A uniform layer of water (n 5 1.33) lies on a glass plate

In a Youngs double-slit experiment, the wavelength of the light used is 520 nm (in vacuum), and the separation between the slits is 1.4 3 1026 m. Determine the angle that locates (a) the dark fringe for which m 5 0, (b) the bright fringe for which m 5 1, (c) the dark fringe for which m 5 1, and (d) the bright fringe for which m 5 2.

In a Youngs double-slit experiment, the angle that locates the second dark fringe on either side of the central bright fringe is 5.48. Find the ratio d/l of the slit separation d to the wavelength l of the light.

Two in-phase sources of waves are separated by a distance of 4.00 m. These sources produce identical waves that have a wavelength of 5.00 m. On the line between them, there are two places at which the same type of interference occurs. (a) Is it constructive or destructive interference, and (b) where are the places located?

The dark fringe for m 5 0 in a Youngs double-slit experiment is located at an angle of u 5 158. What is the angle that locates the dark fringe for m 5 1?

In a Youngs double-slit experiment, the seventh dark fringe is located 0.025 m to the side of the central bright fringe on a fl at screen, which is 1.1 m away from the slits. The separation between the slits is 1.4 3 1024 m. What is the wavelength of the light being used?

Two parallel slits are illuminated by light composed of two wavelengths. One wavelength is lA 5 645 nm. The other wavelength is lB and is unknown. On a viewing screen, the light with wavelength lA 5 645 nm produces its third-order bright fringe at the same place where the light with wavelength lB produces its fourth dark fringe. The fringes are counted relative to the central or zeroth-order bright fringe. What is the unknown wavelength?

In a setup like that in Figure 27.7, a wavelength of 625 nm is used in a Youngs double-slit experiment. The separation between the slits is d 5 1.4 3 1025 m. The total width of the screen is 0.20 m. In one version of the setup, the separation between the double slit and the screen is LA 5 0.35 m, whereas in another version it is LB 5 0.50 m. On one side of the central bright fringe, how many bright fringes lie on the screen in the two versions of the setup? Do not include the central bright fringe in your counting

At most, how many bright fringes can be formed on either side of the central bright fringe when light of wavelength 625 nm falls on a double slit whose slit separation is 3.76 3 1026 m?

In a Youngs double-slit experiment the separation y between the second-order bright fringe and the central bright fringe on a fl at screen is 0.0180 m when the light has a wavelength of 425 nm. Assume that the angles that locate the fringes on the screen are small enough so that sin u < tan u. Find the separation y when the light has a wavelength of 585 nm.

In Youngs experiment a mixture of orange light (611 nm) and blue light (471 nm) shines on the double slit. The centers of the fi rst-order bright blue fringes lie at the outer edges of a screen that is located 0.500 m away from the slits. However, the fi rst-order bright orange fringes fall of the screen. By how much and in which direction (toward or away from the slits) should the screen be moved so that the centers of the fi rst-order bright orange fringes will just appear on the screen? It may be assumed that u is small, so that sin u < tan u.

A sheet that is made of plastic (n 5 1.60) covers one slit of a double slit (see the drawing). When the double slit is illuminated by monochromatic light (lvacuum 5 586 nm), the center of the screen appears dark rather than bright. What is the minimum thickness of the plastic?

You are standing in air and are looking at a fl at piece of glass (n 5 1.52) on which there is a layer of transparent plastic (n 5 1.61). Light whose wavelength is 589 nm in vacuum is incident nearly perpendicularly on the coated glass and refl ects into your eyes. The layer of plastic looks dark. Find the two smallest possible nonzero values for the thickness of the layer.

A nonrefl ective coating of magnesium fl uoride (n 5 1.38) covers the glass (n 5 1.52) of a camera lens. Assuming that the coating prevents refl ection of yellow-green light (wavelength in vacuum 5 565 nm), determine the minimum nonzero thickness that the coating can have.

When monochromatic light shines perpendicularly on a soap fi lm (n 5 1.33) with air on each side, the second smallest nonzero fi lm thickness for which destructive interference of refl ected light is observed is 296 nm. What is the vacuum wavelength of the light in nm?

