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All Textbook Solutions for Physics for Scientists and Engineers with Modern Physics

39AP 40AP 41AP 42AP 43AP 44AP Review. (a) A homeowner has a solar water heater installed on the roof of his house (Fig. P33.45). The heater is a flat, closed box with excellent thermal insulation. Its interior is painted black, and its front face is made of insulating glass. Its emissivity for visible light is 0.900, and its emissivity for infrared light is 0.700. Light from the noontime Sun is incident perpendicular to the glass with an intensity of 1 000 W/m2, and no water enters or leaves the box. Find the steady-state temperature of the boxs interior. (b) What If? The homeowner builds an identical box with no water tubes. It lies flat on the ground in front of the house. He uses it as a cold frame, where he plants seeds in early spring. Assuming the same noontime Sun is at an elevation angle of 50.0, find the steady-state temperature of the interior of the box when its ventilation slots are tightly closed. Figure P33.45 46AP 47AP 48AP 49AP 50CP 51CP 34.1QQ If beam is the incoming beam in Figure 34.10b, which of the other four red lines are reflected beams and which are refracted beams? Figure 34.10 (a) The wave under refraction model. (b) Light incident on the Lucite block refracts both when it enters the block and when it leases the block.Light passes from a material with index of refraction 1.3 into one with index of refraction 1.2. Compared to the incident ray, what happens to the refracted ray? (a) It bends toward the normal. (b) It is undeflected. (c) It bends away from the normal.34.4QQ 34.5QQ In an experiment to measure the speed of light using the apparatus of Armand H. L. Fizeau (see Fig. 34.2), the distance between light source and mirror was 11.45 km and the wheel had 720 notches. The experimentally determined value of c was 2.998 108 m/s when the outgoing light passed through one notch and then returned through the next notch. Calculate the minimum angular speed of the wheel for this experiment.2P As a result of his observations, Ole Roemer concluded that eclipses of Io by Jupiter were delayed by 22 min during a six-month period as the Earth moved from the point in its orbit where it is closest to Jupiter to the diametrically opposite point where it is farthest from Jupiter. Using the value 1.50 108 km as the average radius of the Earths orbit around the Sun, calculate the speed of light from these data.4P You are working for an optical research company during a summer break. Part of the apparatus in one particular experiment is shown in Figure 34.7b. In fact, the experimenter used this text book to set up this part of the experiment, and also used the result in the What If? section to determine the angular change in direction of the light beam: =3602(1) The experimenter is constantly grumbling that the measuring device to determine the angle on the inside of the mirrors is constantly getting in the wav of the light beam and making his life difficult. You quickly draw Figure P34.5 and then say. Then why dont you use the measuring device to measure the angle outside the mirror, and then your device wont get in the way of the light? The experimenter, who has never thought of this, tries to save face and says to you. Well, Smarty, then tell me how angle depends on angle ! You provide him the answer quickly. Figure P34.5 6P 7P 8P 9P A ray of light strikes a flat block of glass (n = 1.50) of thickness 2.00 cm at an angle of 30.0 with the normal. Trace the light beam through the glass and find the angles of incidence and refraction at each surface.11P 12P 13P 14P When you look through a window, by what time interval is the light you see delayed by having to go through glass instead of air? Make an order-of-magnitude estimate on the basis of data you specify. By how many wavelengths is it delayed?16P You have just installed a new bathroom in your home. Your shower doors have frosted glass to provide privacy for the person using the shower. The frosted surface is on the outside of the shower door, facing the rest of the bathroom. The frosting is done by acid etching the surface so that light incident on the rough surface is scattered in all directions. Proud of your new bathroom, you take a photo of it with your smartphone. You notice in the photograph that you can see a reflection of the flash in the shower doors and the reflection is surrounded by a halo of light. Curious, you turn on a laser pointer and aim it at the shower door. Looking closely at the reflection, you again see a halo that consists of a dark area surrounding the reflection of the pointer and then an area of brightness outside this dark ring. You grab a micrometer and a ruler and measure the thickness of the glass to be 6.35 mm and the inner radius of the bright halo to be 10.7 mm. From these measurements, you determine the index of refraction of the glass.18P 19P 20P 21P A submarine is 300 m horizontally from the shore of a fresh water lake and 100 m beneath the surface of the water. A laser beam is sent from the submarine so that the beam strikes the surface of the water 210 m from the shore. A building stands on the shore, and the laser beam hits a target at the top of the building. The goal is to find the height of the target above sea level. (a) Draw a diagram of the situation, identifying the two triangles that are important in finding the solution. (b) Find the angle of incidence of the beam striking the