Solved: Two light rays, initially in phase and with a

Does the spacing between fringes in a two-slit interference pattern increase, decrease, or stay the same if (a) the slit separation is increased, (b) the color of the light is switched from red to blue, and (c) the whole apparatus is submerged in cooking sherry? (d) If the slits are illuminated with white light, then at any side maximum, does the blue component or the red component peak closer to the central maximum?

(a) If you move from one bright fringe in a two-slit interference pattern to the next one farther out, (b) does the path length difference ilL increase or decrease and (c) by how much does it change, in wavelengths A?

Figure 35-22 shows two light rays that are initially exactly in phase and that reflect from several glass surfaces. Neglect the slight slant in the path of the light in the second arrangement. (a) What is the path length difference of the rays? In wavelengths A, (b) what should that path length difference equal if the rays are to be exactly out of phase when they emerge, and (c) what is the smallest value of d that will allow that final phase difference?

In Fig. 35-23, three pulses of light-a, b, and c-of the same wavelength are sent through layers of plastic having the given indexes of refraction and along the paths indicated. Rank the pulses according to their travel time through the plastic layers, greatest first.

Is there an interference maximum, a minimum, an intermediate state closer to a maximum, or an intermediate state closer to a minimum at point P in Fig. 35-10 if the path length difference of the two rays is (a) 2.2A, (b) 3.5A, (c) 1.8A, and (d) LOA? For each situation, give the value of m associated with the maximum or minimum involved.

Figure 35-24a gives intensity I versus position x on the viewing screen for the central portion of a two-slit interference pattern. The other parts of the figure give phasor diagrams for the electric field components of the waves arriving at the screen from the two slits (as in Fig. 35-13a). Which numbered points on the screen best correspond to which phasor diagram?

Figure 35-25 shows two sources Sl and Sz that emit radio waves of wavelength A in all directions. The sources are exactly in phase and are separated by a distance equal to 1.5;\. The vertical broken line is the perpendicular bisector of the distance between the sources. (a) If we start at the indicated start point and travel along path 1, does the interference produce a maximum all along the path, a minimum all along the path, or alternating maxima and minima? Repeat for (b) path 2 (along an axis through the sources) and (c) path 3 (along a perpendicular to that axis).

Figure 35-26 shows two rays of light, of wavelength 600 nm, that reflect from glass surfaces separated by 150 nm. The rays are initially in phase. (a) What is the path length difference of the rays? (b) When they have cleared the reflection region, are the rays exactly in phase, exactly out of phase, or in some intermediate state?

Light travels along the length of a 1500-nm-Iong nanostructure. When a peak of the wave is at one end of the nanostructure, is there a peak or a valley at the other end if the wavelength is (a) 500 nm and (b) 1000 nm?

Figure 35-27a shows the cross section of a vertical thin film whose width increases downward because gravitation causes slumping. Figure 35-27b is a face-on view of the film, showing four bright (red) interference fringes that result when the film is illuminated with a perpendicular beam of red light. Points in the cross section corresponding to the bright fringes are labeled. In terms of the wavelength of the light inside the film, what is the difference in film thickness between (a) points a and band (b) points band d?

Figure 35-28 shows four situations in which light reflects perpendicularly from a thin film of thickness L sandwiched between much thicker materials. The indexes of refraction are given. In which situations does Eq. 35-36 correspond to the reflections yielding maxima (that is, a bright film)?

Figure 35-29 shows the transmission of light through a thin film in air by a perpendicular beam (tilted in the figure for clarity). (a) Did ray r3 undergo a phase shift due to reflection? (b) In wavelengths, what is the reflection phase shift for ray r4? (c) lfthe film thickness is L, what is the path length Incident difference between rays r3 and r4?

Tho waves of light in air, of wavelength .A = 600.0 nm, are initially in phase. They then both travel through a layer of plastic as shown in Fig. 35-35, with L j = 4.00 !Lm, L2 = 3.50 !Lm, /11 = 1.40, and /12 = 1.60. (a) What mUltiple of .A gives their phase difference after they both have emerged from the layers? (b) If the waves later arrive Fig. 35-35 Problem 13. at some common point with the same amplitude, is their interference fully constructive, fully destructive, intermediate but closer to fully constructive, or intermediate but closer to fully destructive?

