Solution: At what angle should the axes of two Polaroids be

Problem 3Q Why is light sometimes described as rays and sometimes as waves?

Problem 4Q We can hear sounds around corners but we cannot see around corners; yet both sound and light are waves. Explain the difference.

Problem 5Q If Young's double-slit experiment were submerged in water, how would the fringe pattern be changed?

Problem 6Q Monochromatic red light is incident on a double slit, and the interference pattern is viewed on a screen some distance away. Explain how the fringe pattern would change if the red light source is replaced by a blue light source.

Problem 7Q Two rays of light from the same source destructively interfere if their path lengths differ by how much?

Problem 10P In a double-slit experiment, the third-order maximum for light of wavelength 500 nm is located 12 mm from the central bright spot on a screen 1.6 m from the slits. Light of wavelength 650 nm is then projected through the same slits. How far from the central bright spot will the second-order maximum of this light be located?

(II) Suppose a thin piece of glass is placed in front of the lower slit in Fig. so that the two waves enter the slits \(18^{\circ}\) out of phase (Fig. ). Describe in detail the interference pattern on the screen. Equation Transcription: Text Transcription: 18^\circ

Problem 10Q Why doesn't the light from the two headlights of a distant car produce an interference pattern?

Problem 14Q For diffraction by a single slit, what is the effect of increasing (a) the slit width, (b) the wavelength?

Problem 15Q Hold one hand close to your eye and focus on a distant light source through a narrow slit between two fingers. (Adjust your fingers to obtain the best pattern.) Describe the pattern that you see.

Problem 16Q What happens to the diffraction pattern of a single slit if the whole apparatus is immersed (a) water, (b) a vacuum, instead of in air.

Problem 17Q Describe the single-slit diffraction pattern produced when white light falls on a slit having a width of (a) 60 nm, (£) 60,000 nm.

Problem 18Q What is the difference in the interference patterns formed by two slits 10-4 apart as compared to a diffraction grating containing! 04 slits_cm?

Problem 19Q For a diffraction grating, what is the advantage of (a) many slits, (b) closely spaced slits?

Problem 20Q White light strikes (a) a diffraction grating and (6) a prism. A rainbow appears on a wall just below the direction of the horizontal incident beam in each case What is the color of the top of the rainbow in each case? Explain.

Problem 22Q Why are interference fringes noticeable only for a thin film like a soap bubble and not for a thick piece of glass?

Why are Newton's rings (Fig. 24-31) closer together farther from the center?

Problem 25Q Some coated lenses appear greenish yellow when seen by reflected light. What reflected wavelengths do you suppose the coating is designed to eliminate completely?

Problem 26Q A drop of oil on a pond appears bright at its edges, where its thickness is much less than the wavelengths of visible light .What can you say about the index of refraction of the oil compared to that of water?

Problem 27Q What does polarization tell us about the nature of light?

Problem 28Q Explain the advantage of polarized sunglasses over plain tinted sunglasses.

Problem 29Q How can you tell if a pair of sunglasses is polarizing or not?

What would be the color of the sky if the Earth had no atmosphere?

Problem 32Q If the Earth's atmosphere were 50 times denser than it is, would sunlight still be white, or would it be some other color?

Problem 35P (II) Two (and only two) full spectral orders can be seen on either side of the central maximum when white light is sent through a diffraction grating. What is the maximum number of slits per cm for the grating?

Problem 37P A He-Ne gas laser which produces monochromatic light of a known wavelength ? = 6.328 × 10?7m is used to calibrate a reflection grating in a spectroscope. The first-order diffraction line is found at an angle of 21.5° to the incident beam. How many lines per meter are there on the grating?