waterair interface. (c) Find the angle of refraction. (d) What angle docs the refracted beam make with the horizontal? (e) Find the height of the target above sea level.23P 24P 25P 26P 27P 28P 29P 30P An optical fiber has an index of refraction n and diameter d. It is surrounded by vacuum. Light is sent into the fiber along its axis as shown in Figure P34.31. (a) Find the smallest outside radius Rmin permitted for a bend in the fiber if no light is to escape. (b) What If? What result does part (a) predict as d approaches zero? Is this behavior reasonable? Explain. (c) As n increases? (d) As n approaches 1? (c) Evaluate Rmin assuming the fiber diameter is 100 m and its index of refraction is 1.40. Figure P34.31 32AP How many times will the incident beam in Figure P34.33 (page 922) be reflected by each of the parallel mirrors? Figure P34.33 34AP 35AP 36AP 37AP 38AP 39AP A light ray enters the atmosphere of a planet and descends vertically to the surface a distance h below. The index of refraction where the light enters the atmosphere is 1.00, and it increases linearly with distance to have the value n at the planet surface. (a) Over what time interval does the light traverse this path? (b) By what fraction is the time interval larger than that required in the absence of an atmosphere?41AP 42AP 43AP 44AP 45AP 46AP 47AP 48AP 49AP Figure P34.50 shows a top view of a square enclosure. The inner surfaces are plane mirrors. A ray of light enters a small hole in the center of one mirror. (a) At what angle must the ray enter if it exits through the hole after being reflected once by each of the other three mirrors? (b) What If? Are there other values of for which the ray can exit after multiple reflections? If so, sketch one of the rays paths. Figure P34.50 51AP 52CP 53CP Pierre de Fermat (16011665) showed that whenever light travels from one point to another, its actual path is the path that requires the smallest time interval. This statement is known as Fermats principle. The simplest example is for light propagating in a homogeneous medium. It moves in a straight line because a straight line is the shortest distance between two points. Derive Snells law of refraction from Fermats principle. Proceed as follows. In Figure P34.54, a light ray travels from point P in medium 1 to point Q in medium 2. The two points are, respectively, at perpendicular distances a and b from the interface. The displacement from P to Q has the component d parallel to the interface, and we let x represent the coordinate of the point where the ray enters the second medium. Let t = 0 be the instant the light starts from P. (a) Show that the time at which the light arrives at Q is t=r1v1+r2v2=n1a2+x2c+n2b2+(dx)2c (b) To obtain the value of x for which t has its minimum value, differentiate t with respect to x and set the derivative equal to zero. Show that the result implies n1xa2+x2=n2(dx)b2+(dx)2 (c) Show that this expression in turn gives Snells law. n1sin1=n2sin2 Figure P34.54 Problems 54 and 55.55CP 56CP 57CP 35.1QQ You wish to start a fire by reflecting sunlight from a mirror onto some paper under a pile of wood. Which would be the best choice for the type of mirror? (a) flat (b) concave (c) convex Consider the image in the mirror in Figure 35.14. Based on the appearance of this image, would you conclude that (a) the mirror is concave and the image is real, (b) the mirror is concave and the image is virtual, (c) the mirror is convex and the image is real, or (d) the mirror is convex and the image is virtual? Figure 35.14 35.4QQ 35.5QQ What is the focal length of a pane of window glass? (a) zero (b) infinity (c) the thickness of the glass (d) impossible to determine 35.7QQ (a) Does your bathroom mirror show you older or younger than you actually are? (b) Compute an order-of-magnitude estimate for the age difference based on data you specify.Two flat mirrors have their reflecting surfaces facing each other, with the edge of one mirror in contact with an edge of the other, so that the angle between the mirrors is . When an object is placed between the mirrors, a number of images are formed. In general, if the angle is such that n = 360, where n is an integer, the number of images formed is n 1. Graphically, find all the image positions for the case n = 6 when a point object is between the mirrors (but not on the angle bisector).A periscope (Fig. P35.3) is useful for viewing objects that cannot be seen directly. It can be used in submarines and when watching golf matches or parades from behind a crowd of people. Suppose the object is a distance p1 from the upper mirror and the centers of the two flat mirrors are separated by a distance h. (a) What is the distance of the final image from the lower mirror? (b) Is the final image real or virtual? (c) Is it upright or inverted? (d) What is its magnification? (e) Does it appear to be left-right reversed? Figure P35.3 4P 5P 6P An object of height 2.00 cm is placed 30.0 cm from a convex spherical mirror of focal length of magnitude 10.0 cm. (a) Find the location of the image. (b) Indicate whether the image is upright or inverted. (c) Determine the height of the image.8P 9P A concave spherical mirror has a radius of curvature of magnitude 24.0 cm. (a) Determine the object position for which the resulting image is upright and larger than the object by a factor of 3.00. (b) Draw a ray diagram to determine the position of the image. (c) Is the image real or virtual?11P 12P 13P 14P 15P 16P One end of a long glass rod (n = 1.50) is formed into a convex surface with a radius of curvature of magnitude 6.0 cm. An object is located in air along the axis of the rod. Find the image positions corresponding to object distances of (a) 20.0 cm, (b) 10.0 cm, and (c) 3.00 cm from the convex end of the rod.18P 19P Figure P35.20 (page 958) shows a curved surface separating a material with index of refraction n1 from a material with index n2. The surface forms an image I of object O. The ray shown in red passes through the surface along a radial line. Its angles of incidence and refraction are both zero, so its direction does not change at the surface. For the ray shown in blue, the direction changes according to Snells law, n1 sin 1 = n2 sin 2. For paraxial rays, we assume 1, and 2 are small, so we may write n1 tan 1 = n2 tan 2. The magnification is defined as M = h/h. Prove that the magnification is given by M = n1q/n2p. Figure P35.20 To dress up your dorm room, you have purchased a perfectly spherical glass fishbowl to place on the windowsill. After placing the sand, decorations, and water in the bowl of diameter 40.0 cm, you transfer a single tropical fish from a plastic bag into the bowl. As you watch the fish, your roommate comes home. He watches the fish also and notices that the apparent size of the fish changes as it swims around in the bowl. (a) He is not taking a physics course, so he asks you to tell him the range of magnifications of the fish as it swims along a line from the back of the bowl along a line passing through the center of the bowl directly toward the observer. (b) Your roommate also asks you if the fish might be baked if it swims through a point at which the rays of the Sun focus at some point as they pass through the curved sides of the bowl. Should you worry about your fish being baked? Ignore the effect of the thin glass walls of the bowl; take only the water into consideration.You are working for a solar energy company. Your supervisor has asked you to investigate a new idea that has been proposed for a solar collector. A large sphere of glass focuses light on photocells, as shown in Figure P35.22. The photocells are moved by electronics along the curved track to the right of the sphere. Your supervisor would like to build a prototype of a material with index of refraction n, but needs for you to calculate the position at which the Suns rays focus and, therefore, to find where to locale the curved track. Figure P35.22 23P An objects distance from a converging lens is 5.00 times the focal length. (a) Determine the location of the image. Express the answer as a fraction of the focal length. (b) Find the magnification of the image and indicate whether it is (c) upright or inverted and (d) real or virtual.25P 26P A converging lens has a focal length of 10.0 cm. Locate the object if a real image is located at a distance from the lens of (a) 20.0 cm and (b) 50.0 cm. What If? Redo the calculations if the images are virtual and located at a distance from the lens of (c) 20.0 cm and (d) 50.0 cm.28P 29P In Figure P35.30, a thin converging lens of focal length 14.0 cm forms an image of the square abed, which is he = hb = 10.0 cm high and lies between distances of pd = 20.0 cm and pa = 30.0 cm from the lens. Let a, b, c. and d represent the respective corners of the image. Let qa represent the image distance for points a and b, qd represent the image distance for points c and d, hb, represent the distance from point b to the axis, and hc represent the height of c. (a) Find qa, qd, hb, and hc. (b) Make a sketch of the image. (c) The area of the object is 100 cm2. By carrying out the following steps, you will evaluate the area of the image. Let q represent the image distance of any point between a and d, for which the object distance is p. Let h represent the distance from the axis to the point at the edge of the image between b and c at image distance q. Demonstrate that h=10.0q(114.01q) where h and q are in centimeters. (d) Explain why the geometric area of the image is given by qaqdhdq (e) Carry out the integration to find the area of the image. Figure P35.30 31P 32P Two rays traveling parallel to the principal axis strike a large plano-convex lens having a refractive index of 1.60 (Fig. P35.33). If the convex face is spherical, a ray near the edge does not pass through the focal point (spherical aberration occurs). Assume this face has a radius of curvature of R = 20.0 cm and the two rays are at distances h1 = 0.500 cm and h2 = 12.0 cm from the principal axis. Find the difference x in the positions where each crosses the principal axis. Figure P35.33 34P 35P 36P 37P 38P 39P The intensity I of the light reaching the CCD in a camera is proportional to the area of the lens. Because this area is proportional to the square of the diameter D, it follows that I is also proportional to D2. Because the area of the image is proportional to q2 and q = f (when p f, so p can be approximated as infinite), we conclude that the intensity is also proportional to 1/f2 and therefore that I D2/f2. The ratio f/D is called the f-number of a lens. Therefore, I 1/(f-number)2. The f-number is often given as a description of the lenss speed. The lower the f-number, the wider the aperture and the higher the rate at which energy from the light exposes the CCD; therefore, a lens with a low f-number is a fast lens. The conventional notation for an f-number is f/ followed by the actual number. For example, f/4 means an f-number of 4; it does not mean to divide f by 