In a double-slit arrangement the slits are separated by a distance equal to 100 times the wavelength of the light passing through the slits. (a) What is the angular separation in radians between the central maximum and an adjacent maximum? (b) What is the distance between these maxima on a screen 50.0 cm from the slits?

A double-slit arrangement produces interference fringes for sodium light (.A = 589 nm) that have an angular separation of 3.50 X 10-3 rad. For what wavelength would the angular separation be 10.0% greater?

A double-slit arrangement produces interference fringes for sodium light (.A = 589 nm) that are 0.20 apart. What is the angular fringe separation if the entire arrangement is immersed in water (/1 = 1.33)?

In Fig. 35-36, two radio-frequency point sources SI and S2, r--- d----1 -----.~----~e~---x Sj S2 separated by distance d = 2.0 m, are Fig. 35-36 Problems radiating in phase with .A = 0.50 m. A 17 and 22. detector moves in a large circular path around the two sources in a plane containing them. How many maxima does it detect?

In the two-slit experiment of Fig. 35-10, let angle B be 20.0, the slit separation be 4.24 !Lm, and the wavelength be .A = 500 nm. (a) What multiple of .A gives the phase difference between the waves of rays rj and r2 when they arrive at point P on the distant screen? (b) What is the phase difference in radians? (c) Determine where in the interference pattern point P lies by giving the maximum or minimum on which it lies, or the maximum and minimum between which it lies.

Suppose that Young's experiment is performed with blue-green light of wavelength 500 nm. The slits are 1.20 mm apart, and the viewing screen is 5.40 m from the slits. How far apart are the bright fringes near the center of the interference pattern?

Monochromatic green light, of wavelength 550 nm, illuminates two parallel narrow slits 7.70 !Lm apart. Calculate the angular deviation (Bin Fig. 35-10) of the third-order (m = 3) bright fringe (a) in radians and (b) in degrees

In a double-slit experiment, the distance between slits is 5.0 mm and the slits are 1.0 m from the screen. Two interference patterns can be seen on the screen: one due to light of wavelength 480 nm, and the other due to light of wavelength 600 nm. What is the separation on the screen between the third-order (m = 3) bright fringes of the two interference patterns?

In Fig. 35-36, two isotropic point sources S 1 and S2 emit identical light waves in phase at wavelength .A. The sources lie at separation d on an x axis, and a light detector is moved in a circle of large radius around the midpoint between them. It detects 30 points of zero intensity, including two on the x axis, one of them to the left of the sources and the other to the right of the sources. What is the value of d/.A?

In Fig. 35-37, sources A and B emit long-range radio waves of wavelength 400 m, with the phase of the emission from A ahead of that A B from source B by 90. The distance rA Fig. 35-37 Problem 23. from A to detector D is greater than the corresponding distance rB by 100 m. What is the phase difference of the waves at D?

In Fig. 35-38, two isotropic point sources Sl and S2 emit light in phase at wavelength .A and at the same amplitude. The sources are separated by distance 2d = 6.00.A. They lie on an axis that is parallel to an x axis, which runs along a viewing screen at distance D = 20.DA. The origin lies on the perpendicular bisector between the sources. The figure shows two rays reaching point P on the screen, at position Xp. (a) At what value of Xp do the rays have the minimum possible phase o I p x difference? (b) What multiple of .A gives that minimum phase difference? (c) At what value of Xp do the rays have the maximum possible phase difference? What mUltiple of .A gives (d) that maximum phase difference and (e) the phase difference when Xp = 6.00.A? (f) When Xp = Sj ~ __ ~L-__ ~~~ __ ___ 6.00.A, is the resulting intensity at 1_. d+d-l point P maximum, minimum, inter- I ~I mediate but closer to maximum, or Fig. 35-38 Problem 24. intermediate but closer to minimum?

In Fig. 35-39, two isotropic point sources of light (Sl and S2) are separated by distance 2.70 !Lm along a y axis and emit in phase at wavelength 900 nm and at the same amplitude. A light detector is lo-------'--

In a double-slit experiment, the fourth-order maximum for a wavelength of 450 nm occurs at an angle of B = 90. Thus, it is on the verge of being eliminated from the pattern because B cannot exceed 90 in Eq. 35-14. (a) What range of wavelengths in the visible range (400 nm to 700 nm) are not present in the third-order maxima? To eliminate all of the visible light in the fourth-order maximum, (b) should the slit separation be increased or decreased and (c) what least change in separation is needed?