A transparent fi lm (n 5 1.43) is deposited on a glass plate (n 5 1.52) to form a nonrefl ecting coating. The fi lm has a thickness that is 1.07 3 1027 m. What is the longest possible wavelength (in vacuum) of light for which this fi lm has been designed?

A tank of gasoline (n 5 1.40) is open to the air (n 5 1.00). A thin fi lm of liquid fl oats on the gasoline and has a refractive index that is between 1.00 and 1.40. Light that has a wavelength of 625 nm (in vacuum) shines perpendicularly down through the air onto this fi lm, and in this light the fi lm looks bright due to constructive interference. The thickness of the fi lm is 242 nm and is the minimum nonzero thickness for which constructive interference can occur. What is the refractive index of the fi lm?

Review Conceptual Example 4 before beginning this problem. A soap fi lm with diff erent thicknesses at diff erent places has an unknown refractive index n and air on both sides. In refl ected light it looks multicolored. One region looks yellow because destructive interference has removed blue (lvacuum 5 469 nm) from the refl ected light, while another looks magenta because destructive interference has removed green (lvacuum 5 555 nm). In these regions the fi lm has the minimum nonzero thickness t required for the destructive interference to occur. Find the ratio tmagenta/tyellow.

A fi lm of oil lies on wet pavement. The refractive index of the oil exceeds that of the water. The fi lm has the minimum nonzero thickness such that it appears dark due to destructive interference when viewed in red light (wavelength 5 640.0 nm in vacuum). Assuming that the visible spectrum extends from 380 to 750 nm, for which visible wavelength(s) in vacuum will the fi lm appear bright due to constructive interference?

Orange light (lvacuum 5 611 nm) shines on a soap fi lm (n 5 1.33) that has air on either side of it. The light strikes the fi lm perpendicularly. What is the minimum thickness of the fi lm for which constructive interference causes it to look bright in refl ected light?

The drawing shows a cross section of a planoconcave lens resting on a fl at glass plate. (A planoconcave lens has one surface that is a plane and the other that is concave spherical.) The thickness t is 1.37 3 1025 m. The lens is illuminated with monochromatic light (lvacuum 5 550 nm), and a series of concentric bright and dark rings is formed, much like Newtons rings. How many bright rings are there? (Hint: The cross section shown in the drawing reveals that a kind of air wedge exists between the place where the two pieces of glass touch and the top of the curved surface where the distance t is marked.)

A piece of curved glass has a radius of curvature of r 5 10.0 m and is used to form Newtons rings, as in Figure 27.13. Not counting the dark spot at the center of the pattern, there are one hundred dark fringes, the last one being at the outer edge of the curved piece of glass. The light being used has a wavelength of 654 nm in vacuum. What is the radius R of the outermost dark ring in the pattern? (Hint: Note that r is much greater than R, and you may assume that tan u 5 u for small angles, where u must be expressed in radians.)

A uniform layer of water (n 5 1.33) lies on a glass plate (n 5 1.52). Light shines perpendicularly on the layer. Because of constructive interference, the layer looks maximally bright when the wavelength of the light is 432 nm in vacuum and also when it is 648 nm in vacuum. (a) Obtain the minimum thickness of the fi lm. (b) Assuming that the fi lm has the minimum thickness and that the visible spectrum extends from 380 to 750 nm, determine the visible wavelength(s) in vacuum for which the fi lm appears completely dark.

As Section 17.3 discusses, high-frequency sound waves exhibit less diff raction than low-frequency sound waves do. However, even highfrequency sound waves exhibit much more diff raction under normal circumstances than do light waves that pass through the same opening. The highest frequency that a healthy ear can typically hear is 2.0 3 104 Hz. Assume that a sound wave with this frequency travels at 343 m/s and passes through a doorway that has a width of 0.91 m. Determine the angle that locates the fi rst minimum to either side of the central maximum in the diff raction pattern for the sound. This minimum is equivalent to the fi rst dark fringe in a single-slit diff raction pattern for light. (b) Suppose that yellow light (wavelength 5 580 nm in vacuum) passes through a doorway and that the fi rst dark fringe in its diff raction pattern is located at the angle determined in part (a). How wide would this hypothetical doorway have to be?

