1. Returning to Section 9.1, let...And ...where the wavefront shapes
are not explicitly specified, and ... and ... are complex vectors
depending on space and initial phase angle. Show that the interference
term is then given by...You will have to evaluate terms of the
form...for T>> T (take another look at Problem 3.10). Show that
Eq. (9.109) leads to Eq. (9.11) for plane waves. Get solution
2. In Section 9.1 we considered the spatial distribution of energy for two point sources. We mentioned that for the case in which the separation a >> λ, I12 spatially averages to zero. Why is this true? What happens when a is much less than λ? Get solution
3. Return to Fig. 2.22 and prove that if two electromagnetic plane waves making an angle θ have the same amplitude, Eθ, the resulting interference pattern on the yx-plane is a cosine-squared irradiance.. distribution given by...Locate the zeros of irradiance. What is the value of the fringe separation? What happens to the separation as θ increases? Compare your analysis with that leading to Eq. (9.17). [Hint: Begin with the wave expressions given in Section 2.7, which have the proper phases already worked out, and write them as exponentials.] Get solution
4. Will we get an interference pattern in Young's Experiment (Fig. 9.8) if we replace the source slit S by a single long-filament light-bulb? What would occur if we replaced the slits S1, and S2 by these same bulbs? Get solution
5. Figure P.9.5 shows an output pattern that was measured by a • tiny microphone when two small piezo-loudspeakers separated by 15 cm were pointed toward the microphone at a distance of 1.5 m away. Given that the speed of sound at 20°C is 343 m/s, determine the approximate frequency at which the speakers were driven. Discuss the nature of the pattern and explain why it has a central minimum.Figure P.9.5 (Data courtesy of CENCO.)... Get solution
6. Two 1.0-MHz radio antennas emitting in-phase are separated by 600 m along a north-south line. A radio receiver placed 2.0 km east is equidistant from both transmitting antennas and picks up a fairly strong signal. How far north should that receiver be moved if it is again to detect a signal nearly as strong? Get solution
7. Get solution
8. Get solution
9. An expanded beam of red light from a He-Ne laser (λ0 = 632.8nm) is incident on a screen containing two very narrow horizontal slits separated by 0.200 mm. A fringe pattern appears on a white screen held 1.00 m away.(a) How far (in radians and millimeters) above and below the central axis ate the first zeros of irradiance?(b) How far (in mm) from the axis is die fifth bright band?(c) Compare these two results. Get solution
10. Get solution
11. Red plane waves from a ruby laser (λ0 = 694.3nm) in air impinge on two parallel slits in an opaque screen. A fringe pattern forms on a distant wall, and we see the fourth bright band 1.0° above the central axis. Kindly calculate the separation between the slits. Get solution
12. A 3 × 5 card containing two pinholes, 0.08 mm in diameter and separated center to center by 0.10 mm, is illuminated by parallel rays of blue light from an argon ion laser (λ0 = 487.99 nm). If the fringes on an observing screen are to be 10mm apart, how far away should the screen be? Get solution
13. White light falling on two long narrow slits emerges and is observed on a distant screen. If red light (λ0 = 780 nm) in the first-order fringe overlaps violet in the second-order fringe, what is the latter's wavelength? Get solution
14. Get solution
15. Get solution
16. Considering the double-slit experiment, derive an equation for the distance ym from the central axis to the m'th irradiance minimum, such that the first dark bands on either side of the central maximum correspond to m' = ±1. Identify and justify all your approximations. Get solution
17. Get solution
18. With regard to Young's Experiment, derive a general expression for the shift in the vertical position of the mth maximum as a result of placing a thin parallel sheet of glass of index n and thickness d directly over one of the slits. Identify your assumptions. Get solution
19. Plane waves of monochromatic light impinge at an angle θi on a screen containing two narrow slits separated by a distance a. Derive" an equation for the angle measured from the central axis which locates the mth maximum. Get solution
20. Sunlight incident on a screen containing two long narrow slits 0.20 mm apart casts a pattern on a white sheet of paper 2.0 m beyond. What is the distance separating the violet (λ0 = 400 nm) in the first-order band from the red (λ0 = 600 nm) in the second-order band? Get solution
