Solution: A plane monochromatic light wave is incident on a

A thin layer of oil (n 5 1.25) is floating on water (n 5 1.33). What is the minimum nonzero thickness of the oil in the region that strongly reflects green light (l 5 530 nm)? (a) 500 nm (b) 313 nm (c) 404 nm (d) 212 nm (e) 285 nm

A monochromatic light beam having a wavelength of 5.0 3 102 nm illuminates a double slit having a slit separation of 2.0 3 1025 m. What is the angle of the second-order bright fringe? (a) 0.050 rad (b) 0.025 rad (c) 0.10 rad (d) 0.25 rad (e) 0.010 rad

A Fraunhofer diffraction pattern is produced on a screen located 1.00 m from a single slit. If a light source of wavelength 5.00 3 1027 m is used and the distance from the center of the central bright fringe to the first dark fringe is 5.00 3 1023 m, what is the slit width? (a) 0.010 0 mm (b) 0.100 mm (c) 0.200 mm (d) 1.00 mm (e) 0.005 00 mm

If plane-polarized light is sent through two polarizers, the first polarizer at 45 to the original plane of polarization and the second polarizer at 90 to the original plane of polarization, what fraction of the original polarized intensity gets through the last polarizer? (a) 0 (b) 0.25 (c) 0.50 (d) 0.125 (e) 0.10

A plane monochromatic light wave is incident on a double-slit as illustrated in Figure 24.4. As the slit separation decreases, what happens to the separation between the interference fringes on the screen? (a) It decreases. (b) It increases. (c) It remains the same. (d) It may increase or decrease, depending on the wavelength of the light. (e) More information is required.

A plane monochromatic light wave is incident on a double-slit as illustrated in Figure 24.4. If the viewing screen is moved away from the double slit, what happens to the separation between the interference fringes on the screen? (a) It increases. (b) It decreases. (c) It remains the same. (d) It may increase or decrease, depending on the wavelength of the light. (e) More information is required.

In one experiment light of wavelength l1 passes through a single slit and forms an interference pattern on a screen. In a second experiment light of wavelength l2 passes through the same slit and forms another interference pattern on the screen. If l2 . l1, what is the separation between bright lines in the two experiments? (a) Its the same for both experiments. (b) Its greater in the first experiment. (c) Its greater in the second experiment. (d) Its dependent on the intensity of the light. (e) More information is required.

What happens to light when it is reflected off a shiny surface on a sunny day? (a) It undergoes refraction. (b) It is totally polarized. (c) It is unpolarized. (d) It tends to be partially polarized. (e) More information is required.

What causes the brightly colored patterns sometimes seen on wet streets covered with a layer of oil? Choose the best answer. (a) diffraction and polarization (b) interference and deflection (c) polarization and reflection (d) refraction and diffraction (e) reflection and interference

Consider a wave passing through a single slit. What happens to the width of the central maximum of its diffraction pattern as the slit is made half as wide? (a) It becomes one-fourth as wide. (b) It becomes one-half as wide. (c) Its width does not change. (d) It becomes twice as wide. (e) It becomes four times wider.

A film of oil on a puddle in a parking lot shows a variety of bright colors in swirled patches. What can you say about the thickness of the oil film? (a) It is much less than the wavelength of visible light. (b) It is of the same order of magnitude as the wavelength of visible light. (c) It is much greater than the wavelength of visible light. (d) It might have any relationship to the wavelength of visible light.

When you receive a chest x-ray at a hospital, the x-rays pass through a set of parallel ribs in your chest. Do your ribs act as a diffraction grating for x-rays? (a) Yes. They produce diffracted beams that can be observed separately. (b) Not to a measurable extent. The ribs are too far apart. (c) Essentially not. The ribs are too close together. (d) Essentially not. The ribs are too few in number. (e) Absolutely not. X-rays cannot diffract.

Why is it so much easier to perform interference experiments with a laser than with an ordinary light source?

A soap film is held vertically in air and is viewed in reflected light as in Figure CQ24.14. Explain why the film appears to be dark at the top.

Waves from a radio station have a wavelength of 300 m. They travel by two paths to a home receiver 20.0 km from the transmitter. One path is a direct path, and the second is by reflection from a mountain directly behind the home receiver. What is the minimum distance from the mountain to the receiver that produces destructive interference at the receiver? (Assume that no phase change occurs on reflection from the mountain.)