(II) A lens appears greenish yellow (\(\lambda=570 \mathrm{~nm}\) is strongest) when white light reflects from it. What minimum thickness of coating \((n=1.25)\) do you think is used on such a glass \((n=1.52)\) lens, and why? Equation Transcription: Text Transcription: \lambda=570 nm (n=1.25) (n=1.52)

Problem 45P (II) How thick (minimum) should the air layer be between two flat glass surfaces if the glass is to appear bright when 450-nm light is incident normally? What if the glass is to appear dark?

(II) What would Brewster's angle be for reflections off the surface of water for light coming from beneath the surface? Compare to the angle for total internal reflection, and to Brewster's angle from above the surface.

Problem 71GP Show that the second- and third-order spectra of white light produced by a diffraction grating always overlap. What wavelengths overlap?

Problem 76GP Calculate the minimum thickness needed for an anti reflective coating (n=1.38) applied to a glass lens in order to eliminate (a) blue (450 nm), or (h) red (720 nm) reflections for light at normal incidence.

Problem 79GP At what angle above the horizon is the Sun when light reflecting off a smooth lake is polarized most strongly?

Problem 80GP At what angle should the axes of two Polaroids be placed so as to reduce the intensity of the incident unpolarized light by an additional factor (after the first Polaroid cuts it in half) of (a) 4, (b) 10, (c) 100?

Problem 85GP A thin film of soap (n=1.34) coats a piece of flat glass (n=1.52). How thick is the film if it reflects 643-nm red light most strongly when illuminated normally by white light?

Problem 8P Water waves having parallel crests 2.5 cm apart pass through two openings 5.0 cm apart in a board. At a point 2.0 m beyond the board, at what angle relative to the “straight-through” direction would there be little or no wave action?

Problem 33P (II) A source produces first-order lines when incident normally on a 9800-slit/cm diffraction grating at angles 28.8°, 36.7°, 38.6°, and 41.2°. What are the wavelengths?

(I) Monochromatic light falling on two slits \(0.016 \mathrm{~mm}\) apart produces the fifth-order fringe at an \(8.8^{\circ}\) angle. What is the wavelength of the light used?

Problem 1Q Does Huygens’ principle apply to sound waves? To water waves? Explain.

Problem 2P The third-order fringe of 610 nm light is observed at an angle of 18° when the light falls on two narrow slits. How far apart are the slits?

Problem 2Q What is the evidence that light is energy?

Problem 3P Monochromatic light falls on two very narrow slits 0.048 mm apart. Successive fringes on a screen 5.00 m away are 6.5 cm apart near the center of the pattern. Determine the wavelength and frequency of the light.

Problem 4P A parallel beam of light from a He-Ne laser, with a wavelength 656 nm, falls on two very narrow slits 0.060 mm apart. How far apart are the fringes in the center of the pattern on a screen 3.6 in away?

Problem 5P Light of wavelength 680 nm falls on two slits and produces an interference pattern in which the fourth-order fringe is 38 mm from the central fringe on a screen 2.0 m away. What is the separation of the two slits?

(II) In a double-slit experiment, it is found that blue light of wavelength 460 nm gives a second-order maximum at a certain location on the screen. What wavelength of visible light would have a minimum at the same location?

Problem 6P If 720-nm and 660-nm light passes through two slits 0.58 mm apart, how far apart are the second-order fringes for these two wavelengths on a screen 1.0 m away?

Problem 8Q Why was the observation of the double-slit interference pattern more convincing evidence for the wave theory of light than the observation of diffraction?

Compare a double-slit experiment for sound waves to that for light waves. Discuss the similarities and differences.

Problem 11P Two narrow slits separated by 1.0 mm are illuminated by 544 nm light. Find the distance between adjacent bright fringes on a screen 5.0 m from the slits.

Suppose white light falls on the two slits of Fig. 24–7, but one slit is covered by a red filter (700 nm) and the other by a blue filter (450 nm). Describe the pattern on the screen.

Problem 12P Light of wavelength 480 nm in air falls on two slits 6.00 × 10?2 mm apart. The slits are immersed in water, as is a viewing screen 40.0 cm away. How far apart are the fringes on the screen?