21. To examine the conditions under which the approximations of H3q. (9.23) are valid:(a) Apply the law of cosines to triangle S1S2P in Fig. 9.8c to get...(b) Expand this in a Maclaurin series yielding...(c) In light of Eq. (9.17), show that if (r1, - r2) is to equal a sin θ, it is required that r1, >> a2/λ. Get solution
22. A stream of electrons, each having an energy of 0.5 eV, impinges on a pair of extremely thin slits separated by 10-2mm. What is the distance between adjacent minima on a screen 20 m behind the slits? (me, = 9.108 × 10-31kg, 1 eV = 1.602 × 10-19J.) Get solution
23. It is our intention to produce interference fringes by illuminating some sort of arrangement (Young's Experiment, a thin film, the Michelson Interferometer, etc.) with light at a mean wavelength of 500 nm, having a linewidth of 2.5 X 10-3 nm. At approximately what optical path length difference can you expect the fringes to vanish? [Hint: Think about the coherence length and revisit Problem 7.39.] Get solution
24. Imagine that you have an opaque screen with three horizontal 1 very narrow parallel slits in it. The second slit is a center-to-center distance a beneath the first, and the third is a distance 5a/2 beneath the first, (a) Write a complex exponential expression in terms of δ for the amplitude of the electric field at some point P at an elevation θ on a distant screen where δ = ka sin θ. Prove that...Verify that at θ =0,I(θ) = I (0). Get solution
25. Get solution
26. In the Fresnel double mirror s = 2 m, λ0 = 589 nm, and the separation of the fringes was found to be 0.5 mm. What is the angle of inclination of the mirrors, if the perpendicular distance of the actual point source to the intersection of the two mirrors is I m? Get solution
27. Show that a for the Fresnel biprism of Fig. 9.13 is given by a = 2d(n - l)α. Get solution
28. The Fresnel biprism is used to obtain fringes from a point source that is placed 2 m from the screen, and the prism is midway between the source and the screen. Let the wavelength of the light be λ0 = 500 nm and the index of refraction of the glass be n = 1.5. What is the prism angle, if the separation of the fringes is 0.5 mm? Get solution
29. What is the general expression for the separation of the fringes of a Fresnel biprism of index n immersed in a medium having an index of refraction n'? Get solution
30. Get solution
31. Using Lloyd's mirror, X-ray fringes were observed, the spacing of which was found to be 0.ÖÖ2 5 cm. The wavelength used was 8.33 Ǻ. If the source-screen distance was 3 m, how high above the min or plane was the point source of X-rays placed? Get solution
32. Imagine that we have an antenna at the edge of a lake picking up a signal from a distant radio star (Fig. P.9.24), which is just coming up above the horizon. Write expressions for δ and for the angular position of the star when the antenna detects its first maximum.Figure P.9.24... Get solution
33. If the plate in Fig. 9.17 is glass in air, show that the amplitudes of Elr, E2r, and E3r are respectively 0.2 E0i, 0.192 E0i. and 0.008E0i where E0i is the incident amplitude. Make use of the Fresnel coefficients at normal incidence, assuming no absorption. You might repeal the calculation for a water film in air. Get solution
34. A soap film surrounded by air has an index of refraction of 1.34. If a region of the film appears bright red (λ0 = 633 nm) in normally reflected light, what is its minimum thickness there? Get solution
35. A thin film of ethyl alcohol (n = 1.36) spread on a flat glass plate and illuminated with white light shows a color pattern in reflection. If a region of the film reflects only green light (500 nm) strongly, how thick is it? Get solution
36. A soap film of index 1.34 has a region where it is 550.0 nm thick. Detennine the vacuum wavelengths of the radiation that is not reflected when the film is illuminated from above with sunlight. Get solution
37. Get solution
38. Consider the circular pattern of Haidinger's fringes resulting from a film with a thickness of 2 mm and an index of refraction of 1.5. For monochromatic illumination of λ0 = 600 nm, find the value of m for the central fringe (θt, = 0). Will it be bright or dark? Get solution