A soap bubble (n 5 1.33) having a wall thickness of 120 nm is floating in air. (a) What is the wavelength of the visible light that is most strongly reflected? (b) Explain how a bubble of different thickness could also strongly reflect light of this same wavelength. (c) Find the two smallest film thicknesses larger than the one given that can produce strongly reflected light of this same wavelength.

A thin layer of liquid methylene iodide (n 5 1.756) is sandwiched between two flat, parallel plates of glass (n 5 1.50). What is the minimum thickness of the liquid layer if normally incident light with l 5 6.00 3 102 nm in air is to be strongly reflected?

A thin film of oil (n 5 1.25) is located on smooth, wet pavement. When viewed from a direction perpendicular to the pavement, the film reflects most strongly red light at 640 nm and reflects no green light at 512 nm. (a) What is the minimum thickness of the oil film? (b) Let m1 correspond to the order of the constructive interference and m2 to the order of the destructive interference. Obtain a relationship between m1 and m2 that is consistent with the given data.

A thin film of glass (n 5 1.52) of thickness 0.420 mm is viewed under white light at near normal incidence. What wavelength of visible light is most strongly reflected by the film when surrounded by air?

A transparent oil with index of refraction 1.29 spills on the surface of water (index of refraction 1.33), producing a maximum of reflection with normally incident orange light (wavelength 600 nm in air). Assuming the maximum occurs in the first order, determine the thickness of the oil slick.

A possible means for making an airplane invisible to radar is to coat the plane with an antireflective polymer. If radar waves have a wavelength of 3.00 cm and the index of refraction of the polymer is n 5 1.50, how thick would you make the coating?

An oil film (n 5 1.45) floating on water is illuminated by white light at normal incidence. The film is 2.80 3 102 nm thick. Find (a) the wavelength and color of the light in the visible spectrum most strongly reflected and (b) the wavelength and color of the light in the visible spectrum most strongly transmitted. Explain your reasoning.

Astronomers observe the chromosphere of the Sun with a filter that passes the red hydrogen spectral line of wavelength 656.3 nm, called the Ha line. The filter consists of a transparent dielectric of thickness d held between two partially aluminized glass plates. The filter is kept at a constant temperature. (a) Find the minimum value of d that will produce maximum transmission of perpendicular Ha light if the dielectric has an index of refraction of 1.378. (b) If the temperature of the filter increases above the normal value increasing its thickness, what happens to the transmitted wavelength? (c) The dielectric will also pass what near-visible wavelength? One of the glass plates is colored red to absorb this light.

Two rectangular optically flat plates (n 5 1.52) are in contact along one end and are separated along the other end by a 2.00-mmthick spacer (Fig. P24.24). The top plate is illuminated by monochromatic light of wavelength 546.1 nm. Calculate the number of dark parallel bands crossing the top plate (including the dark band at zero thickness along the edge of contact between the plates).

An investigator finds a fiber at a crime scene that he wishes to use as evidence against a suspect. He gives the fiber to a technician to test the properties of the fiber. To measure the diameter of the fiber, the technician places it between two flat glass plates at their ends as in Figure P24.24. When the plates, of length 14.0 cm, are illuminated from above with light of wavelength 650 nm, she observes bright interference bands separated by 0.580 mm. What is the diameter of the fiber?

A plano-convex lens with radius of curvature R 5 3.0 m is in contact with a flat plate of glass. A light source and the observers eye are both close to the normal, as shown in Figure 24.8a. The radius of the 50th bright Newtons ring is found to be 9.8 mm. What is the wavelength of the light produced by the source?

A plano-convex lens rests with its curved side on a flat glass surface and is illuminated from above by light of wavelength 500 nm. (See Fig. 24.8.) A dark spot is observed at the center, surrounded by 19 concentric dark rings (with bright rings in between). How much thicker is the air wedge at the position of the 19th dark ring than at the center?

Nonreflective coatings on camera lenses reduce the loss of light at the surfaces of multilens systems and prevent internal reflections that might mar the image. Find the minimum thickness of a layer of magnesium fluoride (n 5 1.38) on flint glass (n 5 1.66) that will cause destructive interference of reflected light of wavelength 550 nm near the middle of the visible spectrum.

A thin film of glycerin (n 5 1.473) of thickness 524 nm with air on both sides is illuminated with white light at near normal incidence. What wavelengths will be strongly reflected in the range 300 nm to 700 nm?

