(a) Calculate the angle at which a 2.00-?m -wide slit

Problem 1PE Show that when light passes from air to water, its wavelength decreases to 0.750 times its original value.

Problem 1CQ What type of experimental evidence indicates that light is a wave?

Problem 100PE Integrated Concepts (a) On a day when the intensity of sunlight is 1.00 kW / m2 , a circular lens 0.200 m in diameter focuses light onto water in a black beaker. Two polarizing sheets of plastic are placed in front of the lens with their axes at an angle of 20.0º. Assuming the sunlight is unpolarized and the polarizers are 100% efficient, what is the initial rate of heating of the water in ºC / s , assuming it is 80.0% absorbed? The aluminum beaker has a mass of 30.0 grams and contains 250 grams of water. (b) Do the polarizing filters get hot? Explain.

Problem 2CQ Give an example of a wave characteristic of light that is easily observed outside the laboratory.

Problem 3CQ How do wave effects depend on the size of the object with which the wave interacts? For example, why does sound bend around the corner of a building while light does not?

Problem 2PE Find the range of visible wavelengths of light in crown glass.

Problem 3PE What is the index of refraction of a material for which the wavelength of light is 0.671 times its value in a vacuum? Identify the likely substance.

Problem 4CQ Under what conditions can light be modeled like a ray? Like a wave?

Problem 4PE Analysis of an interference effect in a clear solid shows that the wavelength of light in the solid is 329 nm. Knowing this light comes from a He-Ne laser and has a wavelength of 633 nm in air, is the substance zircon or diamond?

Problem 5PE What is the ratio of thicknesses of crown glass and water that would contain the same number of wavelengths of light?

Problem 6PE At what angle is the first-order maximum for 450-nm wavelength blue light falling on double slits separated by 0.0500 mm?

Problem 5CQ Go outside in the sunlight and observe your shadow. It has fuzzy edges even if you do not. Is this a diffraction effect? Explain.

Problem 7CQ At what angle is the first-order maximum for 450-nm wavelength blue light falling on double slits separated by 0.0500 mm?

Problem 6CQ Why does the wavelength of light decrease when it passes from vacuum into a medium? State which attributes change and which stay the same and, thus, require the wavelength to decrease.

Problem 7PE Calculate the angle for the third-order maximum of 580-nm wavelength yellow light falling on double slits separated by 0.100 mm.

Problem 8CQ Young’s double slit experiment breaks a single light beam into two sources. Would the same pattern be obtained for two independent sources of light, such as the headlights of a distant car? Explain.

Problem 9CQ Suppose you use the same double slit to perform Young’s double slit experiment in air and then repeat the experiment in water. Do the angles to the same parts of the interference pattern get larger or smaller? Does the color of the light change? Explain.

Problem 9PE Find the distance between two slits that produces the first minimum for 410-nm violet light at an angle of 45.0º .

Problem 8PE What is the separation between two slits for which 610-nm orange light has its first maximum at an angle of 30.0º ?

Problem 10PE Calculate the wavelength of light that has its third minimum at an angle of 30.0º when falling on double slits separated by 3.00 ?m . Explicitly, show how you follow the steps in Problem-Solving Strategies for Wave Optics.

Problem 10CQ Is it possible to create a situation in which there is only destructive interference? Explain.

Figure 27.55 shows the central part of the interference pattern for a pure wavelength of red light projected onto a double slit. The pattern is actually a combination of single slit and double slit interference. Note that the bright spots are evenly spaced. Is this a double slit or single slit characteristic? Note that some of the bright spots are dim on either side of the center. Is this a single slit or double slit characteristic? Which is smaller, the slit width or the separation between slits? Explain your responses. Figure 27.55 This double slit interference pattern also shows signs of single slit interference. (credit: PASCO)

Problem 11PE What is the wavelength of light falling on double slits separated by 2.00 ?m if the third-order maximum is at an angle of 60.0º ?

Problem 12CQ What is the advantage of a diffraction grating over a double slit in dispersing light into a spectrum?

Problem 12PE At what angle is the fourth-order maximum for the situation in Exercise 27.6? Reference Exercise 27.6: At what angle is the first-order maximum for 450-nm wavelength blue light falling on double slits separated by 0.0500 mm?

Problem 13CQ What are the advantages of a diffraction grating over a prism in dispersing light for spectral analysis?