A dark fringe in the diff raction pattern of a single slit is located at an angle of uA 5 348. With the same light, the same dark fringe formed with another single slit is at an angle of uB 5 568. Find the ratio WA/WB of the widths of the two slits

A diff raction pattern forms when light passes through a single slit. The wavelength of the light is 675 nm. Determine the angle that locates the fi rst dark fringe when the width of the slit is (a) 1.8 3 1024 m and (b) 1.8 3 1026 m

A slit has a width of W1 5 2.3 3 1026 m. When light with a wavelength of l1 5 510 nm passes through this slit, the width of the central bright fringe on a fl at observation screen has a certain value. With the screen kept in the same place, this slit is replaced with a second slit (width W2), and a wavelength of l2 5 740 nm is used. The width of the central bright fringe on the screen is observed to be unchanged. Find W2.

Light that has a wavelength of 668 nm passes through a slit 6.73 3 1026 m wide and falls on a screen that is 1.85 m away. What is the distance on the screen from the center of the central bright fringe to the third dark fringe on either side?

Light shines through a single slit whose width is 5.6 3 1024 m. A diff raction pattern is formed on a fl at screen located 4.0 m away. The distance between the middle of the central bright fringe and the fi rst dark fringe is 3.5 mm. What is the wavelength of the light?

Light waves with two diff erent wavelengths, 632 nm and 474 nm, pass simultaneously through a single slit whose width is 7.15 3 1025 m and strike a screen 1.20 m from the slit. Two diff raction patterns are formed on the screen. What is the distance (in cm) between the common center of the diff raction patterns and the fi rst occurrence of a dark fringe from one pattern falling on top of a dark fringe from the other pattern?

The central bright fringe in a single-slit diff raction pattern has a width that equals the distance between the screen and the slit. Find the ratio l/W of the wavelength l of the light to the width W of the slit.

How many dark fringes will be produced on either side of the central maximum if light (l 5 651 nm) is incident on a single slit that is 5.47 3 1026 m wide?

In a single-slit diff raction pattern, the central fringe is 450 times as wide as the slit. The screen is 18 000 times farther from the slit than the slit is wide. What is the ratio l/W, where l is the wavelength of the light shining through the slit and W is the width of the slit? Assume that the angle that locates a dark fringe on the screen is small, so that sin u < tan u.

Two stars are 3.7 3 1011 m apart and are equally distant from the earth. A telescope has an objective lens with a diameter of 1.02 m and just detects these stars as separate objects. Assume that light of wavelength 550 nm is being observed. Also assume that diff raction eff ects, rather than atmospheric turbulence, limit the resolving power of the telescope. Find the maximum distance that these stars could be from the earth.

It is claimed that some professional baseball players can see which way the ball is spinning as it travels toward home plate. One way to judge this claim is to estimate the distance at which a batter can fi rst hope to resolve two points on opposite sides of a baseball, which has a diameter of 0.0738 m. (a) Estimate this distance, assuming that * * * ** the pupil of the eye has a diameter of 2.0 mm and the wavelength of the light is 550 nm in vacuum. (b) Considering that the distance between the pitchers mound and home plate is 18.4 m, can you rule out the claim based on your answer to part (a)?

Late one night on a highway, a car speeds by you and fades into the distance. Under these conditions the pupils of your eyes have diameters of about 7.0 mm. The taillights of this car are separated by a distance of 1.2 m and emit red light (wavelength 5 660 nm in vacuum). How far away from you is this car when its taillights appear to merge into a single spot of light because of the eff ects of diff raction?

An inkjet printer uses tiny dots of red, green, and blue ink to produce an image. Assume that the dot separation on the printed page is the same for all colors. At normal viewing distances, the eye does not resolve the individual dots, regardless of color, so that the image has a normal look. The wavelengths for red, green, and blue are lred 5 660 nm, lgreen 5 550 nm, and lblue 5 470 nm. The diameter of the pupil through which light enters the eye is 2.0 mm. For a viewing distance of 0.40 m, what is the maximum allowable dot separation?