A lens made of glass (ng 5 1.52) is coated with a thin film of MgF2 (ns 5 1.38) of thickness t. Visible light is incident normally on the coated lens as in Figure P24.30. (a) For what minimum value of t will the reflected light of wavelength 540 nm (in air) be missing? (b) Are there other values of t that will minimize the reflected light at this wavelength? Explain.

Light of wavelength 5.40 3 102 nm passes through a slit of width 0.200 mm. (a) Find the width of the central maximum on a screen located 1.50 m from the slit. (b) Determine the width of the first-order bright fringe.

Light of wavelength 600 nm falls on a 0.40-mm-wide slit and forms a diffraction pattern on a screen 1.5 m away. (a) Find the position of the first dark band on each side of the central maximum. (b) Find the width of the central maximum.

Light of wavelength 587.5 nm illuminates a slit of width 0.75 mm. (a) At what distance from the slit should a screen be placed if the first minimum in the diffraction pattern is to be 0.85 mm from the central maximum? (b) Calculate the width of the central maximum.

Microwaves of wavelength 5.00 cm enter a long, narrow window in a building that is otherwise essentially opaque to the incoming waves. If the window is 36.0 cm wide, what is the distance from the central maximum to the first-order minimum along a wall 6.50 m from the window?

A beam of monochromatic light is diffracted by a slit of width 0.600 mm. The diffraction pattern forms on a wall 1.30 m beyond the slit. The width of the central maximum is 2.00 mm. Calculate the wavelength of the light.

A screen is placed 50.0 cm from a single slit that is illuminated with light of wavelength 680 nm. If the distance between the first and third minima in the diffraction pattern is 3.00 mm, what is the width of the slit?

A slit of width 0.50 mm is illuminated with light of wavelength 500 nm, and a screen is placed 120 cm in front of the slit. Find the widths of the first and second maxima on each side of the central maximum.

The second-order dark fringe in a single-slit diffraction pattern is 1.40 mm from the center of the central maximum. Assuming the screen is 85.0 cm from a slit of width 0.800 mm and assuming monochromatic incident light, calculate the wavelength of the incident light.

Three discrete spectral lines occur at angles of 10.1, 13.7, and 14.8, respectively, in the first-order spectrum of a diffraction-grating spectrometer. (a) If the grating has 3 660 slits/cm, what are the wavelengths of the light? (b) At what angles are these lines found in the second-order spectra?

Intense white light is incident on a diffraction grating that has 600 lines/mm. (a) What is the highest order in which the complete visible spectrum can be seen with this grating? (b) What is the angular separation between the violet edge (400 nm) and the red edge (700 nm) of the first-order spectrum produced by the grating?

The hydrogen spectrum has a red line at 656 nm and a violet line at 434 nm. What angular separation between these two spectral lines is obtained with a diffraction grating that has 4 500 lines/cm?

Consider an array of parallel wires with uniform spacing of 1.30 cm between centers. In air at 20.0C, ultrasound with a frequency of 37.2 kHz from a distant source is incident perpendicular to the array. (Take the speed of sound to be 343 m/s.) (a) Find the number of directions on the other side of the array in which there is a maximum of intensity. (b) Find the angle for each of these directions relative to the direction of the incident beam.

A heliumneon laser (l 5 632.8 nm) is used to calibrate a diffraction grating. If the first-order maximum occurs at 20.5, what is the spacing between adjacent grooves in the grating?

White light is spread out into its spectral components by a diffraction grating. If the grating has 2 000 lines per centimeter, at what angle does red light of wavelength 640 nm appear in the first-order spectrum?

Light from an argon laser strikes a diffraction grating that has 5 310 grooves per centimeter. The central and first-order principal maxima are separated by 0.488 m on a wall 1.72 m from the grating. Determine the wavelength of the laser light.

Take red light at 700 nm and violet at 400 nm as the ends of the visible spectrum and consider the continuous spectrum of white light formed by a diffraction grating with a spacing of d meters between adjacent lines. Show that the interval uv 2 # u # ur2 of the continuous spectrum in second order must overlap the interval uv 3 # u # ur3 of the third-order spectrum. Note: uv2 is the angle of the violet light in second order, and ur2 is the angle made by red light in second order.