Problem 12Q When white light passes through a flat piece of window glass, it is not broken down into colors as it is by a prism. Explain.

Problem 13P A very thin sheet of plastic (n = 1.60) covers one slit of a double-slit apparatus illuminated by 640-nm light. The center point on the screen, instead of being a maximum, is dark. What is the (minimum) thickness of the plastic?

Problem 13Q For both converging and diverging lenses, discuss how the focal length for red light differs from that for violet light.

(I) By what percent, approximately, does the speed of red light (700 nm) exceed that of violet light (400 nm) in silicate flint glass? (See Fig. 24–14.)

Problem 15P A light beam strikes a piece of glass at a 60.00° incident angle. The beam contains two wavelengths, 450.0 nm and 700.0 nm, for which the index of refraction of the glass is 1.4820 and 1.4742, respectively. What is the angle between the two refracted beams?

(III) A parallel beam of light containing two wavelengths, \(\lambda_1=450~ \mathrm{nm}\) and \(\lambda_2=650~ \mathrm{nm}\), enters the silicate flint glass of an equilateral prism as shown in Fig. 24–58. At what angle does each beam leave the prism (give angle with normal to the face)? Equation Transcription: Text Transcription: lambda_1=450 nm lambda_2=650 nm 45.0^o 60^o 60^o 60^o theta_1 theta_2

Problem 17P If 580-nm light falls on a slit 0.0440 mm wide, what is the full angular width of the central diffraction peak?

(I) Monochromatic light falls on a slit that is \(2.60 \times 10^{-3} \mathrm{~mm}\) wide. If the angle between the first dark fringes on either side of the central maximum is \(35.0^{\circ}\) (dark fringe to dark fringe), what is the wavelength of the light used?

Problem 19P Light of wavelength 520 nm falls on a slit that is 3.20 × 10?3 mm wide. Estimate how far the first brightish diffraction fringe is from the strong central maximum if the screen is 10.0 m away.

Problem 20P A single slit 1.0 mm wide is illuminated by 450-nm light. What is the width of the central maximum (in cm) in the diffraction pattern on a screen 5.0 m away?

Problem 21P Monochromatic light of wavelength 653 nm falls on a slit. If the angle between the first bright fringes on either side of the central maximum is 32°, estimate the slit width.

Problem 21Q For light consisting of wavelengths between 400 nm and 700 nm, incident normally on a diffraction grating, for what orders (if any) would there be overlap in the observed spectrum? Does your answer depend on the slit spacing?

Problem 22P How wide is the central diffraction peak on a screen 2.30 m behind a 0.0348-mm-wide slit illuminated by 589-nm light?

(II) When blue light of wavelength 440 nm falls on a single slit, the first dark bands on either side of center are separated by \(55.0^\circ\). Determine the width of the slit.

When a compact disk (CD) is held at an angle in white light, the reflected light is a full spectrum (Fig. 24–56). Explain. What would you expect to see if monochromatic light was used?

Problem 24P When violet light of wavelength 415 nm falls on a single slit, it creates a central diffraction peak that is 9.20 cm wide on a screen that is 2.55 m away. How wide is the slit?

Problem 25P If a slit diffracts 650-nm light so that the diffraction maximum is 4.0 cm wide on a screen 1.50 m away, what will be the width of the diffraction maximum for light of wavelength 420 nm?

Problem 26P For a given wavelength ?, what is the maximum slit width for which there will be no diffraction minima?

Problem 27P At what angle will 560-nm light produce a second-order maximum when falling on a grating whose slits are 1.45 × 10?3 cm apart?

Problem 28P A 3500-line/cm grating produces a third-order fringe at a 28.0° angle. What wavelength of light is being used?

Problem 29P How many lines per centimeter does a grating have if the third-order occurs at an 18.0° angle for 630-nm light?