A thin flake of mica (/1 = 1.58) is used to cover one slit of a double-slit interference arrangement. The central point on the viewing screen is now occupied by what had been the seventh bright side fringe (m = 7). If.A = 550 nm, what is the thickness ofthe mica?

Figure 35-39 shows two isotropic point sources of light (Sj and S2) that emit in phase at wavelength 400 nm and at the same amplitude. A detection point P is shown on an x axis that extends through source Sj. The phase difference between the light arriving at point P from the two sources is to be measured as P is moved along the x axis from x = 0 out to x = +00. The results out to Xs = 10 X 10-7 m are given in Fig. 35-40. On the way out to +00, what is the greatest value of x at which the light arriving at P from SI is exactly out of phase with the light arriving at P from S2?

Two waves of the same frequency have amplitudes 1.00 and 2.00. They interfere at a point where their phase difference is 60.0. What is the resultant amplitude?

Find the sum y of the following quantities: Yl = 10 sin wt and Y2 = 8.0 sin( wt + 30).

Add the quantities Yl = 10 sin wt,Yz = 15 sin(wt + 30), and Y3 = 5.0 sin( wt - 45) using the phasor method

In the double-slit experiment of Fig. 35-10, the electric fields of the waves arriving at point P are given by El = (2.00 ItV/m) sin[(1.26 X 1Q15)t] E2 = (2.00 ItV/m) sin[(1.26 X 1Q15)t + 39.6 rad], where time t is in seconds. (a) What is the amplitude of the resultant electric field at point P? (b) What is the ratio of the intensity Ip at point P to the intensity Icen at the center of the interference pattern? (c) Describe where point P is in the interference pattern by giving the maximum or minimum on which it lies, or the maximum and minimum between which it lies. In a phasor diagram of the electric fields, (d) at what rate would the phasors rotate around the origin and (e) what is the angle between the phasors?

Three electromagnetic waves travel through a certain point P along an x axis. They are polarized parallel to a Y axis, with the following variations in their amplitudes. Find their resultant at P. El = (10.0 ItV/m) sin[(2.0 X 1014 rad/s)t] E2 = (5.00 ItV/m) sin[(2.0 X 1014 rad/s)t + 45.0] E3 = (5.00 ItV/m) sin[(2.0 X 1014 rad/s)t 45.0]

In the double-slit experiment of Fig. 35-10, the viewing screen is at distance D = 4.00 m, point P lies at distance Y = 20.5 cm from the center of the pattern, the slit separation d is 4.50 Itm, and the wavelength A is 580 nm. (a) Determine where point P is in the interference pattern by giving the maximum or minimum on which it lies, or the maximum and minimum between which it lies. (b) What is the ratio of the intensity I p at point P to the intensity Icen at the center of the pattern?

We wish to coat flat glass (n = 1.50) with a transparent material (n = 1.25) so that reflection of light at wavelength 600 nm is eliminated by interference. What minimum thickness can the coating have to do this?

A 600-nm-thick soap film (n = 1.40) in air is illuminated with white light in a direction perpendicular to the film. For how many different wavelengths in the 300 to 700 nm range is there (a) fully constructive interference and (b) fully destructive interference in the reflected light?

The rhinestones in costume jewelry are glass with index of refraction 1.50. To make them more reflective, they are often coated with a layer of silicon monoxide of index of refraction 2.00. What is the minimum coating thickness needed to ensure that light of wavelength 560 nm and of perpendicular incidence will be reflected from the two surfaces of the coating with fully constructive interference?

White light is sent downward onto a horizontal thin film that is sandwiched between two materials. The indexes of refraction are 1.80 for the top material, 1.70 for the thin film, and 1.50 for the bottom material. The film thickness is 5.00 X 10-7 m. Of the visible wavelengths (400 to 700 nm) that result in fully constructive interference at an observer above the film, which is the (a) longer and (b) shorter wavelength? The materials and film are then heated so that the film thickness increases. (c) Does the light resulting in fully constructive interference shift toward longer or shorter wavelengths?

Light of wavelength 624 nm is incident perpendicularly on a soap film (n = 1.33) suspended in air. What are the (a) least and (b) second least thicknesses of the film for which the reflections from the film undergo fully constructive interference?