39. Illuminate a microscope slide (or even belter, a thin cover-glass slide). Colored fringes can easily be seen with an ordinary fluorescent lamp (although some of the newer versions don't work well at all) serving as a broad source or a mercury street light as a point source. Describe the fringes. Now rotate the glass. Does the pattern change? Duplicate the conditions shown in Figs. 9.18 and 9.19. Try it again with a sheet of plastic food wrap stretched across the lop of a cup. Get solution
40. Fringes are observed when a parallel beam of light of wavelength 500 nm is incident perpendicularly onto a wedge-shaped film with an index of refraction of 1.5. What is the angle of the wedge if the fringe separation is ... cm?Figure P.9.31... Get solution
41. Suppose a wedge-shaped air film is made between two sheets of glass, with a piece of paper 7.618 X 10-5 m thick used as the spacer at their very ends. If light of wavelength 500 nm comes down from directly above, determine the number of bright fringes that will be seen across the wedge. Get solution
42. Get solution
43. Figure P.9.31 illustrates a setup used for testing lenses. Show thatd = x2(R1 – R1,)/2R1R2when d1 and d2 are negligible in comparison with 2R1, and 2R2, respectively. (Recall the theorem from plane geometry that relates the products of the segments of intersecting chords.) Prove that the radius of the mth dark fringe is then...How does this relate to Eq. (9.43)? Get solution
44. Newton rings are observed on a film with quasimonochromatic that has a wavelength of 500 nm. If the 20th bright ring has a radius of 1 cm, what is the radius of curvature of the lens forming one part of the interfering system? Get solution
45. Get solution
46. Get solution
47. Get solution
48. One of the mirrors of a Michelson Interferometer is moved, and 1000 fringe-pairs shift past the hairline in a viewing telescope during the process. If the device is illuminated wilh 500-nm light, how far was the mirror moved? Get solution
49. Get solution
50. Suppose we place a chamber 10.0 cm long with flat parallel windows in one arm of a Michelson Interferometer that is being illuminated by 600-nm light. If the refractive index of air is 1.000 29 and all the air is pumped out of the cell, how many fringe-pairs will shift by in the process? Get solution
51. Get solution
52. A form of the Jamin Interferometer is illustrated in Fig. P.9.38. How does it work? To what use might it be put? Get solution
53. Starting with Eq. (9.53) for the transmitted wave, compute the flux density, thai is, Eq. (9.54). Get solution
54. Given that the mirrors of a Fabry-Perol Interferometer have an...amplitude reflection coefficient of r = 0.8944, find(a) the coefficient of finesse,(b) the half-width,(c) the finesse, and,(d) the contrast factor defined by... Get solution
55. To fill in some of the details in the derivation of the smallest phase increment separating two resolvable Fabry-Perot fringes, that is,...satisfy yourself that...Show that Eq. (9.72) can be rewritten as...When F is large γ is small, and sin (∆δ) = ∆δ. Prove that Eq. (9.73) then follows. Get solution
56. Consider the interference pattern of the Michelson Interferometer as arising from two beams of equal flux density. Using Eq.(9.17), compute the half-width. What is the separation, in S, between adjacent maxima? What then is the finesse? Get solution
57. Satisfy yourself of the fact that a film of thickness λf/4 and index n1, will always reduce the reflectance of the substrate on which it is deposited, as long as ns, > n1, >n0. Consider the simplest case of normal incidence and n0 = 1. Show that this is equivalent to saying that the waves reflected back from the two interfaces cancel one another. Get solution
58. Verify that the reflectance of a substrate can be increased by coating it with a λf/4, high-index layer, that is, n1 > ns* Show that the reflected waves interfere constructively. The quarter-wave stack g(HL)"'Ha can be thought of as a series of such structures. Get solution
59. Determine the refractive index and thickness of a film to be deposited on a glass surface (ng = 1.54) such that no normally incident light of wavelength 540 nm is reflected. Get solution
60. A glass microscope lens having an index of 1.55 is to be coated with a magnesium fluoride film to increase the transmission of normally incident yellow light (λ0 = 550 nm). What minimum thickness should be deposited on the lens? Get solution
61. A glass camera lens with an index of 1.55 is to be coated with a cryolite film (n ≈ 1.30) to decrease the reflection of normally incident green light (λ0 = 500 nm). What thickness should be deposited on the lens? Get solution
