 277

10. In a double slit experiment, the distance between the slits is 0.2 mm and the distance to the screen is 150 cm. What is the phase difference (in degrees) between the waves from the two slits arriving at a point P when the angular distance of P is 10° relative to the central peak, and the wavelength is 500 nm? (Convert your result so the angle is between 0 and 360°.)

a. 145° b. 155° c. 165° d. 135° e. 95°

11. In a double slit experiment, the distance between the slits is 0.2 mm and the distance to the screen is 100 cm. What is the phase difference (in degrees) between the waves from the two slits arriving at a point 5 mm from the central maximum when the wavelength is 400 nm? (Convert your result so the angle is between 0 and 360°.)

a. 90° b. 180° c. 270° d. 360° e. 160°

12. The electric fields arriving at a point P from three coherent sources are described by E1 = E0 sin ωt, E2 = E0 sin (ωt + π/4) and E3 = E0 sin (ωt + π/2). Assume the resultant field is represented by Ep = ER sin (ωt + α). The amplitude of the resultant wave at P is

a. E0. b. 1.5E0. c. 1.7E0. d. 2.7E0. e. 2.9E0.

278 CHAPTER 37

13. An interference pattern is produced at point P on a screen as a result of direct rays and rays reflected off a mirror as shown in the figure. If the source is 100 m to the left of the screen, 1.0 cm above the mirror, and the source is monochromatic (λ = 500 nm), find the distance y (in mm) to the first dark band above the screen.

y

P

O

Source

Mirror

Screenθd

a. 1.0 b. 2.0 c. 1.5 d. 2.5 e. 5.0

14. An interference pattern is produced at point P on a screen as a result of direct rays and rays reflected off a mirror as shown in the figure. If the source is 100 m to the left of the screen, 1.0 cm above the mirror, and the source is monochromatic (λ = 500 nm), find the distance y (in mm) to the first bright fringe.

y

P

O

Source

Screen

Mirror

θd

a. 1.75 b. 1.50 c. 1.25 d. 2.00 e. 3.75


15. An interference pattern is produced at point P on a screen as a result of direct rays and rays reflected off a mirror as shown in the figure. If the source is 100 m to the left of the screen, 1.0 cm above the mirror, and the source is a distance d above the mirror, monochromatic (λ = 500 nm), find the condition for maximum intensity (constructive interference) on the screen in terms of θ, λ, and d.

y

P

O

Source

Mirror

Screenθd

a. 2d sin θ = mλ b. 2d sin θ = (m + 1/2)λ c. d sin θ = mλ d. d sin θ = (m + 1/2)λ e. none of these

16. An interference pattern is produced at point P on a screen as a result of direct rays and rays reflected off a mirror as shown in the figure. If the source is 100 m to the left of the screen, 1 cm above the mirror, and the source is monochromatic (λ = 500 nm), find the conditions for minimum brightness on the screen in terms of θ, λ, and d.

y

P

O

Source

Screen

Mirror

θd

a. 2d sin θ = (m + 1/2)λ b. 2d sin θ = mλ c. d sin θ = (m + 1/2)λ d. d sin θ = mλ e. none of these

17. Monochromatic light (λ = 500 nm) is incident on a soap bubble (n = 1.40). What is the wavelength of the light (in nm) in the bubble film?

a. 255 b. 500 c. 700 d. 357 e. 422

280 CHAPTER 37

18. Monochromatic light (λ = 500 nm) is incident on a soap bubble (n = 1.4) that is 50 mm thick. What is the change of phase of the light reflected from the front surface?

a. 0 b. 180° c. λ/2 d. π/2 e. 55°

19. Monochromatic light (λ = 500 nm) is incident on a soap bubble (n = 1.4) that is 500-nm thick. Calculate the change of phase of the light that penetrates the front surface, reflects from the second surface, and emerges through the first surface as an angle between 0° and 360°?

a. 280° b. 160° c. 220° d. 100° e. 290°

20. Monochromatic light (λ = 500 nm) is incident on a soap bubble (n = 1.40). How thick is the bubble (in nm) if destructive interference occurs in the reflected light?