A thin film of acetone (n = 1.25) coats a thick glass plate (n = 1.50). White light is incident normal to the film. In the reflections, fully destructive interference occurs at 600 nm and fully constructive interference at 700 nm. Calculate the thickness of the acetone film.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

Reflection by thin layers. In Fig. 35-41, light is incident perpendicularly 1'1 1'2 on a thin layer of material 2 that lies be- -::t:::::!==;;;;;;;:::p~-l tween (thicker) materials 1 and 3. (The rays are tilted only for clarity.) The waves of rays rl and r2 interfere, and here we consider the type of interfer- Fig. 35-41 Problems ence to be either maximum (max) or 41 through 52. minimum (min). For this situation, each problem in Table 35-2 refers to the indexes of refraction n 1> nZ, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third least thickness as indicated.

The reflection of perpendicularly incident white light by a soap film in air has an interference maximum at 600 nm and a minimum at 450 nm, with no minimum in between. If n = 1.33 for the film, what is the film thickness, assumed uniform?

A plane wave of monochromatic light is incident normally on a uniform thin film of oil that covers a glass plate. The wave- length of the source can be varied continuously. Fully destructive interference of the reflected light is observed for wavelengths of 500 and 700 nm and for no wavelengths in between. If the index of refraction of the oil is 1.30 and that of the glass is 1.50, find the thickness of the oil film.

A disabled tanker leaks kerosene (n = 1.20) into the Persian Gulf, creating a large slick on top of the water (n = 1.30). (a) If you are looking straight down from an airplane, while the Sun is overhead, at a region of the slick where its thickness is 460 nm, for which wavelength(s) of visible light is the reflection brightest because of constructive interference? (b) If you are scuba diving directly under this same region of the slick, for which wavelength(s) of visible light is the transmitted intensity strongest?

A thin film, with a thickness of 272.7 nm and with air on both sides, is illuminated with a beam of white light. The beam is perpendicular to the film and consists of the full range of wavelengths for the visible spectrum. In the light reflected by the film, light with a wavelength of 600.0 nm undergoes fully constructive interference. At what wavelength does the reflected light undergo fully destructive interference? (Hint: You must make a reasonable assumption about the index of refraction.)

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

Transmission through thin layers. In Fig. 35-42, light is incident perpendicularly on a thin layer of material 2 that lies between (thicker) materials 1 and 3. (The rays are tilted only for clarity.) Part of the light ends up in material 3 as ray 1'3 (the light does not reflect inside material 2) and 1'4 (the light reflects twice inside material 2). The waves of 1'3 and 1'4 interfere, and here we consider the nj type of interference to be either maximum (max) or minimum (min). For this situation, each problem in Table 35-3 refers to the indexes of refraction nj, nz, and n3, the type of interference, the thin-layer thickness L in nanometers, and the wavelength A in nanometers of the light as measured in air. Where A is missing, give the wavelength that is in the visible range. Where L is missing, give the second least thickness or the third nj 57 1.55 58 1.32 59 1.68 60 1.50 61 1.32 62 1.68 63 1.40 64 1.40 65 1.60 66 1.60 67 1.50 68 1.55 Fig. 35-42 Problems 57 through 68. least thickness as indicated.

n Fig. 35-43, a broad beam of light of wavelength 630 nm is incident at 900 on a thin, wedge-shaped film with index of refraction 1.50. Transmission gives 10 bright and 9 dark fringes along the film's length. What is the left -to-right change in film thickness?

In Fig. 35-44, a broad beam of light of wavelength 620 nm is sent directly downward through the top plate of a pair of glass plates touching at the left end. The air between the plates acts as a thin film, and an interference pattern can be seen from above the L 285 3rd 415 380 325 2nd 2nd 210 2nd 200 2nd 3rd Incident light 382 342 482 632 587 612 plates. Initially, a dark fringe lies at the Fig. 35-44 Problems left end, a bright fringe lies at the right 70-74. end, and nine dark fringes lie between those two end fringes. The plates are then very gradually squeezed together at a constant rate to decrease the angle between them. As a result, the fringe at the right side changes between being bright to being dark every 15.0 s. (a) At what rate is the spacing between the plates at the right end being changed? (b) By how much has the spacing there changed when both left and right ends have a dark fringe and there are five dark fringes between them?

