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All Textbook Solutions for University Physics Volume 3
Check Your Understanding Table 1.1 shows that ethanol and fresh water have very similar indices of refraction. By what percentage do the speeds of light in these liquids differ?Check Your Understanding In Table 1.1, the solid with the next highest index of refraction after diamond is zircon. If the diamond in Example 1.3 were replaced with a piece of zircon, what would be the new angle of refraction?Check Your Understanding At the surface between air and water, light rays can go from air to water and from water to air. For which ray is there no possibility of total internal reflection?Check Your Understanding In the preceding example. how much distance inside the block of flint glass would the red and the violet rays have to progress before they are separated b 1.0 mm?Check Your Understanding in Example 1.6, we had n1n2. If n2were decreased such thatn1n2and the speed of light in medium 2 is faster than in medium 1, what would happen to the length of AA’ ? What would happen to the wave front A’B and the direction of the refracted ray?Check Your Understanding Although we did no specify the direction in Example 1.7, let’s say the polarizing filter was rotated clockwise by 71.6° to reduce the light intensity by 90.0%. What would be the intensity reduction if the polarizing filter were rotated counterclockwise by 71.6°’Check Your Understanding What happens at Brewster’s angle if the original incident light is already 100% vertically polarized?Under what conditions can light be modeled like a ray? Like a wave?Why is the index of refraction always greater than or equal to 1?Does the fact that the light flash from lightning reaches you before its sound prove that the speed of light is extremely large or simply that it is greater than the speed of sound? Discuss how you could use this effect to get an estimate of the speed of light.Speculate as to s1at physical process might be responsible for light traveling more slowly in a medium than in a vacuum.Using the law of reflection, explain how powder takes the shine off of a person’s nose. What is the name of the optical effect?Diffusion by reflection from a rough surface is described in this chapter. Light can also be diffused by refraction. Describe how this occurs in a specific situation, such as light interacting with crushed ice.Will 1igt change direction toward or away from the perpendicular when it goes from air to water? Water to glass? Glass to air?Exp1ain why an object in water always appears to be at a depth shallower than it actually is?Explain why a person’s legs appear very short when wading in a pool. Justify your explanation with a ray diagram showing the path of rays from the feet to the eye of an observer who is out of the water.Explain why an oar that is partially submerged in water appears bent.A ring with a colorless gemstone is dropped into water.The gemstone becomes invisible when submerged. Can it be a diamond? Explain.The most common type of mirage is an illusion that light from faraway objects is reflected by a pool of water that is not really there. Mirages are generally observed in deserts, 1en there is a hot layer of air near the ground. Given that the refractive index of air is lower for air at higher temperatures, explain how mirages can be formed.How can you use total internal reflection to estimate the index of refraction of a medium?Is it possible that total internal reflection plays a role in rainbows? Explain in terms of indices of refraction and angles, perhaps referring to that shown below. Some of us have seen the formation of a double rainbow; is it physically possible to observe a t - - rainbow? (credit “Chad”/Flickr)A high-quality diamond may be quite clear and colorless, transmitting all visible wavelengths with little absorption. Explain how it can sparkle with flashes of brilliant color when illuminated by white light.How do wave effects depend on the size of the object with which the wave interacts? For example, why does sound bend around the corner of a building sil1e 1igJt does not’Does Huygens’s principle apply to all types of waves?If diffraction is observed for some phenomenon, it is evidence that the phenomenon is a wave. Does the reverse hold true? That Is, if diffraction is not observed, does that mean the phenomenon is not a wave?Can a sound wave in air be polarized? Explain.No light passes through two perfect polarizing filters with perpendicular axes. However, if a third polarizing filter is placed between the orignal two, some light can pass. Why is this? Under what circumstances does most of the light pass?Explain what happens to the energy carried by light that it is dimmed by passing it through two crossed polarizing filters.When particles scattering light are much smaller than its wavelength, the amount of scattering is proportional to 1 . Does this mean there is more scattering for small than large ? How does this relate to the fact that the sky is blue?Using the information given in the preceding question, explain why sunsets are red.When light is reflected at Brewster’s angle from a smooth surface, it Is 100’ polarized parallel to the surface. Pail of the light sill be refracted into the surface. Describe how you would do an experiment to determine the