62. Using Fig. 9.60, which depicts the geometry of the Shuttle -radar interferometer, show thatz(x) = h – r1, cos θThen use (he Law of Cosines to establish that Eq. (9.108) is correct. Get solution
2. In Section 9.1 we considered the spatial distribution of energy for two point sources. We mentioned that for the case in which the separation a >> λ, I12 spatially averages to zero. Why is this true? What happens when a is much less than λ? Get solution
3. Return to Fig. 2.22 and prove that if two electromagnetic plane waves making an angle θ have the same amplitude, Eθ, the resulting interference pattern on the yx-plane is a cosine-squared irradiance.. distribution given by...Locate the zeros of irradiance. What is the value of the fringe separation? What happens to the separation as θ increases? Compare your analysis with that leading to Eq. (9.17). [Hint: Begin with the wave expressions given in Section 2.7, which have the proper phases already worked out, and write them as exponentials.] Get solution
4. Will we get an interference pattern in Young's Experiment (Fig. 9.8) if we replace the source slit S by a single long-filament light-bulb? What would occur if we replaced the slits S1, and S2 by these same bulbs? Get solution
5. Figure P.9.5 shows an output pattern that was measured by a • tiny microphone when two small piezo-loudspeakers separated by 15 cm were pointed toward the microphone at a distance of 1.5 m away. Given that the speed of sound at 20°C is 343 m/s, determine the approximate frequency at which the speakers were driven. Discuss the nature of the pattern and explain why it has a central minimum.Figure P.9.5 (Data courtesy of CENCO.)... Get solution
6. Two 1.0-MHz radio antennas emitting in-phase are separated by 600 m along a north-south line. A radio receiver placed 2.0 km east is equidistant from both transmitting antennas and picks up a fairly strong signal. How far north should that receiver be moved if it is again to detect a signal nearly as strong? Get solution
7. Get solution
8. Get solution
9. An expanded beam of red light from a He-Ne laser (λ0 = 632.8nm) is incident on a screen containing two very narrow horizontal slits separated by 0.200 mm. A fringe pattern appears on a white screen held 1.00 m away.(a) How far (in radians and millimeters) above and below the central axis ate the first zeros of irradiance?(b) How far (in mm) from the axis is die fifth bright band?(c) Compare these two results. Get solution
10. Get solution
11. Red plane waves from a ruby laser (λ0 = 694.3nm) in air impinge on two parallel slits in an opaque screen. A fringe pattern forms on a distant wall, and we see the fourth bright band 1.0° above the central axis. Kindly calculate the separation between the slits. Get solution
12. A 3 × 5 card containing two pinholes, 0.08 mm in diameter and separated center to center by 0.10 mm, is illuminated by parallel rays of blue light from an argon ion laser (λ0 = 487.99 nm). If the fringes on an observing screen are to be 10mm apart, how far away should the screen be? Get solution
13. White light falling on two long narrow slits emerges and is observed on a distant screen. If red light (λ0 = 780 nm) in the first-order fringe overlaps violet in the second-order fringe, what is the latter's wavelength? Get solution
14. Get solution
15. Get solution
16. Considering the double-slit experiment, derive an equation for the distance ym from the central axis to the m'th irradiance minimum, such that the first dark bands on either side of the central maximum correspond to m' = ±1. Identify and justify all your approximations. Get solution
17. Get solution
18. With regard to Young's Experiment, derive a general expression for the shift in the vertical position of the mth maximum as a result of placing a thin parallel sheet of glass of index n and thickness d directly over one of the slits. Identify your assumptions. Get solution
19. Plane waves of monochromatic light impinge at an angle θi on a screen containing two narrow slits separated by a distance a. Derive" an equation for the angle measured from the central axis which locates the mth maximum. Get solution
20. Sunlight incident on a screen containing two long narrow slits 0.20 mm apart casts a pattern on a white sheet of paper 2.0 m beyond. What is the distance separating the violet (λ0 = 400 nm) in the first-order band from the red (λ0 = 600 nm) in the second-order band? Get solution