a. 102 b. 179 c. 54 d. 1 e. 89

21. The light reflected from a soap bubble (n = 1.40) appears red (λ = 640 nm) at its center. What is the minimum thickness (in nm)?

a. 124 b. 104 c. 114 d. 134 e. 234

22. A thin sheet of plastic (n = 1.60) is inserted between two panes of glass to reduce infrared (λ = 700 nm) losses. What thickness (in nm) is necessary to produce constructive interference in the reflected infrared radiation?

a. 218 b. 109 c. 55 d. 318 e. 443


23. In a Newton’s rings apparatus, find the phase difference (in radians) when an air wedge of 500 nm thickness is illuminated with red light (λ = 640 nm).

a. 13 b. 11 c. 9 d. 7 e. 3

24. The figure shows two point sources of light, A and B, that emit light waves in phase with each other. A is distant 3λ from point P. B is distant 5λ from P. (λ is the wavelength.) The phase difference between the waves arriving at P from A and B is

A B

3λ5λ

P a. 0 rad. b. π rad. c. 2π rad. d. 3π rad. e. 4π rad.

25. The figure shows two point sources of light, A and B. B emits light waves that are +π radians out of phase with the waves from A. A is 3λ from P. B is 5λ from P. (λ is the wavelength.) The phase difference between waves arriving at P from A and B is

A B

3λ5λ

P a. 0 rad. b. π rad. c. 2π rad. d. 3π rad. e. 4π rad.

282 CHAPTER 37

26. The bright and dark bands you see in a photograph of a double slit interference pattern represent

a. the respective positions of the crests and the troughs of the light wave. b. an interference pattern that is not present unless it is produced by the

camera lens. c. the respective positions of constructive and destructive interference of light

from the two sources. d. the respective positions of destructive and constructive interference of light

from the two sources. e. the respective positions of bright and dark particles of light.

27. In an interference pattern, the wavelength and frequency are

a. the same in both the regions of constructive interference and the regions of destructive interference.

b. greater in regions of constructive interference than in regions of destructive interference.

c. smaller in regions of constructive interference than in regions of destructive interference.

d. unchanged in regions of destructive interference but greater in regions of constructive interference.

e. unchanged in regions of destructive interference but smaller in regions of constructive interference.

28. A planar cross section through two spherical waves emanating from the sources S1 and S2 in the plane is shown in the figure. S1 and S2 are in phase. The black circles are one and two wavelengths from their respective sources. The lighter circles are one half and one and a half wavelengths distant from their respective sources. If the waves shown arriving at P1 both arrive with amplitude A, the resultant amplitude at point P1 is

S1 S2

P1

P2

a. 0.

A21

. b.

c. A.

d. A23

.

e. 2A.


29. A planar cross section through two spherical waves emanating from the sources S1 and S2 in the plane is shown in the figure. S1 and S2 are in phase. The black circles are one and two wavelengths from their respective sources. The lighter circles are one half and one and a half wavelengths distant from their respective sources. If the waves shown arriving at P2 both arrive with amplitude A, the resultant amplitude at point P2 is

S1 S2

P1

P2

a. 0.

b. A21

.

c. A.

d. A23

.

e. 2A.

284 CHAPTER 37

30. A planar cross section through two spherical waves emanating from the sources S1 and S2 in the plane is shown in the figure. The black circles are one and two wavelengths from their respective sources. The lighter circles are one half and one and a half wavelengths distant from their respective sources. If the phase at S1 and S2 is zero at this instant, and the waves shown arriving at P1 both arrive with amplitude A, the phase angle of each wave at point P1 (in radians) is

S1 S2

P1

P2

a. 0. b. π. c. 2π. d. 3π. e. π/2.

31. A planar cross section through two spherical waves emanating from the sources S1 and S2 in the plane is shown in the figure. The black circles are one and two wavelengths from their respective sources. The lighter circles are one half and one and a half wavelengths distant from their respective sources. If the phase at S1 and S2 is zero at this instant, and the waves shown arriving at P2 both arrive with amplitude A, the difference in phase angle at point P2 (in radians) is

S1 S2

P1

P2

a. 0. b. π/2. c. π. d. 3π/2. e. 2π.