polarization of the refracted light. What direction would you expect the polarization to have and would you expect it to be 1000%?If you lie on a beach looking at the water with yow head tipped slightly sideways, your polarized sunglasses do no work very well. Why not?What is the speed of light in water? In glycerine?What is the speed of light in air? In crown glass?Calculate the index of refraction for a medium in which the speed of light is 2.120108m/s , and identify the most likely substance based on Table 1.1.In what substance in Table 1.1 is the speed of light 2.290108 m/s?There was a major collision of an asteroid with the Moon in medieval times. It was described by monks at Canterbury Cathedral in England as a red glow on and around the Moon. How long after the asteroid hit the Moon, which is 3.84105 km away, would the light first arrive on Earth?Components of some computers communicate with each other through optical fibers having an index of retraction n=1.55 . What time in nanoseconds is required for a signal to travel 0.200 m through such a fiber?Compare the time it takes for light to travel 1000 m on the surface of Earth and in outer space.How far does light travel underwater during a time interval of 1.50106 s?Suppose a man stands in front of a mm-or as show below. His eyes are 1.65 m above the floor and the top of his head is 0.13 m higher. Find the height above the floor of the top and bottom of the smallest minor in which he can see both the top of his head and his feet. How is this distance related to the man’s height?Show that when light reflects from two mirrors that meet each other at a right angle, the outgoing ray is parallel to the incoming ray, as illustrated below.On the Moon’s surface, lunar astronauts placed a comet reflector, off which a laser beam Is periodically reflected. The distance to the Moon is calculated from the round-trip time. What percent correction Is needed to account for the delay in time due to the slowing of light in Earth’s atmosphere? Assume the distance to the Moon is precisely 3.84108 m and Earth’s atmosphere (which varies in density with altitude) is equivalent to a layer 30.0 km thick with a constant index of refraction n= l.000293.A flat mirror Is neither converging nor diverging. To prove this, consider two rays originating from the same point and diverging at an angle (I (see below). Show that after striking a plane mirror, the angle between their directions remains 0. 1.3 Refraction Unless otherwise specified, for problems 1 through 10, the indices of refraction of glass and water should be taken to be 1.50 and 1.333, respectively.A light beam in air has an angle of incidence of 35° at the surface of a glass plate. What are the angles of reflection and refraction?A light beam in air is incident on the surface of a pond, making an angle of 20° with respect to the surface. What are the angles of reflection and refraction?When a light ray crosses from water into glass, it emerges at an angle of 30with respect to the normal of the interface. What is its angle of incidence?A pencil flashlight submerged in water sends a light beam toward the surface at an angle of incidence of 30 .What is the angle of refraction in air?Light rays from the Sun make a 30° angle to the vertical when seen from below the surface of a body of water. At what angle above the horizon is the Sun?The path of a light beam in air goes from an angle of incidence of 35° to an angle of refraction of 22° when it enters a rectangular block of plastic. What is the index of refraction of the plastic?A scuba diver training in a pool looks at his instructor as shown below. What angle does the ray from the instructor’s face make with the perpendicular to the water at the point water the ray enters? The angle between the ray in the water and the perpendicular to the water is 25.0 .(a) Using information in the preceding problem, find the height of the instructor’s head above the water, noting that you will first have to calculate the angle of incidence. (b) Find the apparent depth of the diver’s head below water as seen by the instructor.Verify that the critical angle for light going from water to air is 48.6°, as discussed at the end of Example 1.4, regarding the critical angle for lig1it traveling in a polystyrene (a type of plastic) pipe surrounded by air.(a) At the end of Example 1.4, it was stated that the critical angle for light going from diamond to air is 24.4°. Verify this. (b) What is the critical angle for light going from zircon to air?An optical fiber uses flint glass clad with crown glass. What is the critical angle?At that minimum angle will you get total internal reflection of light traveling in water and reflected from ice?Suppose you are using total internal reflection to make an efficient corner reflector. If there is air outside and the incident angle is 45.0 , what must be the minimum index of refraction of the material from which the reflector is made?You can determine the index of refraction of a substance by determining its critical angle. (a) What is the index of refraction of a substance that has a critical angle of 68.4° when submerged in water? What Is the substance, based on Table 1.1? (b) What would the critical angle be for this substance in air?A ray of light, emitted beneath the surface of an unknown liquid with air above it, undergoes total internal reflection as shown below. What is the index of refraction for the liquid and its likely identification?Light rays fall normally on the vertical surface of the glass prism (n=1.50)shown below. (a) What is the largest value for such that the ray is totally reflected at the slanted face? (b) Repeat the calculation of part (a) if the prism is immersed in water.