A grating has 8300 lines/cm. How many complete spectral orders can be seen (400 nm to 700 nm) when it is illuminated by white light?

Problem 30Q Two polarized sheets rotated at an angle of 90° with respect to each other will not let any light through. Three polarized sheets, each rotated at an angle of 45° with respect to each other, will let some light through. What will happen to unpolarized light if you align four polarized sheets, each rotated at an angle of 30° with respect to the one in front of it?

Problem 31P The first-order line of 589-nm light falling on a diffraction grating is observed at a 15.5° angle. How far apart are the slits? At what angle will the third order be observed?

Problem 32P A diffraction grating has 6.0 × 105 lines/m. Find the angular spread in the second-order spectrum between red light of wavelength 7.0 × 10?7m and blue light of wavelength 4.5 × 10?7 m.

Problem 34P What is the highest spectral order that can be seen if a grating with 6000 lines per cm is illuminated with 633-nm laser light? Assume normal incidence.

Problem 36P White light containing wavelengths from 410 nm to 750 nm falls on a grating with 8500 lines/cm. How wide is the first-order spectrum on a screen 2.30 m away?

Problem 38P Two first-order spectrum lines are measured by a 9500-line/cm spectroscope at angles, on each side of center, of +26°38?, +41°08? and ?26°48?, ?41°19?. What are the wavelengths?

Problem 39P If a soap bubble is 120 nm thick, what wavelength is most strongly reflected at the center of the outer surface when illuminated normally by white light? Assume that n = 1.34.

(I) How far apart are the dark fringes in Example 24–8 if the glass plates are each 26.5 cm long?

Problem 41P What is the smallest thickness of a soap film (n = 1.42) that would appear black if illuminated with 480-nm light? Assume there is air on both sides of the soap film.

(II) A total of 31 bright and 31 dark Newton’s rings (not counting the dark spot at the center) are observed when 550-nm light falls normally on a planoconvex lens resting on a flat glass surface (Fig. 24–31). How much thicker is the center than the edges?

(II) A fine metal foil separates one end of two pieces of optically flat glass, as in Fig. 24–33. When light of wavelength 670 nm is incident normally, 28 dark lines are observed (with one at each end). How thick is the foil?

Problem 46P A piece of material, suspected of being a stolen diamond (n = 2.42), is submerged in oil of refractive index 1.43 and illuminated by unpolarized light. It is found that the reflected light is completely polarized at an angle of 59°. Is it diamond?

Problem 47P A thin film of alcohol (n = 1.36) lies on a flat glass plate (n = 1.51). When monochromatic light, whose wavelength can be changed, is incident normally, the reflected light is a minimum for ? = 512 nm and a maximum for ? = 640 nm. What is the minimum thickness of the film?

(III) When a Newton’s ring apparatus (Fig. 24–31) is immersed in a liquid, the diameter of the eighth dark ring decreases from 2.92 cm to 2.48 cm. What is the refractive index of the liquid?

Problem 49P What is the wavelength of the light entering an inter ferometer if 644 bright fringes are counted when the movable mirror moves 0.225 mm?

Problem 50P A micrometer is connected to the movable mirror of an interferometer. When the micrometer is tightened down on a thin metal foil, the net number of bright fringes that move, compared to the empty micrometer, is 272. What is the thickness of the foil? The wavelength of light used is 589 nm.

Problem 51P How far must the mirror M1 in a Michelson interferometer be moved if 850 fringes of 589-nm light are to pass by a reference line?

(III) One of the beams of an interferometer (Fig. 24–59) passes through a small glass container containing a cavity 1.30 cm deep. When a gas is allowed to slowly fill the container, a total of 236 dark fringes are counted to move past a reference line. The light used has a wavelength of 610 nm. Calculate the index of refraction of the gas, assuming that the interferometer is in vacuum. Equation Transcription: Text Transcription: M_1 M_S M_2

Problem 53P Two polarizers are oriented at 65° to one another. Unpolarized light falls on them. What fraction of the light intensity is transmitted?