Problem 14CQ Can the lines in a diffraction grating be too close together to be useful as a spectroscopic tool for visible light? If so, what type of EM radiation would the grating be suitable for? Explain.

Problem 15CQ If a beam of white light passes through a diffraction grating with vertical lines, the light is dispersed into rainbow colors on the right and left. If a glass prism disperses white light to the right into a rainbow, how does the sequence of colors compare with that produced on the right by a diffraction grating?

Problem 14PE Find the largest wavelength of light falling on double slits separated by 1.20 ?m for which there is a first-order maximum. Is this in the visible part of the spectrum?

Problem 15PE What is the smallest separation between two slits that will produce a second-order maximum for 720-nm red light?

Problem 13PE What is the highest-order maximum for 400-nm light falling on double slits separated by 25.0 ?m ?

Problem 16CQ Suppose pure-wavelength light falls on a diffraction grating. What happens to the interference pattern if the same light falls on a grating that has more lines per centimeter? What happens to the interference pattern if a longer-wavelength light falls on the same grating? Explain how these two effects are consistent in terms of the relationship of wavelength to the distance between slits.

Problem 16PE (a) What is the smallest separation between two slits that will produce a second-order maximum for any visible light? (b) For all visible light?

Problem 17CQ Suppose a feather appears green but has no green pigment. Explain in terms of diffraction.

Problem 17PE (a) If the first-order maximum for pure-wavelength light falling on a double slit is at an angle of 10.0º , at what angle is the second-order maximum? (b) What is the angle of the first minimum? (c) What is the highest-order maximum possible here?

Problem 18CQ It is possible that there is no minimum in the interference pattern of a single slit. Explain why. Is the same true of double slits and diffraction gratings?

Figure 27.56 shows a double slit located a distance \(x\) from a screen, with the distance from the center of the screen given by \(y\) When the distance \(d\) between the slits is relatively large, there will be numerous bright spots, called fringes. Show that, for small angles (where \(\sin \theta \approx \theta\) with \(\theta\) in radians), the distance between fringes is given by \(\Delta y=x \lambda / d\). Figure 27.56 The distance between adjacent fringes is \(\Delta y=x \lambda / d\) assuming the slit separation \(d\) is large compared with \(\lambda\). Equation Transcription: Text Transcription: x y d sin theta approx theta theta Delta y=x lambda/d Delta y=x lambda/d d

Problem 19CQ As the width of the slit producing a single-slit diffraction pattern is reduced, how will the diffraction pattern produced change?

Using the result of the problem above, calculate the distance between fringes for \(633-nm\) light falling on double slits separated by \(0.0800 \ mm\), located \(3.00 \ m\) from a screen as in Figure 27.56. Equation Transcription: Text Transcription: 633-nm 0.0800 mm 3.00 m

Problem 20CQ A beam of light always spreads out. Why can a beam not be created with parallel rays to prevent spreading? Why can lenses, mirrors, or apertures not be used to correct the spreading?

Using the result of the problem two problems prior, find the wavelength of light that produces fringes \(7.50 \ mm\) apart on a screen \(2.00 \ m\) from double slits separated by \(0.120 \ mm\) (see Figure 27.56). Equation Transcription: Text Transcription: 7.50 mm 2.00 m 0.120 mm

Problem 21PE A diffraction grating has 2000 lines per centimeter. At what angle will the first-order maximum be for 520-nmwavelength green light?

Problem 22PE Find the angle for the third-order maximum for 580-nmwavelength yellow light falling on a diffraction grating having 1500 lines per centimeter.

Problem 23CQ Is there a phase change in the light reflected from either surface of a contact lens floating on a person’s tear layer? The index of refraction of the lens is about 1.5, and its top surface is dry.

Problem 22CQ How is the difference in paths taken by two originally in-phase light waves related to whether they interfere constructively or destructively? How can this be affected by reflection? By refraction?

Problem 23PE How many lines per centimeter are there on a diffraction grating that gives a first-order maximum for 470-nm blue light at an angle of 25.0º ?

Problem 24PE What is the distance between lines on a diffraction grating that produces a second-order maximum for 760-nm red light at an angle of 60.0º ?

Problem 24CQ In placing a sample on a microscope slide, a glass cover is placed over a water drop on the glass slide. Light incident from above can reflect from the top and bottom of the glass cover and from the glass slide below the water drop. At which surfaces will there be a phase change in the reflected light?