(a) What is the ratio of the speed of red light to violet light in diamond, based on Table 1.2? (b) What is this ratio in polystyrene? (c) Which is more dispersive?A beam of white light goes from air into water at an incident angle of 75.0. At what angles are the red (660 nm) and violet (410 nm) parts of the light refracted?By how much do the critical angles for red (660 nm) and violet (410 nm) light differ in a diamond surrounded by air?(a) A narrow beam of light containing yellow (580 nm) and green (550 nm) wavelengths goes from polystyrene to air, striking the surface at a 30.0incident angle. What is the angle between the colors when they emerge? (b) How far would they have to travel to be separated by 1.00 mm?A parallel beam of light containing orange (610 nm) and violet (4.10wm) wavelengths goes from fused quartz to water, striking the surface between them at a 60.00 incident angle. What is the angle between the two colors in water?A ray of 610-nm light goes from air into fused quartz at an incident angle of 55.0. At what incident angle must 470 nm light enter flint glass to have the same angle of refraction?A narrow beam of light containing red (660 nm) and blue (470 nm) wavelengths travels from air through a 1.00-cm-thick flat piece of crown glass and back to air again. The beam strikes at a 30.0incident angle. (a) At what angles do the two colors emerge? (b) By what distance are the red and blue separated when they emerge?A narrow beam of white light enters a prism made of crown glass at a 45.0incident angle, as shown below. At what angles, Rand do v red (660 nm) and violet (410 nm) components of the light emerge from the prism?What angle is needed between the direction of polarized light and the axis of a polarizing filter to cut its intensity in half?The angle been the axes of two polarizing filters is 45.00. By how much does the second filter reduce the intensity of the light coming through the first?Two polarizing sheets P1 and P2 are placed together with their transmission axes oriented at an angle to each other. What is when only 25% of the maximum transmitted light intensity passes through them?Suppose that in the preceding problem the light incident on P1 is unpolarized. At the determined value of , shat fraction of the incident light passes through the combination?If you have completely polarized light of intensity 150 W/m2 , what will its intensity be after passing through a polarizing filter with its axis at an 89.0 angle to the light’s polarization direction?What angle would the axis of a polarizing filter need to make with the direction of polarized light of intensity 1.00kW/m2 to reduce the intensity to 10.0W/m2 ?At the end of Example 17, it was stated that the intensity of polarized light is reduced to 90.0% of its original value by passing through a polarizing filter with its axis at an angle of 18.4° to the direction of polarization. Verify this statement.Show that if you have three polarizing filters, with the second at an angle of 45.0to the first and at third at an angle of 90.0to the first, the intensity of light passed by the first will be reduced to 25.0% of its value. (This is in contrast to having only the first and third, which reduces the intensity to zero, so that placing the second between them increases the intensity of the transmitted light.)Three polarizing sheets are placed together such that the transmission axis of the second sheet is oriented at 25.0° to the axis of the first, whereas the transmission axis of the third sheet is oriented at 40.0° (in the same sense) to the axis of the first. What fraction of an intensity of an incident unpolarized beam is transmitted by the combination?In order to rotate the polarization axis of a beam of linearly polarized light by 90.00, a student places sheets P1 and P, with their transmission axes at 45.0° 90.0°, respectively, to the beam’s axis of polarization. (a) What fraction of the incident light passes through P1 and(b) through the combination? (C) Repeat your calculations for part (b) for transmission-axis angles of 30.00 and 90.0 , respectively.It is found that when light traveling in water falls on a plastic block, Brewster’s angle is 50.0°. What is the refractive index of the plastic?At what angle ill lig1t reflected from diamond be completely polarized?What is Brewster’s angle for light traveling in water that is reflected from crown glass?A scuba diver sees light reflected from the water’s surface. At what angle relative to the water’s surface will this light be completely polarized?From his measurements, Roemer estimated that it took 22 mm for light to travel a distance equal to the diameter of Earth’s orbit around the Sun. (a) Use this estimate along with the known diameter of Earth’s orbit to obtain a rough value of the speed of light. (b) Light actually takes 16.5 mm to travel this distance. Use this time to calculate the speed of light.Cornu performed Fizeau’s measurement of the speed of light