A hunter who is a bit of a braggart claims that from a distance of 1.6 km he can selectively shoot either of two squirrels who are sitting ten centimeters apart on the same branch of a tree. Whats more, he claims that he can do this without the aid of a telescopic sight on his rifl e. (a) Determine the diameter of the pupils of his eyes that would be required for him to be able to resolve the squirrels as separate objects. In this calculation use a wavelength of 498 nm (in vacuum) for the light. (b) State whether his claim is reasonable, and provide a reason for your answer. In evaluating his claim, consider that the human eye automatically adjusts the diameter of its pupil over a typical range of 2 to 8 mm, the larger values coming into play as the lighting becomes darker. Note also that under dark conditions, the eye is most sensitive to a wavelength of 498 nm.

Review Conceptual Example 8 as background for this problem. In addition to the data given there, assume that the dots in the painting are separated by 1.5 mm and that the wavelength of the light is lvacuum 5 550 nm. Find the distance at which the dots can just be resolved by (a) the eye and (b) the camera.

Astronomers have discovered a planetary system orbiting the star Upsilon Andromedae, which is at a distance of 4.2 3 1017 m from the earth. One planet is believed to be located at a distance of 1.2 3 1011 m from the star. Using visible light with a vacuum wavelength of 550 nm, what is the minimum necessary aperture diameter that a telescope must have so that it can resolve the planet and the star?

The pupil of an eagles eye has a diameter of 6.0 mm. Two fi eld mice are separated by 0.010 m. From a distance of 176 m, the eagle sees them as one unresolved object and dives toward them at a speed of 17 m/s. Assume that the eagles eye detects light that has a wavelength of 550 nm in vacuum. How much time passes until the eagle sees the mice as separate objects?

Consult Multiple-Concept Example 7 to see a model for solving this kind of problem. You are using a microscope to examine a blood sample. Recall from Section 26.12 that the sample should be placed just outside the focal point of the objective lens of the microscope. (a) If the specimen is being illuminated with light of wavelength l and the diameter of the objective equals its focal length, determine the closest distance between two blood cells that can just be resolved. Express your answer in terms of l. (b) Based on your answer to (a), should you use light with a longer wavelength or a shorter wavelength if you wish to resolve two blood cells that are even closer together?

Two concentric circles of light emit light whose wavelength is 555 nm. The larger circle has a radius of 4.0 cm, and the smaller circle has a radius of 1.0 cm. When taking a picture of these lighted circles, a camera admits light through an aperture whose diameter is 12.5 mm. What is the maximum distance at which the camera can (a) distinguish one circle from the other and (b) reveal that the inner circle is a circle of light rather than a solid disk of light?

A diff raction grating is 1.50 cm wide and contains 2400 lines. When used with light of a certain wavelength, a third-order maximum is formed at an angle of 18.08. What is the wavelength (in nm)?

The light shining on a diff raction grating has a wavelength of 495 nm (in vacuum). The grating produces a second-order bright fringe whose position is defi ned by an angle of 9.348. How many lines per centimeter does the grating have?

For a wavelength of 420 nm, a diff raction grating produces a bright fringe at an angle of 268. For an unknown wavelength, the same grating produces a bright fringe at an angle of 418. In both cases the bright fringes are of the same order m. What is the unknown wavelength?

Two diff raction gratings, A and B, are located at the same distance from the observation screens. Light with the same wavelength l is used for each. The separation between adjacent principal maxima for grating A is 2.7 cm, and for grating B it is 3.2 cm. Grating A has 2000 lines per meter. How many lines per meter does grating B have? (Hint: The diff raction angles are small enough that the approximation sin u < tan u can be used.)

The wavelength of the laser beam used in a compact disc player is 780 nm. Suppose that a diff raction grating produces fi rst-order tracking beams that are 1.2 mm apart at a distance of 3.0 mm from the grating. Estimate the spacing between the slits of the grating

The fi rst-order principle maximum produced by a grating is located at an angle of u 5 18.08. What is the angle for the third-order maximum with the same light?

A diff raction grating has 2604 lines per centimeter, and it produces a principal maximum at u 5 30.08. The grating is used with light that contains all wavelengths between 410 and 660 nm. What is (are) the wavelength(s) of the incident light that could have produced this maximum?