Problem 25PE Calculate the wavelength of light that has its second-order maximum at 45.0º when falling on a diffraction grating that has 5000 lines per centimeter.

Problem 26CQ While contemplating the food value of a slice of ham, you notice a rainbow of color reflected from its moist surface. Explain its origin.

Problem 25CQ Answer the above question if the fluid between the two pieces of crown glass is carbon disulfide.

Problem 27CQ An inventor notices that a soap bubble is dark at its thinnest and realizes that destructive interference is taking place for all wavelengths. How could she use this knowledge to make a non-reflective coating for lenses that is effective at all wavelengths? That is, what limits would there be on the index of refraction and thickness of the coating? How might this be impractical?

Problem 27PE (a) What do the four angles in the above problem become if a 5000-line-per-centimeter diffraction grating is used? (b) Using this grating, what would the angles be for the secondorder maxima? (c) Discuss the relationship between integral reductions in lines per centimeter and the new angles of various order maxima.

Problem 26PE An electric current through hydrogen gas produces several distinct wavelengths of visible light. What are the wavelengths of the hydrogen spectrum, if they form first-order maxima at angles of 24.2º , 25.7º , 29.1º , and 41.0º when projected on a diffraction grating having 10,000 lines per centimeter? Explicitly show how you follow the steps in Problem-Solving Strategies for Wave Optics

Problem 28CQ A non-reflective coating like the one described in Example 27.6 works ideally for a single wavelength and for perpendicular incidence. What happens for other wavelengths and other incident directions? Be specific. Example 27.6: Calculating Non-reflective Lens Coating Using Thin Film Interference Sophisticated cameras use a series of several lenses. Light can reflect from the surfaces of these various lenses and degrade image clarity. To limit these reflections, lenses are coated with a thin layer of magnesium fluoride that causes destructive thin film interference. What is the thinnest this film can be, if its index of refraction is 1.38 and it is designed to limit the reflection of 550-nm light, normally the most intense visible wavelength? The index of refraction of glass is 1.52.

Problem 28PE What is the maximum number of lines per centimeter a diffraction grating can have and produce a complete firstorder spectrum for visible light?

Problem 29PE The yellow light from a sodium vapor lamp seems to be of pure wavelength, but it produces two first-order maxima at 36.093º and 36.129º when projected on a 10,000 line per centimeter diffraction grating. What are the two wavelengths to an accuracy of 0.1 nm?

Problem 30CQ Under what circumstances is the phase of light changed by reflection? Is the phase related to polarization?

Problem 30PE Under what circumstances is the phase of light changed by reflection? Is the phase related to polarization?

Problem 31CQ Can a sound wave in air be polarized? Explain.

Problem 31PE Structures on a bird feather act like a reflection grating having 8000 lines per centimeter. What is the angle of the first-order maximum for 600-nm light?

Problem 29CQ Why is it much more difficult to see interference fringes for light reflected from a thick piece of glass than from a thin film? Would it be easier if monochromatic light were used?

Problem 32CQ No light passes through two perfect polarizing filters with perpendicular axes. However, if a third polarizing filter is placed between the original two, some light can pass. Why is this? Under what circumstances does most of the light pass?

Problem 33CQ Explain what happens to the energy carried by light that it is dimmed by passing it through two crossed polarizing filters.

Problem 34CQ When particles scattering light are much smaller than its wavelength, the amount of scattering is proportional to 1 / ?4 . Does this mean there is more scattering for small ? than large ? ? How does this relate to the fact that the sky is blue?

An opal such as that shown in Figure 27.17 acts like a reflection grating with rows separated by about \(8\ \mu m\). If the opal is illuminated normally, (a) at what angle will red light be seen and (b) at what angle will blue light be seen? Equation Transcription: Text Transcription: 8 mu m

Problem 33PE At what angle does a diffraction grating produces a second-order maximum for light having a first-order maximum at 20.0º ?

Problem 35CQ Using the information given in the preceding question, explain why sunsets are red.

Problem 34PE Show that a diffraction grating cannot produce a secondorder maximum for a given wavelength of light unless the first-order maximum is at an angle less than 30.0º .

Problem 35PE If a diffraction grating produces a first-order maximum for the shortest wavelength of visible light at 30.0º , at what angle will the first-order maximum be for the longest wavelength of visible light?