32. When a central dark fringe is observed in reflection in a circular interference pattern, waves reflected from the upper and lower surfaces of the medium must have a phase difference, in radians, of

a. 0. b. π/2. c. π. d. 3π/2. e. 2π.

33. A film of index of refraction n1 coats a surface with index of refraction n2. When , the condition for constructive interference for reflected monochromatic

light of wavelength λ in air is 21 nn >

a. 1n

mtλ

= .

b. 1

1nλ

⎟⎠⎞⎛

2mt ⎜

⎝+= .

c. 1

2n

mtλ

= .

d. 1

1nλ

⎟⎠⎞⎛

22 mt ⎜

⎝+= .

e. 1n

4 mtλ

= .

34. A film of index of refraction n1 coats a surface with index of refraction n2. When , the condition for destructive interference for reflected monochromatic

light of wavelength λ in air is 1n >

a.

2n

1nmt

λ= .

λb.

21

mt ⎟⎠⎞

⎜⎝⎛ += .

1n

1

2n

mtλ

= . c.

λd.

21

2 mt ⎟⎠⎞

⎜⎝⎛ += .

1n

e. 1

4n

mtλ

= .

286 CHAPTER 37

35. A film of index of refraction n1 coats a surface with index of refraction n2. When , the condition for constructive interference for reflected monochromatic

light of wavelength λ in air is 21 nn <

a. 1n

mtλ

= .

b. 12

1n

mtλ

⎟⎠⎞

⎜⎝⎛ += .

c. 1

2n

mtλ

= .

d. 12

12

nmt

λ⎟⎠⎞

⎜⎝⎛ += .

e. 1

4n

mtλ

= .

36. A film of index of refraction n1 coats a surface with index of refraction n2. When , the condition for destructive interference for reflected monochromatic

light of wavelength λ in air is 21 nn <

a. λ

1nmt = .

b. 12

1n

mtλ

⎟⎠⎞

⎜⎝⎛ += .

c. λ

2 mt = . 1n

d. 12

12

nmt

λ⎟⎠⎞

⎜⎝⎛ += .

e. 1

4n

mtλ

= .

37. The superposition of two waves )sin( tEE 01 ω= and )sin(02 φω += tEE arriving at the same point in space at the same time is = E

)cos()sin(2 20 ωtEcos()sin(2 0

φ

). a.

b. φωtE .

)cos()sin(2 220φφω +tEc. .

)cos()sin(2 20 φω φ+tEd. .

e. )cos()cos(2 220φφω +tE .


38. Ray says that interference effects cannot be observed with visible light because random phase changes occur in time intervals less than a nanosecond. Stacy says that doesn’t matter if collimated light from a single source reaches multiple openings. (They are arguing about a light source 50.0 cm away from two 0.0100 mm-wide slits, 2.00 mm apart, with a screen 1.00 m away from the slits.) Which one, if either, is correct, and why?

a. Ray, because the phases at the two slits will be random and different. b. Ray, because it takes light over 3 ns to travel 1.00 m to the screen. c. Stacy, because the difference in time of travel from the source to the slits is

no more than about 7 × 10−12 s . d. Stacy, but only if a lens is placed in front of the slits. e. Both, because interference of light never occurs outside a physics lab.

39. Bright and dark fringes are seen on a screen when light from a single source reaches two narrow slits a short distance apart. The locations of bright and dark fringes can be interchanged if a thin film is placed in front of one of the slits. The minimum thickness of this film must be

λd = air

2a. .

λair

2nfilm

d = . b.

λair

(nfilm − 1)d = . c.

λd = air

− 1)2(nfilm

. d.

λd = air

n film

. e.

40. Bright and dark fringes are seen on a screen when light from a single source reaches two narrow slits a short distance apart. Each bright fringe will shift to the location of the adjacent bright fringe if a thin film is placed in front of one of the slits. The minimum thickness of this film must be

d =λair

2a. .

d =λair

2nfilm

. b.

d =λair

(nfilm − 1). c.

d =λair

− 1)2(nfilm

. d.

λd = air

n film

. e.

288 CHAPTER 37

41. Bright and dark fringes are seen on a screen when light from a single source reaches two narrow slits a short distance apart. The number of fringes per unit length on the screen can be doubled

a. if the distance between the slits is doubled.

b. if the wavelength is changed to ′λ =λ2

.

c. if the distance between the slits is one quarter of the original distance and the wavelength is changed to 2′λ = λ .

d. if any of the above occurs.

e. only if the width of the slits is changed to ′w =w2

.