(I) What is Brewster’s angle for an air–glass (n = 1.52) surface?

(II) What is Brewster’s angle for a diamond submerged in water if the light is hitting the diamond (n = 2.42) while traveling in the water?

Problem 56P Two Polaroids are aligned so that the light passing through them is a maximum. At what angle should one of them be placed so that the intensity is subsequently reduced by half?

(II) At what angle should the axes of two Polaroids be placed so as to reduce the intensity of the incident unpolarized light to (a) \(\frac{1}{3}\), (b) \(\frac{1}{10}\)?

(II) Two polarizers are oriented at \(40^\circ\) to each other and plane-polarized light is incident on them. If only 15% of the light gets through both of them, what was the initial polarization direction of the incident light?

Problem 59P Two polarizers are oriented at 38.0° to one another. Light polarized at a 19.0° angle to each polarizer passes through both. What percent reduction in intensity takes place?

Problem 61P Unpolarized light passes through five successive Polaroid sheets, each of whose axis makes a 45° angle with the previous one. What is the intensity of the transmitted beam?

Problem 62GP Light of wavelength 5.0 × 10?7 m passes through two parallel slits and falls on a screen 4.0 m away. Adjacent bright bands of the interference pattern are 2.0 cm apart. (a) Find the distance between the slits. (b) The same two slits are next illuminated by light of a different wavelength, and the fifth-order minimum for this light occurs at the same point on the screen as the fourth-order minimum for the previous light. What is the wavelength of the second source of light?

Television and radio waves reflecting from mountains or airplanes can interfere with the direct signal from the station. (a) What kind of interference will occur when 75-MHz television signals arrive at a receiver directly from a distant station, and are reflected from a nearby airplane 118 m directly above the receiver? Assume \(\frac{1}{2} \lambda\) change in phase of the signal upon reflection. (b) What kind of interference will occur if the plane is 22 m closer to the receiver? Equation Transcription: Text Transcription: {1 over 2} lambda

Problem 64GP Red light from three separate sources passes through a diffraction grating with 3.00 × 105lines/m. The wavelengths of the three lines are 6.56 × 10?7 m (hydrogen), 6.50 × 10?7 m (neon), and 6.97 × 10?7 m (argon). Calculate the angles for the first-order diffraction lines of each of these sources.

Problem 65GP Light of wavelength 590 nm passes through two narrow slits 0.60 mm apart. The screen is 1.70 m away. A second source of unknown wavelength produces its second-order fringe 1.33 mm closer to the central maximum than the 590-nm light. What is the wavelength of the unknown light?

A radio station operating at 102.1 MHz broadcasts from two identical antennae at the same elevation but separated by an 8.0-m horizontal distance d, Fig. 24–60. A maximum signal is found along the midline, perpendicular to d at its midpoint and extending horizontally in both directions. If the midline is taken as \(0^\circ\), at what other angle(s) \(\theta\) is a maximum signal detected? A minimum signal? Assume all measurements are made much farther than 8.0 m from the antenna towers.

Problem 67GP A teacher stands well back from an outside doorway 0.88 m wide, and blows a whistle of frequency 750 Hz. Ignoring reflections, estimate at what angle(s) it is not possible to hear the whistle clearly on the playground outside the doorway.

If parallel light falls on a single slit of width D at a \(30^{\circ}\) angle to the normal, describe the diffraction pattern.

Problem 69GP The wings of a certain beetle have a series of parallel lines across them. When normally incident 460-mn light is reflected from the wing, the wing appears bright when viewed at an angle of 51°. How far apart are the lines?

Problem 70GP How many lines per centimeter must a grating have if there is to be no second-order spectrum for any visible wavelength?