Problem 36CQ When light is reflected at Brewster’s angle from a smooth surface, it is 100% polarized parallel to the surface. Part of the light will be refracted into the surface. Describe how you would do an experiment to determine the polarization of the refracted light. What direction would you expect the polarization to have and would you expect it to be 100% ?

Problem 36PE (a) Find the maximum number of lines per centimeter a diffraction grating can have and produce a maximum for the smallest wavelength of visible light. (b) Would such a grating be useful for ultraviolet spectra? (c) For infrared spectra?

Problem 37PE (a) Show that a 30,000-line-per-centimeter grating will not produce a maximum for visible light. (b) What is the longest wavelength for which it does produce a first-order maximum? (c) What is the greatest number of lines per centimeter a diffraction grating can have and produce a complete secondorder spectrum for visible light?

Problem 38CQ A bright white light under water is collimated and directed upon a prism. What range of colors does one see emerging?

Problem 38PE A He–Ne laser beam is reflected from the surface of a CD onto a wall. The brightest spot is the reflected beam at an angle equal to the angle of incidence. However, fringes are also observed. If the wall is 1.50 m from the CD, and the first fringe is 0.600 m from the central maximum, what is the spacing of grooves on the CD?

Problem 37CQ Explain how microscopes can use wave optics to improve contrast and why this is important.

Problem 40PE Unreasonable Results Red light of wavelength of 700 nm falls on a double slit separated by 400 nm. (a) At what angle is the first-order maximum in the diffraction pattern? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

The analysis shown in the figure below also applies to diffraction gratings with lines separated by a distance \(d\). What is the distance between fringes produced by a diffraction grating having \(125\) lines per centimeter for \(600-nm\) light, if the screen is \(1.50 \ m\) away? Figure 27.57 The distance between adjacent fringes is \(\Delta y=x \lambda / d\), assuming the slit separation \(d\) is large compared with \(\lambda\). Equation Transcription: Text Transcription: d 125 600-nm 1.50 m Delta y = x lambda / d d lambda

Problem 41PE Unreasonable Results (a) What visible wavelength has its fourth-order maximum at an angle of 25.0º when projected on a 25,000-line-percentimeter diffraction grating? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Problem 42PE Construct Your Own Problem Consider a spectrometer based on a diffraction grating. Construct a problem in which you calculate the distance between two wavelengths of electromagnetic radiation in your spectrometer. Among the things to be considered are the wavelengths you wish to be able to distinguish, the number of lines per meter on the diffraction grating, and the distance from the grating to the screen or detector. Discuss the practicality of the device in terms of being able to discern between wavelengths of interest.

Problem 44PE (a) Calculate the angle at which a 2.00-?m -wide slit produces its first minimum for 410-nm violet light. (b) Where is the first minimum for 700-nm red light?

Problem 45PE (a) How wide is a single slit that produces its first minimum for 633-nm light at an angle of 28.0º ? (b) At what angle will the second minimum be?

Problem 43PE (a) At what angle is the first minimum for 550-nm light falling on a single slit of width 1.00 ?m ? (b) Will there be a second minimum?

Problem 46PE (a) What is the width of a single slit that produces its first minimum at 60.0º for 600-nm light? (b) Find the wavelength of light that has its first minimum at 62.0º .

Problem 47PE Find the wavelength of light that has its third minimum at an angle of 48.6º when it falls on a single slit of width 3.00 ?m .

Problem 48PE Calculate the wavelength of light that produces its first minimum at an angle of 36.9º when falling on a single slit of width 1.00 ?m .

Problem 49PE (a) Sodium vapor light averaging 589 nm in wavelength falls on a single slit of width 7.50 ?m . At what angle does it produces its second minimum? (b) What is the highest-order minimum produced?

Problem 51PE (a) Find the angle between the first minima for the two sodium vapor lines, which have wavelengths of 589.1 and 589.6 nm, when they fall upon a single slit of width 2.00 ?m . (b) What is the distance between these minima if the diffraction pattern falls on a screen 1.00 m from the slit? (c) Discuss the ease or difficulty of measuring such a distance.

Problem 52PE (a) What is the minimum width of a single slit (in multiples of ? ) that will produce a first minimum for a wavelength ? ? (b) What is its minimum width if it produces 50 minima? (c) 1000 minima?