Sunlight is incident on a diffraction grating that has 2 750 lines/cm. The second-order spectrum over the visible range (400700 nm) is to be limited to 1.75 cm along a screen that is a distance L from the grating. What is the required value of L?

A diffraction grating has 4.200 3 103 rulings per centimeter. The screen is 2.000 m from the grating. In parts (a) through (e), round each result to four digits, using the rounded values for subsequent calculations. (a) Compute the value of d, the distance between adjacent rulings. Express the answer in meters. (b) Calculate the angle of the second-order maximum made by the 589.0-nm wavelength of sodium. (c) Find the position of this second-order maximum on the screen. (d) Repeat parts (b) and (c) for the second-order maximum of the 589.6-nm wavelength of sodium, finding the angle and position on the screen. (e) What is the distance between the two second-order maxima on the screen? (f) Now find the distance between the two second-order maxima without rounding, carrying all digits in calculator memory. How many significant digits are in agreement? What can you conclude about rounding and the use of intermediate answers in this case?

Light of wavelength 500 nm is incident normally on a diffraction grating. If the third-order maximum of the diffraction pattern is observed at 32.0, (a) what is the number of rulings per centimeter for the grating? (b) Determine the total number of primary maxima that can be observed in this situation.

Light containing two different wavelengths passes through a diffraction grating with 1 200 slits/cm. On a screen 15.0 cm from the grating, the third-order maximum of the shorter wavelength falls midway between the central maximum and the first side maximum for the longer wavelength. If the neighboring maxima of the longer wavelength are 8.44 mm apart on the screen, what are the wavelengths in the light? Hint: Use the small-angle approximation.

The angle of incidence of a light beam in air onto a reflecting surface is continuously variable. The reflected ray is found to be completely polarized when the angle of incidence is 48.0. (a) What is the index of refraction of the reflecting material? (b) If some of the incident light (at an angle of 48.0) passes into the material below the surface, what is the angle of refraction?

Unpolarized light passes through two Polaroid sheets. The transmission axis of the analyzer makes an angle of 35.0 with the axis of the polarizer. (a) What fraction of the original unpolarized light is transmitted through the analyzer? (b) What fraction of the original light is absorbed by the analyzer?

The index of refraction of a glass plate is 1.52. What is the Brewsters angle when the plate is (a) in air and (b) in water? (See Problem 57.)

At what angle above the horizon is the Sun if light from it is completely polarized upon reflection from water?

A light beam is incident on a piece of fused quartz (n 5 1.458) at the Brewsters angle. Find (a) the value of Brewsters angle and (b) the angle of refraction for the transmitted ray.

The critical angle for total internal reflection for sapphire surrounded by air is 34.4. Calculate the Brewsters angle for sapphire if the light is incident from the air.

Equation 24.14 assumes the incident light is in air. If the light is incident from a medium of index n1 onto a medium of index n2, follow the procedure used to derive Equation 24.14 to show that tan up 5 n2/n1.

Plane-polarized light is incident on a single polarizing disk, with the direction of E0 parallel to the direction of the transmission axis. Through what angle should the disk be rotated so that the intensity in the transmitted beam is reduced by a factor of (a) 2.00, (b) 4.00, and (c) 6.00?

Three polarizing plates whose planes are parallel are centered on a common axis. The directions of the transmission axes relative to the common vertical direction are shown in Figure P24.59. A linearly polarized beam of light with plane of polarization parallel to the vertical reference direction is incident from the left onto the first disk with intensity Ii 5 10.0 units (arbitrary). Calculate the transmitted intensity If when u1 5 20.0, u2 5 40.0, and u3 5 60.0. Hint: Make repeated use of Maluss law.

Light of intensity I0 and polarized parallel to the transmission axis of a polarizer is incident on an analyzer. (a) If the transmission axis of the analyzer makes an angle of 45 with the axis of the polarizer, what is the intensity of the transmitted light? (b) What should the angle between the transmission axes be to make I/I0 5 1/3?

Light with a wavelength in vacuum of 546.1 nm falls perpendicularly on a biological specimen that is 1.000 mm thick. The light splits into two beams polarized at right angles, for which the indices of refraction are 1.320 and 1.333, respectively. (a) Calculate the wavelength of each component of the light while it is traversing the specimen. (b) Calculate the phase difference between the two beams when they emerge from the specimen.

Light from a heliumneon laser (l 5 632.8 nm) is incident on a single slit. What is the maximum width of the slit for which no diffraction minima are observed?