In Fig. 35-44, two microscope slides touch at one end and are separated at the other end. When light of wavelength 500 nm shines vertically down on the slides, an overhead observer sees an interference pattern on the slides with the dark fringes separated by 1.2 mm. What is the angle between the slides?

In Fig. 35-44, a broad beam of monochromatic light is directed perpendicularly through two glass plates that are held together at one end to create a wedge of air between them. An observer intercepting light reflected from the wedge of air, which acts as a thin film, sees 4001 dark fringes along the length of the wedge. When the air between the plates is evacuated, only 4000 dark fringes are seen. Calculate to six significant figures the index of refraction of air from these data.

In Fig. 35-44, a broad beam of light of wavelength 683 nm is sent directly downward through the top plate of a pair of glass plates. The plates are 120 mm long, touch at the left end, and are separated by 48.0 p,m at the right end. The air between the plates acts as a thin film. How many bright fringes will be seen by an observer looking down through the top plate?

Tho rectangular glass plates (n = 1.60) are in contact along one edge and are separated along the opposite edge (Fig. 35-44). Light with a wavelength of 600 nm is incident perpendicularly onto the top plate. The air between the plates acts as a thin film. Nine dark fringes and eight bright fringes are observed from above the top plate. If the distance between the two plates along the separated edges is increased by 600 nm, how many dark fringes will there then be across the top plate?

Figure 35-45a shows a lens with radius of curvature R lying on a flat glass plate and illuminated from above by light with wavelength A. Figure 35-45b (a photograph taken from above the lens) shows that circular interference fringes (called Newton's rings) appear, associated with the variable thickness d of (a) Incident light Air Glass R Fig.35-45 Problems 75-77. (Courtesy Bausch & Lomb) PROBLEMS 987 the air film between the lens and the plate. Find the radii r of the interference maxima assuming rlR q 1.

In a Newton's rings experiment (see Problem 75), the radius of curvature R of the lens is 5.0 m and the lens diameter is 20 mm. (a) How many bright rings are produced? Assume that A = 589 nm. (b) How many bright rings would be produced if the arrangement were immersed in water (n = 1.33)?

A Newton's rings apparatus is to be used to determine the radius of curvature of a lens (see Fig. 35-45 and Problem 75). The radii of the nth and (n + 20)th bright rings are measured and found to be 0.162 and 0.368 cm, respectively, in light of wavelength 546 nm. Calculate the radius of curvature of the lower surface of the lens.

A thin film of liquid is held in a horizontal circular ring, with air on both sides of the film. A beam of light at wavelength 550 nm is directed perpendicularly onto the film, and the intensity I of its reflection is monitored. Figure 35-46 gives intensity I as a function of time t; the horizontal scale is set by ts = 20.0 s. The intensity changes because of evaporation from the two sides of the film. Assume that the film is flat and has parallel sides, a radius of 1.80 cm, and an index of refraction of 1.40. Also assume that the film's volume decreases at a constant rate. Find that rate.

If mirror M2 in a Michelson interferometer (Fig. 35-21) is moved through 0.233 mm, a shift of 792 bright fringes occurs. What is the wavelength of the light producing the fringe pattern?

A thin film with index of refraction n = 1.40 is placed in one arm of a Michelson interferometer, perpendicular to the optical path. If this causes a shift of 7.0 bright fringes of the pattern produced by light of wavelength 589 nm, what is the film thickness?

In Fig. 35-47, an airtight chamber of length d = 5.0 cm is placed in one of the arms of a Michelson interferometer. (The glass window on each end of the chamber has negligible thickness.) Light of wavelength A = 500 nm is used. Evacuating the air from the chamber causes a shift of 60 bright fringes. From these data and to six significant figures, find the index of refraction of air at atmospheric pressure.

The element sodium can pump emit light at two wavelengths, Al = Fig. 35-47 Problem 81. 588.9950 nm and A2 = 589.5924 nm. Light from sodium is being used in a Michelson interferometer (Fig. 35-21). Through what distance must mirror M2 be moved if the shift in the fringe pattern for one wavelength is to be 1.00 fringe more than the shift in the fringe pattern for the other wavelength?