21. To examine the conditions under which the approximations of H3q. (9.23) are valid:(a) Apply the law of cosines to triangle S1S2P in Fig. 9.8c to get...(b) Expand this in a Maclaurin series yielding...(c) In light of Eq. (9.17), show that if (r1, - r2) is to equal a sin θ, it is required that r1, >> a2/λ. Get solution
22. A stream of electrons, each having an energy of 0.5 eV, impinges on a pair of extremely thin slits separated by 10-2mm. What is the distance between adjacent minima on a screen 20 m behind the slits? (me, = 9.108 × 10-31kg, 1 eV = 1.602 × 10-19J.) Get solution
23. It is our intention to produce interference fringes by illuminating some sort of arrangement (Young's Experiment, a thin film, the Michelson Interferometer, etc.) with light at a mean wavelength of 500 nm, having a linewidth of 2.5 X 10-3 nm. At approximately what optical path length difference can you expect the fringes to vanish? [Hint: Think about the coherence length and revisit Problem 7.39.] Get solution
24. Imagine that you have an opaque screen with three horizontal 1 very narrow parallel slits in it. The second slit is a center-to-center distance a beneath the first, and the third is a distance 5a/2 beneath the first, (a) Write a complex exponential expression in terms of δ for the amplitude of the electric field at some point P at an elevation θ on a distant screen where δ = ka sin θ. Prove that...Verify that at θ =0,I(θ) = I (0). Get solution
25. Get solution
26. In the Fresnel double mirror s = 2 m, λ0 = 589 nm, and the separation of the fringes was found to be 0.5 mm. What is the angle of inclination of the mirrors, if the perpendicular distance of the actual point source to the intersection of the two mirrors is I m? Get solution
27. Show that a for the Fresnel biprism of Fig. 9.13 is given by a = 2d(n - l)α. Get solution
28. The Fresnel biprism is used to obtain fringes from a point source that is placed 2 m from the screen, and the prism is midway between the source and the screen. Let the wavelength of the light be λ0 = 500 nm and the index of refraction of the glass be n = 1.5. What is the prism angle, if the separation of the fringes is 0.5 mm? Get solution
29. What is the general expression for the separation of the fringes of a Fresnel biprism of index n immersed in a medium having an index of refraction n'? Get solution
30. Get solution
31. Using Lloyd's mirror, X-ray fringes were observed, the spacing of which was found to be 0.ÖÖ2 5 cm. The wavelength used was 8.33 Ǻ. If the source-screen distance was 3 m, how high above the min or plane was the point source of X-rays placed? Get solution
32. Imagine that we have an antenna at the edge of a lake picking up a signal from a distant radio star (Fig. P.9.24), which is just coming up above the horizon. Write expressions for δ and for the angular position of the star when the antenna detects its first maximum.Figure P.9.24... Get solution
33. If the plate in Fig. 9.17 is glass in air, show that the amplitudes of Elr, E2r, and E3r are respectively 0.2 E0i, 0.192 E0i. and 0.008E0i where E0i is the incident amplitude. Make use of the Fresnel coefficients at normal incidence, assuming no absorption. You might repeal the calculation for a water film in air. Get solution
34. A soap film surrounded by air has an index of refraction of 1.34. If a region of the film appears bright red (λ0 = 633 nm) in normally reflected light, what is its minimum thickness there? Get solution
35. A thin film of ethyl alcohol (n = 1.36) spread on a flat glass plate and illuminated with white light shows a color pattern in reflection. If a region of the film reflects only green light (500 nm) strongly, how thick is it? Get solution
36. A soap film of index 1.34 has a region where it is 550.0 nm thick. Detennine the vacuum wavelengths of the radiation that is not reflected when the film is illuminated from above with sunlight. Get solution
37. Get solution
38. Consider the circular pattern of Haidinger's fringes resulting from a film with a thickness of 2 mm and an index of refraction of 1.5. For monochromatic illumination of λ0 = 600 nm, find the value of m for the central fringe (θt, = 0). Will it be bright or dark? Get solution
39. Illuminate a microscope slide (or even belter, a thin cover-glass slide). Colored fringes can easily be seen with an ordinary fluorescent lamp (although some of the newer versions don't work well at all) serving as a broad source or a mercury street light as a point source. Describe the fringes. Now rotate the glass. Does the pattern change? Duplicate the conditions shown in Figs. 9.18 and 9.19. Try it again with a sheet of plastic food wrap stretched across the lop of a cup. Get solution