Laser light with a wavelength of 632.8 nm is directed through one slit or two slits and allowed to fall on a screen 2.60 m beyond. Figure P24.63 shows the pattern on the screen, with a centimeter ruler below it. Did the light pass through one slit or two slits? Explain how you can tell. If the answer is one slit, find its width. If the answer is two slits, find the distance between their centers.

In a Youngs interference experiment, the two slits are separated by 0.150 mm and the incident light includes two wavelengths: l1 5 540 nm (green) and l2 5 450 nm (blue). The overlapping interference patterns are observed on a screen 1.40 m from the slits. (a) Find a relationship between the orders m1 and m2 that determines where a bright fringe of the green light coincides with a bright fringe of the blue light. (The order m1 is associated with l1, and m2 is associated with l2.) (b) Find the minimum values of m1 and m2 such that the overlapping of the bright fringes will occur and find the position of the overlap on the screen.

Light of wavelength 546 nm (the intense green line from a mercury source) produces a Youngs interference pattern in which the second minimum from the central maximum is along a direction that makes an angle of 18.0 min of arc with the axis through the central maximum. What is the distance between the parallel slits?

The two speakers are placed 35.0 cm apart. A single oscillator makes the speakers vibrate in phase at a frequency of 2.00 kHz. At what angles, measured from the perpendicular bisector of the line joining the speakers, would a distant observer hear maximum sound intensity? Minimum sound intensity? (Take the speed of sound to be 340 m/s.)

Interference effects are produced at point P on a screen as a result of direct rays from a 500-nm source and reflected rays off a mirror, as shown in Figure P24.67. If the source is L 5 100 m to the left of the screen and h 5 1.00 cm above the mirror, find the distance y (in millimeters) to the first dark band above the mirror.

Many cells are transparent and colorless. Structures of great interest in biology and medicine can be practically invisible to ordinary microscopy. An interference microscope reveals a difference in refractive index as a shift in interference fringes to indicate the size and shape of cell structures. The idea is exemplified in the following problem: An air wedge is formed between two glass plates in contact along one edge and slightly separated at the opposite edge. When the plates are illuminated with monochromatic light from above, the reflected light has 85 dark fringes. Calculate the number of dark fringes that appear if water (n 5 1.33) replaces the air between the plates.

Figure P24.69 shows a radio-wave transmitter and a receiver, both h 5 50.0 m above the ground and d 5 600 m apart. The receiver can receive signals directly from the transmitter and indirectly from signals that bounce off the ground. If the ground is level between the transmitter and receiver and a l/2 phase shift occurs upon reflection, determine the longest wavelengths that interfere (a) constructively and (b) destructively.

Three polarizers, centered on a common axis and with their planes parallel to one another, have transmission axes oriented at angles of u1, u2, and u3 from the vertical, as shown in Figure P24.59. Light of intensity Ii, polarized with its plane of polarization oriented vertically, is incident from the left onto the first polarizer. What is the ratio If /Ii of the final transmitted intensity to the incident intensity if (a) u1 5 45, u2 5 90, and u3 5 0? (b) u1 5 0, u2 5 45, and u3 5 90?

The transmitting antenna on a submarine is 5.00 m above the water when the ship surfaces. The captain wishes to transmit a message to a receiver on a 90.0-m-tall cliff at the ocean shore. If the signal is to be completely polarized by reflection off the ocean surface, how far must the ship be from the shore?

A plano-convex lens (flat on one side, convex on the other) with index of refraction n rests with its curved side (radius of curvature R) on a flat glass surface of the same index of refraction with a film of index nfilm between them. The lens is illuminated from above by light of wavelength l. Show that the dark Newton rings that appear have radii of r < "mlR/nfilm where m is an integer.

A diffraction pattern is produced on a screen 140 cm from a single slit, using monochromatic light of wavelength 500 nm. The distance from the center of the central maximum to the first-order maximum is 3.00 mm. Calculate the slit width. Hint: Assume that the first-order maximum is halfway between the firstand second-order minima.

A flat piece of glass is supported horizontally above the flat end of a 10.0-cm-long metal rod that has its lower end rigidly fixed. The thin film of air between the rod and the glass is observed to be bright when illuminated by light of wavelength 500 nm. As the temperature is slowly increased by 25.0C, the film changes from bright to dark and back to bright 200 times. What is the coefficient of linear expansion of the metal?