Problem 50PE (a) Find the angle of the third diffraction minimum for 633-nm light falling on a slit of width 20.0 ?m . (b) What slit width would place this minimum at 85.0º ? Explicitly show how you follow the steps in Problem-Solving Strategies for Wave Optics

Problem 53PE (a) If a single slit produces a first minimum at 14.5º , at what angle is the second-order minimum? (b) What is the angle of the third-order minimum? (c) Is there a fourth-order minimum? (d) Use your answers to illustrate how the angular width of the central maximum is about twice the angular width of the next maximum (which is the angle between the first and second minima).

Problem 54PE A double slit produces a diffraction pattern that is a combination of single and double slit interference. Find the ratio of the width of the slits to the separation between them, if the first minimum of the single slit pattern falls on the fifth maximum of the double slit pattern. (This will greatly reduce the intensity of the fifth maximum.)

Problem 55PE Integrated Concepts A water break at the entrance to a harbor consists of a rock barrier with a 50.0-m-wide opening. Ocean waves of 20.0-m wavelength approach the opening straight on. At what angle to the incident direction are the boats inside the harbor most protected against wave action?

Problem 56PE Integrated Concepts An aircraft maintenance technician walks past a tall hangar door that acts like a single slit for sound entering the hangar. Outside the door, on a line perpendicular to the opening in the door, a jet engine makes a 600-Hz sound. At what angle with the door will the technician observe the first minimum in sound intensity if the vertical opening is 0.800 m wide and the speed of sound is 340 m/s?

Problem 58PE Assuming the angular resolution found for the Hubble Telescope in Example 27.5, what is the smallest detail that could be observed on the Moon? Example 27.5: Calculating Diffraction Limits of the Hubble Space Telescope The primary mirror of the orbiting Hubble Space Telescope has a diameter of 2.40 m. Being in orbit, this telescope avoids the degrading effects of atmospheric distortion on its resolution. (a) What is the angle between two just-resolvable point light sources (perhaps two stars)? Assume an average light wavelength of 550 nm. (b) If these two stars are at the 2 million light year distance of the Andromeda galaxy, how close together can they be and still be resolved? (A light year, or ly, is the distance light travels in 1 year.)

Problem 59PE Diffraction spreading for a flashlight is insignificant compared with other limitations in its optics, such as spherical aberrations in its mirror. To show this, calculate the minimum angular spreading of a flashlight beam that is originally 5.00 cm in diameter with an average wavelength of 600 nm.

Problem 60PE (a) What is the minimum angular spread of a 633-nm wavelength He-Ne laser beam that is originally 1.00 mm in diameter? (b) If this laser is aimed at a mountain cliff 15.0 km away, how big will the illuminated spot be? (c) How big a spot would be illuminated on the Moon, neglecting atmospheric effects? (This might be done to hit a corner reflector to measure the round-trip time and, hence, distance.) Explicitly show how you follow the steps in Problem-Solving Strategies for Wave Optics.

Problem 61PE A telescope can be used to enlarge the diameter of a laser beam and limit diffraction spreading. The laser beam is sent through the telescope in opposite the normal direction and can then be projected onto a satellite or the Moon. (a) If this is done with the Mount Wilson telescope, producing a 2.54-m-diameter beam of 633-nm light, what is the minimum angular spread of the beam? (b) Neglecting atmospheric effects, what is the size of the spot this beam would make on the Moon, assuming a lunar distance of 3.84×108 m ?

Problem 62PE The limit to the eye’s acuity is actually related to diffraction by the pupil. (a) What is the angle between two just-resolvable points of light for a 3.00-mm-diameter pupil, assuming an average wavelength of 550 nm? (b) Take your result to be the practical limit for the eye. What is the greatest possible distance a car can be from you if you can resolve its two headlights, given they are 1.30 m apart? (c) What is the distance between two just-resolvable points held at an arm’s length (0.800 m) from your eye? (d) How does your answer to (c) compare to details you normally observe in everyday circumstances?

Problem 63PE What is the minimum diameter mirror on a telescope that would allow you to see details as small as 5.00 km on the Moon some 384,000 km away? Assume an average wavelength of 550 nm for the light received.

Problem 64PE You are told not to shoot until you see the whites of their eyes. If the eyes are separated by 6.5 cm and the diameter of your pupil is 5.0 mm, at what distance can you resolve the two eyes using light of wavelength 555 nm?