4! Suppose the lens of a digital camera has a focal length of 55 mm and a speed of f/1.8. The correct exposure time for this speed under certain conditions is known to be 1500s. (a) Determine the diameter of the lens. (b) Calculate the correct exposure time if the f-number is changed to f/4 under the same lighting conditions.41P 42P A simple model of the human eye ignores its lens entirely. Most of what the eye does to light happens at the outer surface of the transparent cornea. Assume that this surface has a radius of curvature of 6.00 mm and that the eyeball contains just one fluid with a refractive index of 1.40. Prove that a very distant object will be imaged on the retina, 21.0 mm behind the cornea. Describe the image.44AP 45AP The distance between an object and its upright image is d. If the magnification is M, what is the focal length of the lens being used to form the image?47AP Two converging lenses having focal lengths of f1 = 10.0 cm and f2 = 20.0 cm are placed a distance d = 50.0 cm apart as shown in Figure P35.48. The image due to light passing through both lenses is to be located between the lenses at the position x = 31.0 cm indicated. (a) At what value of p should the object be positioned to the left of the first lens? (b) What is the magnification of the final image? (c) Is the final image upright or inverted? (d) Is the final image real or virtual?Two lenses made of kinds of glass having different indices of refraction n1 and n2 are cemented together to form an optical doublet. Optical doublets are often used to correct chromatic aberrations in optical devices. The first lens of a certain doublet has index of refraction n1, one flat side, and one concave side with a radius of curvature of magnitude R. The second lens has index of refraction n2 and two convex sides with radii of curvature also of magnitude R. Show that the doublet can be modeled as a single thin lens with a focal length described by 1f=2n2n11R 50AP 51AP 52AP 53AP In many applications, it is necessary to expand or decrease the diameter of a beam of parallel rays of light, which can be accomplished by using a converging lens and a diverging lens in combination. Suppose you have a converging lens of focal length 21.0 cm and a diverging lens of focal length 12.0 cm. (a) How can you arrange these lenses to increase the diameter of a beam of parallel rays? (b) By what factor will the diameter increase?55AP A zoom lens system is a combination of lenses that produces a variable magnification of a fixed object as it maintains a fixed image position. The magnification is varied by moving one or more lenses along the axis. Multiple lenses are used in practice, but the effect of zooming in on an object can be demonstrated with a simple two-lens system. An object, two converging lenses, and a screen are mounted on an optical bench. Lens 1, which is to the right of the object, has a focal length of f1 = 5.00 cm, and lens 2, which is to the right of the first lens, has a focal length of f2 = 10.0 cm. The screen is to the right of lens 2. Initially, an object is situated at a distance of 7.50 cm to the left of lens 1, and the image formed on the screen has a magnification of +1.00. (a) Find the distance between the object and the screen. (b) Both lenses are now moved along their common axis while the object and the screen maintain fixed positions until the image formed on the screen has a magnification of +3.00. Find the displacement of each lens from its initial position in part (a). (c) Can the lenses be displaced in more than one way?57CP 58CP Which of the following causes the fringes in a two-slit interference pattern to move farther apart? (a) decreasing the wavelength of the light (b) decreasing the screen distance L (c) decreasing the slit spacing d (d) immersing the entire apparatus in water Using Figure 36.6 as a model, sketch the interference pattern from six slits. Figure 36.6 Multiple-slit interference patterns. As N, the number of slits, is increased, the primary maxima (the tallest peaks in each graph) become narrower but remain fixed in position and the number of secondary maxima increases.One microscope slide is placed on top of another with their left edges in contact and a human hair under the right edge of the upper slide. As a result, a wedge of air exists between the slides. An interference pattern results when monochromatic light is incident on the wedge. What is at the left edges of the slides? (a) a dark fringe (b) a bright fringe (c) impossible to determine Two slits are separated by 0.320 mm. A beam of 500-nm light strikes the slits, producing an interference pattern. Determine the number of maxima observed in the angular range 30.0 30.0.2P A laser beam is incident on two slits with a separation of 0.200 mm. and a screen is placed 5.00 m from the slits. An interference pattern appears on the screen. If the angle from the center fringe to the first bright fringe to the side is 0.181, what is the wavelength of the laser light?4P 5P Light with wavelength 442 nm passes through a double-slit system that has a slit separation d = 0.400 mm. Determine how far away a screen must be placed so that dark fringes appear directly opposite both slits, with only one bright fringe between them.7P A student holds a laser that emits light of wavelength . The laser beam passes through a pair of slits separated by a distance d, in a glass plate attached to the front of the laser. The beam then falls perpendicularly on a screen, creating an interference