Two light rays, initially in phase and with a wavelength of 500 nm, go through different paths by reflecting from the various mirrors shown in Fig. 35-48. (Such a reflection does not itself produce a phase shift.) (a) What least value of distance d will put the rays exactly out of phase when they emerge from the region? (Ignore the slight :--d-+-d~: I I I ,,' I I Ray 1 tilt of the path for ray 2.) (b) Fig. 35-48 Problem 83. Repeat the question assuming that the entire apparatus is immersed in a protein solution with an index of refraction of 1.38.

In Figure 35-49, two isotropic point sources Sl and S2 emit light in phase at wavelength A and at the same amplitude. The sources are separated by distance d = 6.0OA on an x axis. A viewing screen is at distance D = 20.0A from S2 and parallel to the y axis. The figure shows two rays reaching point P on the screen, at height yp. (a) At what value of y p do the rays have the minimum possible phase difference? (b) What multiple of A gives that minimum phase difference? (c) At what value of yp do the rays have the maximum possible phase difference? What multiple of A gives (d) that maximum phase difference and (e) the phase difference when yp = d? (f) When yp = d, is the resulting intensity at point P maximum, minimum, intermediate but closer to maximum, or intermediate but closer to minimum?

A double-slit arrangement produces bright interference fringes for sodium light (a distinct yellow light at a wavelength of A = 589 nm). The fringes are angularly separated by 0.30 near the center of the pattern. What is the angular fringe separation if the entire arrangement is immersed in water, which has an index of refraction of 1.33?

In Fig. 35-50a, the waves along rays 1 and 2 are initially in phase, with the same wavelength A in air. Ray 2 goes through a material with length L and index of refraction n. The rays are then reflected by mirrors to a common point P on a screen. Suppose that we can vary n from n = 1.0 to n = 2.5. Suppose also that, from n = 1.0 to n = ns = 1.5, the intensity I of the light at point P varies with n as given in Fig. 35-50b. At what values of 11 greater than 1.4 is intensity I (a) maximum and (b) zero? (c) What multiple of A gives the phase difference between the rays at point P when n = 2.0?

In Fig. 35-50a, the waves along rays 1 and 2 are initially in phase, with the same wavelength A in air. Ray 2 goes through a material with length L and index of refraction n. The rays are then reflected by mirrors to a common point P on a screen. Suppose that we can vary L from 0 to 2400 nm. Suppose also 0 that, from L = 0 to Ls = 900 nm, the I L, L(nm) intensity I of the light at point P Fig. 35-51 Problem 87. varies with L as given in Fig. 35-51. At what values of L greater than Ls is intensity I (a) maximum and (b) zero? (c) What multiple of A gives the phase difference between ray 1 and ray 2 at common point P when L = 1200 nm?

Light of wavelength 700.0 nm is sent along a route of length 2000 nm. The route is then filled with a medium having an index of refraction of 1.400. In degrees, by how much does the medium phase-shift the light? Give (a) the full shift and (b) the equivalent shift that has a value less than 360.

In Fig. 35-52, a microwave transmitter at height a above the water level of a wide lake transmits microwaves of wavelength A toward a receiver on the opposite shore, a distance x above the water level. The microwaves reflecting from the water interfere with the microwaves arriving directly from the transmitter. Assuming that the lake width D is much greater than a and x, and that A 2: a, find an expression that gives the values of x for which the signal at the receiver is maximum. (Hint: Does the reflection cause a phase change?)

In Fig. 35-53, two isotropic point sources Sl and S2 emit light at wavelength A = 400 nm: Source Sl is located at y = 640 nm; source S2 is located at y = -640 nm. At point PI (at x = 720 nm), the wave from S2 arrives ahead of the wave from Sl by a phase difference of 0.6001T rad. (a) What multiple of A gives the phase difference between the waves from the two sources as the waves arrive at point Pz, which is located at y = 720 nm. (The figure is not drawn to scale.) (b) If the waves arrive at P2 with equal amplitudes, is the interference there fully constructive, fully destructive, intermediate but closer to fully construc----+--~~-x tive, or intermediate but closer to fully destructive?

Ocean waves moving at a speed of 4.0 mls are approaching a beach at angle 01 = 30 to the normal, as shown from above in Fig. 35-54. Suppose the water depth changes abruptly at a certain distance from the beach and the wave speed there drops to 3.0 mls. (a) Close to the beach, what is the angle e2 between the direction of wave motion and the normal? (Assume the same law of refraction as for light.) (b) Explain why most waves come in normal to a shore even though at large distances they approach at a variety of angles.