65PE Problem (a) The planet Pluto and its Moon Charon are separated by 19,600 km. Neglecting atmospheric effects, should the 5.08-m-diameter Mount Palomar telescope be able to resolve these bodies when they are 4.50×109 km from Earth? Assume an average wavelength of 550 nm. (b) In actuality, it is just barely possible to discern that Pluto and Charon are separate bodies using an Earth-based telescope. What are the reasons for this?

Problem 66PE The headlights of a car are 1.3 m apart. What is the maximum distance at which the eye can resolve these two headlights? Take the pupil diameter to be 0.40 cm.

Problem 67PE When dots are placed on a page from a laser printer, they must be close enough so that you do not see the individual dots of ink. To do this, the separation of the dots must be less than Raleigh’s criterion. Take the pupil of the eye to be 3.0 mm and the distance from the paper to the eye of 35 cm; find the minimum separation of two dots such that they cannot be resolved. How many dots per inch (dpi) does this correspond to?

Problem 68PE Unreasonable Results An amateur astronomer wants to build a telescope with a diffraction limit that will allow him to see if there are people on the moons of Jupiter. (a) What diameter mirror is needed to be able to see 1.00 m detail on a Jovian Moon at a distance of 7.50×108 km from Earth? The wavelength of light averages 600 nm. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Problem 69PE Construct Your Own Problem Consider diffraction limits for an electromagnetic wave interacting with a circular object. Construct a problem in which you calculate the limit of angular resolution with a device, using this circular object (such as a lens, mirror, or antenna) to make observations. Also calculate the limit to spatial resolution (such as the size of features observable on the Moon) for observations at a specific distance from the device. Among the things to be considered are the wavelength of electromagnetic radiation used, the size of the circular object, and the distance to the system or phenomenon being observed.

Problem 70PE A soap bubble is 100 nm thick and illuminated by white light incident perpendicular to its surface. What wavelength and color of visible light is most constructively reflected, assuming the same index of refraction as water?

The \(300\)-m-diameter Arecibo radio telescope pictured in Figure 27.28 detects radio waves with a \(4.00 \ cm\) average wavelength. (a) What is the angle between two just-resolvable point sources for this telescope? (b) How close together could these point sources be at the \(2\) million light year distance of the Andromeda galaxy? Equation Transcription: Text Transcription: 300 4.0 cm 2

Problem 71PE An oil slick on water is 120 nm thick and illuminated by white light incident perpendicular to its surface. What color does the oil appear (what is the most constructively reflected wavelength), given its index of refraction is 1.40?

Problem 72PE Calculate the minimum thickness of an oil slick on water that appears blue when illuminated by white light perpendicular to its surface. Take the blue wavelength to be 470 nm and the index of refraction of oil to be 1.40.

Problem 73PE Find the minimum thickness of a soap bubble that appears red when illuminated by white light perpendicular to its surface. Take the wavelength to be 680 nm, and assume the same index of refraction as water.

Problem 74PE A film of soapy water ( n = 1.33 ) on top of a plastic cutting board has a thickness of 233 nm. What color is most strongly reflected if it is illuminated perpendicular to its surface?

Problem 76PE Suppose you have a lens system that is to be used primarily for 700-nm red light. What is the second thinnest coating of fluorite (magnesium fluoride) that would be nonreflective for this wavelength?

Problem 75PE What are the three smallest non-zero thicknesses of oapy water ( n = 1.33 ) on Plexiglas if it appears green (constructively reflecting 520-nm light) when illuminated perpendicularly by white light? Explicitly show how you follow the steps in Problem Solving Strategies for Wave Optics.

Problem 77PE (a) As a soap bubble thins it becomes dark, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the bubble can be and appear dark at all visible wavelengths? Assume the same index of refraction as water. (b) Discuss the fragility of the film considering the thickness found.

Problem 78PE A film of oil on water will appear dark when it is very thin, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the oil can be and appear dark at all visible wavelengths? Oil has an index of refraction of 1.40.