pattern on it. The student begins to walk directly toward the screen at speed v. The central maximum on the screen is stationary. Find the speed of the mth-order maxima on the screen, where m can be very large.Coherent light rays of wavelength strike a pair of slits separated by distance d at an angle 1 with respect to the normal to the plane containing the slits as shown in Figure P36.9. The rays leaving the slits make an angle 2 with respect to the normal, and an interference maximum is formed by those rays on a screen that is a great distance from the slits. Show that the angle 2 is given by 2=sin1(sin1md) where m is an integer. Figure P36.9 In Figure P36.10 (not to scale), let L = 1.20 m and d = 0.120 mm and assume the slit system is illuminated with monochromatic 500-nm light. Calculate the phase difference between the two wave fronts arriving at P when (a) = 0.500 and (b) y = 5.00 mm. (c) What is the value of for which the phase difference is 0.333 rad? (d) What is the value of for which the path difference is /4? Figure P36.10 11P 12P In the double-slit arrangement of Figure P36.13, d = 0.150 mm, L = 140 cm, = 643 nm. and y = 1.80 cm. (a) What is the path difference for the rays from the two slits arriving at P? (b) Express this path difference in terms of . (c) Does P correspond to a maximum, a minimum, or an intermediate condition? Give evidence for your answer. Figure P36.13 Monochromatic light of wavelength is incident on a pair of slits separated by 2.40 104 m and forms an interference pattern on a screen placed 1.80 m from the slits. The first-order bright fringe is at a position ybright = 4.52 mm measured from the center of the central maximum. From this information, we wish to predict where the fringe for n = 50 would be located. (a) Assuming the fringes are laid out linearly along the screen, find the position of the n = 50 fringe by multiplying the position of the n = 1 fringe by 50.0. (b) Find the tangent of the angle the first-order bright fringe makes with respect to the line extending from the point midway between the slits to the center of the central maximum. (c) Using the result of part (b) and Equation 36.2, calculate the wavelength of the light. (d) Compute the angle for the 50th-order bright fringe from Equation 36.2. (e) Find the position of the 50th-order bright fringe on the screen from Equation 36.5. (f) Comment on the agreement between the answers to parts (a) and (e).15P Show that the distribution of intensity in a double-slit pattern is given by Equation 36.9. Begin by assuming that the total magnitude of the electric field at point P on the screen in Figure 36.4 is the superposition of two waves, with electric field magnitudes E1=E0sintE2=E0sin(t+) The phase angle in in E2 is due to the extra path length traveled by the lower beam in Figure 36.4. Recall from Equation 33.27 that the intensity of light is proportional to the square of the amplitude of the electric field. In addition, the apparent intensity of the pattern is the time-averaged intensity of the electromagnetic wave. You will need to evaluate the integral of the square of the sine function over one period. Refer to Figure 32.5 for an easy way to perform this evaluation. You will also need the trigonometric identity sinA+sinB=2sin(A+B2)cos(AB2)17P Monochromatic coherent light of amplitude E0 and angular frequency passes through three parallel slits, each separated by a distance d from its neighbor. (a) Show that the time-averaged intensity as a function of the angle is I()=Imax[1+2cos(2dsin)]2 (b) Explain how this expression describes both the primary and the secondary maxima. (c) Determine the ratio of the intensities of the primary and secondary maxima. Hint: See Problem 16.19P 20P 21P 22P When a liquid is introduced into the air space between the lens and the plate in a Newtons-rings apparatus, the diameter of the tenth ring changes from 1.50 to 1.31 cm. Find the index of refraction of the liquid.24P 25P 26P 27P 28AP 29AP 30AP 31AP 32AP In a Youngs double-slit experiment using light of wavelength , a thin piece of Plexiglas having index of refraction n covers one of the slits. If the center point on the screen is a dark spot instead of a bright spot, what is the minimum thickness of the Plexiglas?34AP Figure P36.35 shows a radio-wave transmitter and a receiver separated by a distance d = 50.0 m and both a distance h = 35.0 m above the ground. The receiver can receive signals both directly from the transmitter and indirectly from signals that reflect from the ground. Assume the ground is level between the transmitter and receiver and a 180 phase shift occurs upon reflection. Determine the longest wavelengths that interfere (a) constructively and (b) destructively. Figure P36.35 Problems 35 and 36.36AP In a Newtons-rings experiment, a plano-convex glass (n = 1.52) lens having radius r = 5.00 cm is placed on a flat plate as shown in Figure P36.37. When light of wavelength = 650 nm is incident normally, 55 bright rings are observed, with the last one precisely on the edge of the lens. (a) What is the radius R of curvature of the convex surface of the lens? (b) What is the focal length of the lens? Figure P36.37 38AP A plano-concave lens having index of refraction 1.50 is placed on a flat glass plate as shown in Figure P36.39. Its curved surface, with radius of curvature 8.00 m, is on the bottom. The lens is illuminated from above with yellow sodium light of wavelength 589 nm, and a series of concentric bright and dark rings is