Light of wavelength 410 nm (in vacuum) is incident on a diffraction grating that has a slit separation of 1.2 3 1025 m. The distance between the grating and the viewing screen is 0.15 m. A diff raction pattern is produced on the screen that consists of a central bright fringe and ** * * higher-order bright fringes (see the drawing). (a) Determine the distance y from the central bright fringe to the second-order bright fringe. (Hint: The diff raction angles are small enough that the approximation tan u < sin u can be used.) (b) If the entire apparatus is submerged in water (nwater 5 1.33), what is the distance y?

Violet light (wavelength 5 410 nm) and red light (wavelength 5 660 nm) lie at opposite ends of the visible spectrum. (a) For each wavelength, fi nd the angle u that locates the fi rst-order maximum produced by a grating with 3300 lines/cm. This grating converts a mixture of all colors between violet and red into a rainbow-like dispersion between the two angles. Repeat the calculation above for (b) the second-order maximum and (c) the third-order maximum. (d) From your results, decide whether there is an overlap between any of the rainbows and, if so, specify which orders overlap.

The distance between adjacent slits of a certain diff raction grating is 1.250 3 1025 m. The grating is illuminated by monochromatic light with a wavelength of 656.0 nm, and is then heated so that its temperature increases by 100.0 C8. Determine the change in the angle of the seventh-order principal maximum that occurs as a result of the thermal expansion of the grating. The coeffi cient of linear expansion for the diffraction grating is 1.30 3 1024 (C8) 21 . Be sure to include the proper algebraic sign with your answer: 1 if the angle increases, 2 if the angle decreases

Two gratings A and B have slit separations dA and dB, respectively. They are used with the same light and the same observation screen. When grating A is replaced with grating B, it is observed that the fi rst-order maximum of A is exactly replaced by the second-order maximum of B. (a) Determine the ratio dB/dA of the spacings between the slits of the gratings. (b) Find the next two principal maxima of grating A and the principal maxima of B that exactly replace them when the gratings are switched. Identify these maxima by their order numbers.

A soap fi lm (n 5 1.33) is 465 nm thick and lies on a glass plate (n 5 1.52). Sunlight, whose wavelengths (in vacuum) extend from 380 to 750 nm, travels through the air and strikes the fi lm perpendicularly. For which wavelength(s) in this range does destructive interference cause the fi lm to look dark in refl ected light?

In a Youngs double-slit experiment, two rays of monochromatic light emerge from the slits and meet at a point on a distant screen, as in Figure 27.6a. The point on the screen where these two rays meet is the eighth-order bright fringe. The diff erence in the distances that the two rays travel is 4.57 3 1026 m. What is the wavelength (in nm) of the monochromatic light?

Point A is the midpoint of one of the sides of a square. On the side opposite this spot, two in-phase loudspeakers are located at adjacent corners, as shown in the fi gure. Standing at point A you hear a loud sound because of constructive interference between the identical sound waves coming from the speakers. As you walk along the side of the square toward either empty corner, the loudness diminishes gradually to nothing and then increases again until you hear a maximally loud sound at the corner. If the length of each side of the square is 4.6 m, fi nd the wavelength of the sound waves.

A fl at observation screen is placed at a distance of 4.5 m from a pair of slits. The separation on the screen between the central bright fringe and the fi rst-order bright fringe is 0.037 m. The light illuminating the slits has a wavelength of 490 nm. Determine the slit separation.

A fl at screen is located 0.60 m away from a single slit. Light with a wavelength of 510 nm (in vacuum) shines through the slit and produces a diff raction pattern. The width of the central bright fringe on the screen is 0.050 m. What is the width of the slit?

A large group of football fans comes to the game with colored cards that spell out the name of their team when held up simultaneously. Most of the cards are colored blue (lvacuum 5 480 nm). When displayed, the average distance between neighboring cards is 5.0 cm. If the cards are to blur together into solid blocks of color when viewed by a spectator at the other end of the stadium (160 m away), what must be the maximum diameter (in mm) of the spectators pupils?