40. Fringes are observed when a parallel beam of light of wavelength 500 nm is incident perpendicularly onto a wedge-shaped film with an index of refraction of 1.5. What is the angle of the wedge if the fringe separation is ... cm?Figure P.9.31... Get solution
41. Suppose a wedge-shaped air film is made between two sheets of glass, with a piece of paper 7.618 X 10-5 m thick used as the spacer at their very ends. If light of wavelength 500 nm comes down from directly above, determine the number of bright fringes that will be seen across the wedge. Get solution
42. Get solution
43. Figure P.9.31 illustrates a setup used for testing lenses. Show thatd = x2(R1 – R1,)/2R1R2when d1 and d2 are negligible in comparison with 2R1, and 2R2, respectively. (Recall the theorem from plane geometry that relates the products of the segments of intersecting chords.) Prove that the radius of the mth dark fringe is then...How does this relate to Eq. (9.43)? Get solution
44. Newton rings are observed on a film with quasimonochromatic that has a wavelength of 500 nm. If the 20th bright ring has a radius of 1 cm, what is the radius of curvature of the lens forming one part of the interfering system? Get solution
45. Get solution
46. Get solution
47. Get solution
48. One of the mirrors of a Michelson Interferometer is moved, and 1000 fringe-pairs shift past the hairline in a viewing telescope during the process. If the device is illuminated wilh 500-nm light, how far was the mirror moved? Get solution
49. Get solution
50. Suppose we place a chamber 10.0 cm long with flat parallel windows in one arm of a Michelson Interferometer that is being illuminated by 600-nm light. If the refractive index of air is 1.000 29 and all the air is pumped out of the cell, how many fringe-pairs will shift by in the process? Get solution
51. Get solution
52. A form of the Jamin Interferometer is illustrated in Fig. P.9.38. How does it work? To what use might it be put? Get solution
53. Starting with Eq. (9.53) for the transmitted wave, compute the flux density, thai is, Eq. (9.54). Get solution
54. Given that the mirrors of a Fabry-Perol Interferometer have an...amplitude reflection coefficient of r = 0.8944, find(a) the coefficient of finesse,(b) the half-width,(c) the finesse, and,(d) the contrast factor defined by... Get solution
55. To fill in some of the details in the derivation of the smallest phase increment separating two resolvable Fabry-Perot fringes, that is,...satisfy yourself that...Show that Eq. (9.72) can be rewritten as...When F is large γ is small, and sin (∆δ) = ∆δ. Prove that Eq. (9.73) then follows. Get solution
56. Consider the interference pattern of the Michelson Interferometer as arising from two beams of equal flux density. Using Eq.(9.17), compute the half-width. What is the separation, in S, between adjacent maxima? What then is the finesse? Get solution
57. Satisfy yourself of the fact that a film of thickness λf/4 and index n1, will always reduce the reflectance of the substrate on which it is deposited, as long as ns, > n1, >n0. Consider the simplest case of normal incidence and n0 = 1. Show that this is equivalent to saying that the waves reflected back from the two interfaces cancel one another. Get solution
58. Verify that the reflectance of a substrate can be increased by coating it with a λf/4, high-index layer, that is, n1 > ns* Show that the reflected waves interfere constructively. The quarter-wave stack g(HL)"'Ha can be thought of as a series of such structures. Get solution
59. Determine the refractive index and thickness of a film to be deposited on a glass surface (ng = 1.54) such that no normally incident light of wavelength 540 nm is reflected. Get solution
60. A glass microscope lens having an index of 1.55 is to be coated with a magnesium fluoride film to increase the transmission of normally incident yellow light (λ0 = 550 nm). What minimum thickness should be deposited on the lens? Get solution
61. A glass camera lens with an index of 1.55 is to be coated with a cryolite film (n ≈ 1.30) to decrease the reflection of normally incident green light (λ0 = 500 nm). What thickness should be deposited on the lens? Get solution
62. Using Fig. 9.60, which depicts the geometry of the Shuttle -radar interferometer, show thatz(x) = h – r1, cos θThen use (he Law of Cosines to establish that Eq. (9.108) is correct. Get solution