observed by reflection. The interference pattern has a dark spot at the center that is surrounded by 50 dark rings, the largest of which is at the outer edge of the lens. (a) What is the thickness of the air layer at the center of the interference pattern? (b) Calculate the radius of the outermost dark ring. (c) Find the focal length of the lens. Figure P36.39 40AP Interference fringes are produced using Lloyds mirror and a source S of wavelength = 606 nm as shown in Figure P36.41. Fringes separated by y = 1.20 mm are formed on a screen a distance L = 2.00 m from the source. Find the vertical distance h of the source above the reflecting surface. Figure P36.41 A plano-convex lens has index of refraction n. The curved side of the lens has radius of curvature R and rests on a flat glass surface of the same index of refraction, with a film of index nflim between them, as shown in Figure P36.42. The lens is illuminated from above by light of wavelength . Show that the dark Newtons rings have radii given approximately by r=mRnfilm where r R and m is an integer. Figure P36.42 43AP 44AP 45AP 46CP 47CP 48CP 49CP 50CP Suppose the slit width in Figure 37.4 is made half as wide. Does the central bright fringe (a) become wider, (b) remain the same, or (c) become narrower? Figure 37.4 (a) Geometry for analyzing the Fraunhofer diffraction pattern of a single slit. (Drawing not to scale.) (b) Simulation of a single-slit Fraunhofer diffraction pattern.Cats eyes have pupils that can be modeled as vertical slits. At night, would cats be more successful in resolving (a) headlights on a distant car or (b) vertically separated lights on the mast of a distant boat?Suppose you are observing a binary star with a telescope and are having difficulty resolving the two stars. You decide to use a colored filter to maximize the resolution. (A filter of a given color transmits only that color of light.) What color filter should you choose? (a) blue (b) green (c) yellow (d) red Ultraviolet light of wavelength 350 nm is incident on a diffraction grating with slit spacing d and forms an interference pattern on a screen a distance L away. The angular positions bright of the interference maxima are large. The locations of the bright fringes are marked on the screen. Now red light of wavelength 700 nm is used with a diffraction grating to form another diffraction pattern on the screen. Will the bright fringes of this pattern be located at the marks on the screen if (a) the screen is moved to a distance 2L from the grating, (b) the screen is moved to a distance L/2 from the grating, (c) the grating is replaced with one of slit spacing 2d, (d) the grating is replaced with one of slit spacing d/2, or (e) nothing is changed?A polarizer for microwaves can be made as a grid of parallel metal wires approximately 1 cm apart. Is the electric field vector for microwaves transmitted through this polarizer (a) parallel or (b) perpendicular to the metal wires?37.6QQ 1P 2P 3P In Figure 37.7, show mathematically how many interference maxima are enclosed by the central diffraction maximum in the pattern. Notice that the diagram is generated by using 650-nm light to illuminate two 3.0-m slits separated by 18 m. Figure 37.7 The combined effects of two-slit and single-slit interference. This pattern is produced when 650-nm light waves pass through two 3.0-m slits that are 18 m apart.5P What If? Suppose light strikes a single slit of width a at an angle from the perpendicular direction as shown in Figure P37.6. Show that Equation 37.1, the condition for destructive interference, must be modified to read sindark=masinm=1,2,3,7P Coherent light of wavelength 501.5 nm is sent through two parallel slits in an opaque material. Each slit is 0.700 m wide. Their centers are 2.80 m apart. The light then falls on a semicylindrical screen, with its axis at the midline between the slits. We would like to describe the appearance of the pattern of light visible on the screen. (a) Find the direction for each two-slit interference maximum on the screen as an angle away from the bisector of the line joining the slits. (b) How many angles are there that represent two-slit interference maxima? (c) Find the direction for each single-slit interference minimum on the screen as an angle away from the bisector of the line joining the slits. (d) How many angles are there that represent single-slit interference minima? (e) How many of the angles in part (d) are identical to those in part (a)? (f) How many bright fringes are visible on the screen? (g) If the intensity of the central fringe is Imax, what is the intensity of the last fringe visible on the screen?9P 10P What is the approximate size of the smallest object on the Earth that astronauts can resolve by eye when they are orbiting 250 km above the Earth? Assume = 500 nm and a pupil diameter of 5.00 mm.12P 13P 14P Impressionist painter Georges Seurat created paintings with an enormous number of dots of pure pigment, each of which was approximately 2.00 mm in diameter. The idea was to have colors such as red and green next to each other to form a scintillating canvas, such as in his masterpiece, A Sunday Afternoon on the Island of La Grande Jatte (Fig. P37.15). Assume = 500 nm and a pupil diameter of 5.00 mm. Beyond what distance would a viewer be unable to discern individual dots on the canvas? Figure P37.15 16P 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. (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.18P A grating with 250 grooves/mm is used with an incandescent light source. Assume the visible spectrum to range in wavelength from 400 nm to 700 nm. In how many orders can one see (a) the entire visible spectrum and (b) the short-wavelength region of the visible spectrum?Show that whenever white light is passed through a diffraction grating of any spacing size, the violet end of the spectrum in the third order on a screen always overlaps the red end of the spectrum in the second order.