Review Conceptual Example 2 before attempting this problem. Two slits are 0.158 mm apart. A mixture of red light (wavelength 5 665 nm) and yellow-green light (wavelength 5 565 nm) falls on the slits. A fl at observation screen is located 2.24 m away. What is the distance on the screen between the third-order red fringe and the third-order yellow-green fringe?

The same diff raction grating is used with two diff erent wavelengths of light, lA and lB. The fourth-order principal maximum of light A exactly overlaps the third-order principal maximum of light B. Find the ratio lA/lB.

A spotlight sends red light (wavelength 5 694.3 nm) to the moon. At the surface of the moon, which is 3.77 3 108 m away, the light * * * strikes a refl ector left there by astronauts. The refl ected light returns to the earth, where it is detected. When it leaves the spotlight, the circular beam of light has a diameter of about 0.20 m, and diff raction causes the beam to spread as the light travels to the moon. In eff ect, the fi rst circular dark fringe in the diff raction pattern defi nes the size of the central bright spot on the moon. Determine the diameter (not the radius) of the central bright spot on the moon.

In a single-slit diff raction pattern on a fl at screen, the central bright fringe is 1.2 cm wide when the slit width is 3.2 3 1025 m. When the slit is replaced by a second slit, the wavelength of the light and the distance to the screen remaining unchanged, the central bright fringe broadens to a width of 1.9 cm. What is the width of the second slit? It may be assumed that u is so small that sin u < tan u.

A beam of light is sent directly down onto a glass plate (n 5 1.5) and a plastic plate (n 5 1.2) that form a thin wedge of air (see the drawing). An observer looking down through the glass plate sees the fringe pattern shown in the lower part of the drawing, with the dark fringes at the ends A and B. The wavelength of the light is 520 nm. Using the fringe pattern shown in the drawing, determine the thickness of the air wedge at B.

There are 5620 lines per centimeter in a grating that is used with light whose wavelength is 471 nm. A fl at observation screen is located at a distance of 0.750 m from the grating. What is the minimum width that the screen must have so the centers of all the principal maxima formed on either side of the central maximum fall on the screen?

A circular drop of oil lies on a smooth, horizontal surface. The drop is thickest in the center and tapers to zero thickness at the edge. When illuminated from above by blue light (l 5 455 nm), 56 concentric bright rings are visible, including a bright fringe at the edge of the drop. In addition, there is a bright spot in the center of the drop. When the drop is illuminated from above by red light (l 5 637 nm), a bright spot again appears at the center, along with a diff erent number of bright rings. Ignoring the bright spot, how many bright rings appear in red light? Assume that the index of refraction of the oil is the same for both wavelengths

A square is 3.5 m on a side, and point A is the midpoint of one of the sides. On the side opposite this spot, two in-phase loudspeakers are located at adjacent corners, as shown in the fi gure. Standing at point A, you hear a loud sound because constructive interference occurs between the identical sound waves coming from the speakers. As you walk along the side of the square toward either empty corner, the loudness diminishes gradually but does not entirely disappear until you reach either empty corner, where you hear no sound at all. Thus, at each empty corner destructive interference occurs. Concepts: (i) Why does constructive interference occur at point A? (ii) What is the general condition that leads to destructive interference? (iii) The condition that leads to destructive interference entails a number of possibilities. Which of them applies at either empty corner of the square? Calculation: Find the wavelength of the sound waves.

A soap fi lm (n 5 1.33) is 375 nm thick and coats a fl at piece of glass (n 5 1.52). Thus, air is on one side of the fi lm and glass is on the other side, as the fi gure illustrates. Sunlight, whose wavelengths * (in vacuum) extend from 380 to 750 nm, travels through air and strikes the fi lm nearly perpendicularly. Concepts: (i) What, if any, phase change occurs when light, traveling in air, refl ects from the airfi lm interface? (ii) What, if any, phase change occurs when light, traveling in the fi lm, refl ects from the fi lmglass interface? (iii) Is the wavelength of the light in the fi lm greater than, smaller than, or equal to the wavelength in a vacuum? Calculations: For what wavelengths in the range of 380 to 750 nm does constructive interference cause the fi lm to look bright in refl ected light?