Figure 27.34 shows two \(7.50\)-cm-long glass slides illuminated by pure \(589-nm\) wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on some debris at the other end, forming a wedge of air. How thick is the debris, if the dark bands are \(1.00 \ mm\) apart? Equation Transcription: Text Transcription: 7.50 589-nm 1.00 mm

Figure 27.34 shows two glass slides illuminated by pure-wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on a \(0.100\)-mm-diameter hair at the other end, forming a wedge of air. (a) How far apart are the dark bands, if the slides are \(7.50 \ cm\) long and \(589-nm\) light is used? (b) Is there any difference if the slides are made from crown or flint glass? Explain. Equation Transcription: Text Transcription: 0.100 7.50 cm 589-nm

Problem 81PE Repeat Exercise 27.70, but take the light to be incident at a 45º angle. Reference Exercise 27.70: A soap bubble is 100 nm thick and illuminated by white light incident perpendicular to its surface. What wavelength and color of visible light is most constructively reflected, assuming the same index of refraction as water?

Problem 83PE Unreasonable Results To save money on making military aircraft invisible to radar, an inventor decides to coat them with a non-reflective material having an index of refraction of 1.20, which is between that of air and the surface of the plane. This, he reasons, should be much cheaper than designing Stealth bombers. (a) What thickness should the coating be to inhibit the reflection of 4.00-cm wavelength radar? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Problem 84PE What angle is needed between the direction of polarized light and the axis of a polarizing filter to cut its intensity in half?

Problem 85PE The angle between the axes of two polarizing filters is 45.0º . By how much does the second filter reduce the intensity of the light coming through the first?

Problem 86PE If you have completely polarized light of intensity 150 W / m2 , what will its intensity be after passing through a polarizing filter with its axis at an 89.0º angle to the light’s polarization direction?

Problem 82PE Repeat Exercise 27.71, but take the light to be incident at a 45º angle. Reference Exercise 27.71: An oil slick on water is 120 nm thick and illuminated by white light incident perpendicular to its surface. What color does the oil appear (what is the most constructively reflected wavelength), given its index of refraction is 1.40?

Problem 87PE What angle would the axis of a polarizing filter need to make with the direction of polarized light of intensity 1.00 kW/m2 to reduce the intensity to 10.0 W/m2 ?

Problem 88PE At the end of Example 27.8, it was stated that the intensity of polarized light is reduced to 90.0% of its original value by passing through a polarizing filter with its axis at an angle of 18.4º to the direction of polarization. Verify this statement. Example 27.8: Calculating Intensity Reduction by a Polarizing Filter What angle is needed between the direction of polarized light and the axis of a polarizing filter to reduce its intensity by 90.0% ?

Problem 89PE Show that if you have three polarizing filters, with the second at an angle of 45º to the first and the third at an angle of 90.0º to the first, the intensity of light passed by the first will be reduced to 25.0% of its value. (This is in contrast to having only the first and third, which reduces the intensity to zero, so that placing the second between them increases the intensity of the transmitted light.)

Problem 90PE Prove that, if I is the intensity of light transmitted by two polarizing filters with axes at an angle ? and I? is the intensity when the axes are at an angle 90.0º??, then I + I? = I0, the original intensity. (Hint: Use the trigonometric identities cos (90.0º??) = sin ? and cos2 ? + sin2 ? = 1. )

Problem 92PE What is Brewster’s angle for light traveling in water that is reflected from crown glass?

Problem 91PE At what angle will light reflected from diamond be completely polarized?

Problem 94PE At what angle is light inside crown glass completely polarized when reflected from water, as in a fish tank?

Problem 93PE A scuba diver sees light reflected from the water’s surface. At what angle will this light be completely polarized?

Problem 95PE Light reflected at 55.6º from a window is completely polarized. What is the window’s index of refraction and the likely substance of which it is made?

Problem 96PE (a) Light reflected at 62.5º from a gemstone in a ring is completely polarized. Can the gem be a diamond? (b) At what angle would the light be completely polarized if the gem was in water?

Problem 97PE If ?b is Brewster’s angle for light reflected from the top of an interface between two substances, and ??b is Brewster’s angle for light reflected from below, prove that ?b + ??b = 90.0º.

Problem 99PE Integrated Concepts Suppose you put on two pairs of Polaroid sunglasses with their axes at an angle of 15.0º . How much longer will it take the light to deposit a given amount of energy in your eye compared with a single pair of sunglasses? Assume the lenses are clear except for their polarizing characteristics.

Problem 98PE Integrated Concepts If a polarizing filter reduces the intensity of polarized light to 50.0% of its original value, by how much are the electric and magnetic fields reduced?

Problem 21CQ What effect does increasing the wedge angle have on the spacing of interference fringes? If the wedge angle is too large, fringes are not observed. Why?