Figure 35-55a shows two light rays that are initially in phase as they travel upward through a block of plastic, with wavelength 400 nm as measured in air. Light ray 1'1 exits directly into air. However, before light ray 1'2 exits into air, it travels through a liquid in a hollow cylinder within the plastic. Initially the height L liq of the liquid is 40.0 tim, but then the liquid begins to evaporate. Let be the phase difference between rays 1'1 and 1'2 once they both exit into the air. Figure 35-55b shows versus the liquid's height L liq until the liquid disappears, with given in terms of wavelength and the horizontal scale set by Ls = 40.00 tim. What are (a) the index of refraction of the plastic and (b) the index of refraction of the liquid?

If the distance between the first and tenth minima of a double-slit pattern is 18.0 mm and the slits are separated by 0.150 mm with the screen 50.0 cm from the slits, what is the wavelength of the light used?

Figure 35-56 shows an optical fiber in which a central plastic core of index of refraction nl = 1.58 is surrounded by a plastic sheath of index of refraction n2 = 1.53. Light can travel along different paths within the central core, leading ..------------- Fig. 35-56 Problem 94. --_.' to different travel times through the fiber. This causes an initially short pulse of light to spread as it travels along the fiber, resulting in information loss. Consider light that travels directly along the central axis of the fiber and light that is repeatedly reflected at the critical angle along the core-sheath interface, reflecting from side to side as it travels down the central core. If the fiber length is 300 m, what is the difference in the travel times along these two routes?

Tho parallel slits are illuminated with monochromatic light of wavelength 500 nm. An interference pattern is formed on a screen some distance from the slits, and the fourth dark band is located 1.68 cm from the central bright band on the screen. (a) What is the path length difference corresponding to the fourth dark band? (b) What is the distance on the screen between the central bright band and the first bright band on either side of the central band? (Hint: The angle to the fourth dark band and the angle to the first bright band are small enough that tan e = sin e.)

A camera lens with index of refraction greater than 1.30 is coated with a thin transparent film of index of refraction 1.25 to eliminate by interference the reflection of light at wavelength A that is incident perpendicularly on the lens. What multiple of A gives the minimum film thickness needed?

Light of wavelength A is used in a Michelson interferometer. Let x be the position of the movable mirror, with x = 0 when the arms have equal lengths d2 = dl Write an expression for the intensity of the observed light as a function of x, letting 1m be the maximum intensity.

In two experiments, light is to be sent along the two paths shown in Fig. 35-34 by reflecting it from the various flat surfaces shown. In the first experiment, rays 1 and 2 are initially in phase and have a wavelength of 620.0 nm. In the second experiment, rays 1 and 2 are initially in phase and have a wavelength of 496.0 nm. What least value of distance L is required such that the 620.0 nm waves emerge from the region exactly in phase but the 496.0 nm waves emerge exactly out of phase?

Figure 35-57 shows the design of a Texas arcade game. Four laser pistols are pointed toward the center of an array of plastic layers where a clay armadillo is the target. The indexes of refraction of the layers are nl = 1.55, n2 = 1.70, n3 = 1.45, n4 = 1.60, ns = 1.45, n6 = 1.61, n7 = 1.59, ns = 1.70, and n9 = 1.60. The layer thicknesses are either 2.00 mm or 4.00 mm, as drawn. What is the travel time through the layers for the laser burst from (a) pistol 1, (b) pistol2, (c) pistol 3, and (d) pistol4? (e) If the pistols are fired simultaneously, which laser burst hits the target first?

A thin film suspended in air is 0.410 tim thick and is illuminated with white light incident perpendicularly on its surface. The index of refraction of the film is 1.50. At what wavelength will visible light that is reflected from the two surfaces of the film undergo fully constructive interference?

Find the slit separation of a double-slit arrangement that will produce interference fringes 0.018 rad apart on a distant screen when the light has wavelength A = 589 nm.

In a phasor diagram for any point on the viewing screen for the two-slit experiment in Fig. 35-10, the resultant-wave phasor rotates 60.0 in 2.50 X 10-16 s. What is the wavelength?