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.22P You are working as a demonstration assistant for a physics professor. For an upcoming lecture on diffraction gratings, he wishes to perform a demonstration where he shines a laser pointer at normal incidence onto the recorded surface of a DVD that is laying flat on the demonstration table. (a) He asks you to determine how many additional maxima beyond the normal reflection (which will be blocked by his hand holding the laser pointer) will be projected onto the ceiling or walls of the room if he uses a laser pointer with a wavelength of 632.8 nm. (b) He also asks you if he can show more maxima by using a laser pointer of another visible color. The tracks of pits on a DVD are separated by 0.800 m.24P 25P 26P 27P 28P 29P 30P 31P 32P 33AP 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 P37.34 shows the pattern on the screen, with a centimeter ruler below it. (a) Did the light pass through one slit or two slits? Explain how you can determine the answer. (b) If one slit, find its width. If two slits, find the distance between their centers. Figure P37.34 35AP 36AP 37AP 38AP 39AP 40AP 41AP 42AP 43AP 44AP 45AP 46AP 47AP 48AP Two closely spaced wavelengths of light are incident on a diffraction grating. (a) Starting with Equation 37.7, show that the angular dispersion of the grating is given by dd=mdcos (b) A square grating 2.00 cm on each side containing 8 000 equally spaced slits is used to analyze the spectrum of mercury. Two closely spaced lines emitted by this element have wavelengths of 579.065 nm and 576.959 nm. What is the angular separation of these two wavelengths in the second-order spectrum?50CP 51CP In Figure P37.52, suppose the transmission axes of the left and right polarizing disks are perpendicular to each other. Also, let the center disk be rotated on the common axis with an angular speed . Show that if unpolarized light is incident on the left disk with an intensity Imax, the intensity of the beam emerging from the right disk is I=116Imax(1cos4t) This result means that the intensity of the emerging beam is modulated at a rate four times the rate of rotation of the center disk. Suggestion: Use the trigonometric identities cos2=12(1+cos2) and sin2=12(1cos2). Figure P37.52 53CP Which observer in Figure 38.1 sees the balls correct path? (a) the observer in the truck (b) the observer on the ground (c) both observers Figure 38.1 Two observers watch the path of a thrown ball and obtain different results.38.2QQ Suppose the observer O on the train in Figure 38.6 aims her flashlight at the far wall of the boxcar and turns it on and off, sending a pulse of light toward the far wall. Both O and O measure the time interval between when the pulse leaves the flashlight and when it hits the far wall. Which observer measures the proper time interval between these two events? (a) O (b) O (c) both observers (d) neither observer Figure 38.6 (a) A mirror is fixed to a moving vehicle, and a light pulse is sent out by observer O at rest in the vehicle. (b) Relative to a stationary observer O standing alongside the vehicle, the mirror and O move with a speed v and the light pulse follows a diagonal path. (c) The right triangle for calculating the relationship between t and tp.38.4QQ 38.5QQ You are observing a spacecraft moving away from you. You measure it to be shorter than when it was at rest on the ground next to you. You also see a clock through the spacecraft window, and you observe that the passage of time on the clock is measured to be slower than that of the watch on your wrist. Compared with when the spacecraft was on the ground, what do you measure if the spacecraft turns around and comes toward you at the same speed? (a) The spacecraft is measured to be longer, and the clock runs faster. (b) The spacecraft is measured to be longer, and the clock runs slower. (c) The spacecraft is measured to be shorter, and the clock runs faster. (d) The spacecraft is measured to be shorter, and the clock runs slower.You are driving on a freeway at a relativistic speed. (i) Straight ahead of you, a technician standing on the ground turns on a searchlight and a beam of light moves exactly vertically upward as seen by the technician. As you observe the beam of light, do you measure the magnitude of the vertical component of its velocity as (a) equal to c, (b) greater than c, or (c) less than c? (ii) If the technician aims the searchlight directly at you instead of upward, do you measure the magnitude of the horizontal component of its velocity as (a) equal to c, (b) greater than c, or (c) less than c?38.8QQ In a laboratory frame of reference, an observer notes that Newtons second law is valid. Assume forces and masses are measured to be the same in any reference frame for speeds small compared with the speed of light. (a) Show that Newtons second law is also valid for an observer moving at a constant speed, small compared with the speed of light, relative to the laboratory frame. (b) Show that Newtons second law is not valid in a