42. Bright and dark fringes are seen on a screen when light from a single source reaches two narrow slits a short distance apart. The number of fringes per unit length on the screen can be halved

′d =d2

. a. if the distance between the slits is changed to

′λ = 2λ . b. if the wavelength is changed to

′d =2d

the wavelength is changed to c. if the distance between the slits is

= 4′λ λ . d. if any of the above occurs.

′w 2w . =e. only if the width of the slits is changed to

43. When illuminated with monochromatic light of wavelength λ , a clear double slit interference pattern with approximately equal intensity for at least a dozen centrally located fringed can be seen only if

λ . a. the distance between the slits is less than λ . b. the width of the slits is equal to or less than

c. the width of the slits is equal to or greater than 12λ . d. the distance between the slits is equal to or greater than 12λ . e. laser light is used and the width of the slits is equal to or greater than 12λ .


44. The figures below represent interference fringes. The distances from the screen to the slits is the same for each figure, and the planes of the screen and the slits are parallel. Which figure(s) represent(s) slits with the smallest spacing d between the slits? The white spaces represent the interference maxima.

I. II.

III. IV. V. a. I. b. II. c. III. d. IV. e. V.

45. The figures below represent interference fringes. The distances from the screen to the slits is the same for each figure, and the planes of the screen and the slits are parallel. Which figure(s) represent(s) slits with the greatest spacing d between the slits? The white spaces represent the interference maxima.

I. II.


290 CHAPTER 37

46. The figures below represent interference fringes. The distances from the screen to the slits is the same for each figure, and the planes of the screen and the slits are parallel. In each figure the spacing d between the slits is the same. Which figure(s) represent(s) slits illuminated with light of the greatest wavelength λ ? The white spaces represent the interference maxima.

I. II.



47. The figures below represent interference fringes. The distances from the screen to the slits is the same for each figure, and the planes of the screen and the slits are parallel. In each figure the spacing d between the slits is the same. Which figure(s) represent(s) slits illuminated with light of the shortest wavelength λ ? The white spaces represent the interference maxima.

I. II.


292 CHAPTER 37

Open-Ended Problems

48. In a double-slit experiment using light of wavelength 486 nm, the slit spacing is 0.600 mm and the screen is 2.00 m from the slits. Find the distance along the screen between adjacent bright fringes.

49. Nonreflective coatings for camera lenses reduce the loss of light at various surfaces of multi-lens systems, as well as preventing internal reflections that might mar the image. Find the minimum thickness of a layer of magnesium fluoride (n = 1.38) on flint glass (n = 1.66) that will cause destructive interference of reflected light of wavelength λ = 550 nm, a wavelength which is near the middle of the visual spectrum.

50. Suppose two flat glass plates 30-cm long are in contact along one end and separated by a human hair at the other end. If the diameter of the hair is 50 μm, find the separation of the interference fringes when the plates are illuminated by green light, λ = 546 nm.

51. A soap bubble (n = 1.35) is floating in air. If the thickness of the bubble wall is 115 nm, what visible light wavelength is most strongly reflected?

52. A uniform film of oil (n = 1.31) is floating on water. When sunlight in air is incident normally on the film, an observer finds that the reflected light has a brightness maximum at λ = 450 nm and a brightness minimum at λ = 600 nm. What is the thickness of the oil film?


Chapter 37

Interference of Light Waves

1. b

2. e

3. d

4. a

5. c

6. c

7. d

8. a

9. a

10. c

11. b

12. c

13. d

14. c

15. b

16. b

17. d

18. b

19. e

20. b

21. c

22. b

23. a

24. e

25. d

26. c

27. a

28. e

29. a

30. d

31. c

32. c

33. d

34. c

35. c

36. d

37. c

38. c

39. d

40. c

41. d

42. d

43. b

44. e

45. d

46. e

47. d

48. 1.62 mm

49. 99.6 nm

50. 1.64 mm

51. 621 nm (red)

52. 343.5 nm

294 CHAPTER 37