When yellow sodium light, \(\lambda=589 \mathrm{~nm}\), falls on a diffraction grating, its first-order peak on a screen \(60.0 \mathrm{~cm}\) away falls \(3.32 \mathrm{~cm}\) from the central peak. Another source produces a line \(3.71 \mathrm{~cm}\) from the central peak. What is the wavelength of the new source? How many lines/ \(\mathrm{cm}\) are on the grating?

Problem 73GP Light is incident on a diffraction grating with 8600 lines/cm, and the pattern is viewed on a screen 2.5 m from the grating. The incident light beam consists of two wavelengths, ?1 = 4.6 × 10?7 m and ?2 = 6.8 × 10?7 m. Calculate the linear distance between the first-order bright fringes of these two wavelengths on the screen.

Problem 74GP What is the index of refraction of a clear material if a minimum of 150 nm thickness of it, when laid on glass, is needed to reduce reflection to nearly zero when light of 600 nm is incident normally upon it? Do you have a choice for an answer?

Problem 75GP Monochromatic light of variable wavelength is incident normally on a thin sheet of plastic film in air. The reflected light is a minimum only for ? = 512 nm and ? = 640 nm in the visible spectrum. What is the thickness of the film (n = 1.58)? [Hint: assume successive values of m.]

Problem 77GP What is the minimum (non-zero) thickness for the air layer between two flat glass surfaces if the glass is to appear dark when 640-nm light is incident normally? What if the glass is to appear bright?

Suppose you viewed the light transmitted through a thin film layered on a flat piece of glass. Draw a diagram, similar to Fig. 24–30 or 24–36, and describe the conditions required for maxima and minima. Consider all possible values of index of refraction. Discuss the relative size of the minima compared to the maxima and to zero.

Problem 81GP Unpolarized light falls on two polarizer sheets whose transmission axes are at right angles. A third polarizer is placed between the first two so that its axis makes a 62° angle with the axis of the first polarizer. (a) What fraction of the incident light intensity is transmitted? (b) What if the third polarizer is in front of the other two?

Problem 82GP Four polarizers are placed in succession with their axes vertical, at 30° to the vertical, at 60° to the vertical, and at 90° to the vertical. (a) Calculate what fraction of the incident unpolarized light is transmitted by the four polarizers. (b) Can the transmitted light be decreased by removing one of the polarizers? If so, which one? (c) Can the transmitted light intensity be extinguished by removing polarizers? If so, which one(s)?

Problem 83GP A laser beam passes through a slit of width 1.0 cm and is pointed at the Moon, which is approximately 380,000 km from the Earth. Assume the laser emits waves of wavelength 630 nm (the red light of a He-Ne laser). Estimate the width of the beam when it reaches the Moon.

A series of polarizers are each placed at a \(10^{\circ}\) interval from the previous polarizer. Unpolarized light is incident on this series of polarizers. How many polarizers does the light have to go through before it is \(\frac{1}{4}\) of its original intensity?

Consider two antennas radiating 6.0-MHz radio waves in phase with each other. They are located at points \(\mathrm {S_1}\) and \(\mathrm {S_2}\), separated by a distance \(d=175 \mathrm {~m}\), Fig. 24–61. What are the first three points on the y axis where the signals from the two sources will be out of phase (crests of one meet troughs of the other)? Equation Transcription: Text Transcription: S_1 S_2 d=175 m S_1 S_2

A parallel beam of light containing two wavelengths, 420 nm and 650 nm, enters a silicate flint glass equilateral prism (Fig. 24–58). (a) What is the angle between the two beams leaving the prism? (b) Repeat part (a) for a diffraction grating with 6200 lines/cm.

A Lucite planoconvex lens has one flat surface and one with R = 18.4 cm. It is used to view an object, located 66.0 cm away from the lens, which is a mixture of red and yellow. The index of refraction of the Lucite is 1.5106 for red light and 1.5226 for yellow light. What are the locations of the red and yellow images formed by the lens? [Hint: see Section 23-10.]