reference frame moving past the laboratory frame with a constant acceleration.2P 3P 4P 5P An astronaut is traveling in a space vehicle moving at 0.500c relative to the Earth. The astronaut measures her pulse rate at 75.0 beats per minute. Signals generated by the astronauts pulse are radioed to the Earth when the vehicle is moving in a direction perpendicular to the line that connects the vehicle with an observer on the Earth. (a) What pulse rate does the Earth-based observer measure? (b) What If? What would be the pulse rate if the speed of the space vehicle were increased to 0.990c?7P You have been hired as an expert witness for an attorney who is representing a speeding driver. The driver of the car was given a ticket for running a red light at an intersection. According to the driver, who has taken some courses in physics, when he was looking at the red light as he approached the intersection, the Doppler shift made the light of wavelength 650 nm appear to be green light of wavelength 520 nm. Therefore, according to the driver, he should not be charged with running a red light because it appeared green to him. What advice do you give the attorney?9P 10P 11P A cube of steel has a volume of 1.00 cm3 and mass 8.00 g when at rest on the Earth. If this cube is now given a speed u = 0.900c, what is its density as measured by a stationary observer? Note that relativistic density is defined as Eg/c2V.Review. In 1963, astronaut Gordon Cooper orbited the Earth 22 times. The press stated that for each orbit, he aged two-millionths of a second less than he would have had he remained on the Earth. (a) Assuming Cooper was 160 km above the Earth in a circular orbit, determine the difference in elapsed time between someone on the Earth and the orbiting astronaut for the 22 orbits. You may use the approximation 11x=1+x2 for small x. (b) Did the press report accurate information? Explain.You have an assistantship with a math professor in a future world where space travel is common and spacecraft regularly achieve near-light speeds. A spacecraft has taken of recently to carry individuals to colonize an Earth-like planet around a nearby star. Your professor, who remains on Earth, is teaching the students on the spacecraft via the future version of distance learning. It is time for the students on the spacecraft to take a math exam. The professor wishes the students to have a time interval tp = 2.00 h to complete the exam, so just as the spacecraft passes Earth on its last trip around the Sun at its constant cruising speed of 0.960c, she sends a signal to the proctor to have the students begin the exam. Knowing of your experience in physics courses, the professor asks you to determine the time interval through which she should wait before sending a radio signal to the departing spacecraft to tell the proctor to have the students stop working on the exam.15P 16P A moving rod is observed to have a length of = 2.00 m and to be oriented at an angle of = 30.0 with respect to the direction of motion as shown in Figure P38.17. The rod has a speed of 0.995c. (a) What is the proper length of the rod? (b) What is the orientation angle in the proper frame? Figure P38.17 18P 19P You have been hired as an expert witness in the future by an attorney representing the driver of a spacecraft. The driver is accused of exceeding the galactic speed limit of 0.700c relative to the Earth while being chased by a galactic police spacecraft. The driver claims he is innocent, that his speed was well below that limit. You have been provided with the following data: the police spacecraft was traveling at 0.600c while chasing the driver and a technician on the police spacecraft measured the suspected spacecraft as traveling at 0.300c relative to the police spacecraft. What advice should you give the attorney?Figure P38.21 shows a jet of material (at the upper right) being ejected by galaxy M87 (at the lower left). Such jets are believed to be evidence of supermassive black holes at the center of a galaxy. Suppose two jets of material from the center of a galaxy are ejected in opposite directions. Both jets move at 0.750c relative to the galaxy center. Determine the speed of one jet relative to the other. Figure P 38.21 22P 23P 24P 25P 26P 27P (a) Find the kinetic energy of a 78.0-kg spacecraft launched out of the solar system with speed 106 km/s by using the classical equation K=12mu2. (b) What If? Calculate its kinetic energy using the relativistic equation. (c) Explain the result of comparing the answers of parts (a) and (b).29P 30P 31P 32P 33P 34P 35P 36P 37P 38P 39P An unstable particle with mass m = 3.34 1027 kg is initially at rest. The particle decays into two fragments that fly off along the x axis with velocity components u1 = 0.987c and u2 = 0.868c. From this information, we wish to determine the masses of fragments 1 and 2. (a) Is the initial system of the unstable particle, which becomes the system of the two fragments, isolated or nonisolated? (b) Based on your answer to part (a), what two analysis models are appropriate for this situation? (c) Find the values of for the two fragments after the decay. (d) Using one of the analysis models in part (b), find a relationship between the masses m1 and m2 of the fragments. (e) Using the second analysis model in part (b). find a second relationship between the masses m1 and m2. (f) Solve the relationships in parts (d) and (c) simultaneously for the masses m1 and m2.

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