using a wheel of diameter 4.00 cm that contained 180 teeth. The distance from the wheel to the mirror was 22.9 km. Assuming he measured the speed of light accurately, what was the angular velocity of the wheel?Suppose you have an unknown clear substance immersed in water, and you wish to identify it by finding its index of refraction. You arrange to have a beam of light enter it at an angle of 45.0°, and you observe the angle of refraction to be 40.3°. What is the index of refraction of the substance and its likely identity?Shown below is a ray of light going from air through crown glass into water, such as going into a fish tank. Calculate the amount the ray is displaced by the glass (x), given that the incident angle is 40.0° and the glass is 1 .00 cm thick.Considering the previous problem, show that 3is the same as it would be if the second medium were not present.At what angle is light inside crown glass completely polarized when reflected from water, as in a fish tank?Light reflected at 55.6° from a window is completely polarized. What is the window’s index of refraction and the likely substance of which it is made?(a) Light reflected at 62.5° from a gemstone in a ring is completely polarized. Can the gem be a diamond? (b) At what angle would the light be completely polarized if the gem was in water?If bis Brewster’s angle for light reflected from the top of an interface between two substances, and bis Brewster’s angle for light reflected from below, prove that b+b=90.0Unreasonable results Suppose light travels from water to another substance, with an angle of incidence of 10.0and an angle of refraction of 14.9 . (a) What is the index of refraction of the other substance? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?Unreasonable results Light traveling from water to a gemstone strikes the surface at an angle of 80.00 and has an angle of refraction of 15.2°. (a) What is the speed of light in the gemstone? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?If a polarizing filter reduces the intensity of polarized light to 50.0% of its original value, by how much are the electric and magnetic fields reduced?Suppose you put on two pairs of polarizing sunglasses with their axes at an angle of 15.0 . . How much longer will it take the light to deposit a given amount of energy in your eye compared with a single pair of sunglasses? Assume the lenses are clear except for their polarizing characteristics.(a) On a day when the intensity of sunlight is 1.00kW/m2 , a circular lens 0.200 m in diameter focuses light onto water in a black beaker. Two polarizing sheets of plastic are placed in front of the lens with their axes at an angle of 20.0°. Assuming the sunlight is unpolarized and the polarizers are 100% efficient, what is the initial rate of heating of the water in °C/s, assuming it is 80.0% absorbed? The aluminum beaker has a mass of 30.0 grams and contains 250 grams of water. (b) Do the polarizing filters get hot? Explain.Light shows staged with lasers use moving mirrors to swing beams and create colorful effects. Show that a light ray reflected from a mirror changes direction by 2 when the mirror is rotated by an angle .Consider sunlight entering Earth’s atmosphere at sunrise and sunset—that is, at 90.0incident angle. Taking the boundary between nearly empty space and the atmosphere to be sudden, calculate the angle of refraction for sunlight. This lengthens the time the Sun appears to be above the horizon, both at sunrise and sunset. Now construct a problem in which you determine the angle of refraction for different models of the atmosphere, such as various layers of varying density. Your instructor may wish to guide you on the level of complexity to consider and on how the index of refraction varies with air density.A light ray entering an optical fiber surrounded by air is first refracted and then reflected as shown below. Show that if the fiber is made from crown glass, any incident ray will be totally internally reflected.A light ray falls on the left face of a prism (see below) at the angle of incidence for which the emerging beam has an angle of refraction at the right face. Show that the index of refraction n of the glass prism is given by n=sin12(+)sin12 where is the vertex angle of the prism and is the angle through which the beam has been deviated. If = 37.0° and the base angles of the prism are each 50.00, what is n?If the apex angle in the previous problem is 20.0° and n = 1.50, what is the value of a?The light incident on polarizing sheet P1is linearly polarized at an angle of 30.00 with respect to the transmission axis of P1 Sheet P2is placed so that its axis is parallel to the polarization axis of the incident light, that is, also at 30.0° with respect to P1 . (a) What fraction of the incident light passes through P1 ? (b) What fraction of the incident light is passed by the combination? (c) By rotating P2 , a maximum in transmitted intensity is obtained. What is the ratio of this maximum intensity to the intensity of transmitted light when P2 is at 30.0° with respect to P1 ?Prove that if I is the intensity of light transmitted by two polarizing filters with axes at an angle and I’ is the intensity when the axes are at an angle 90.0, then I+I=I0, the original intensity. (Hint: Use the trigonometric identities cos 90.0=sinand cos2sin2=1 .)What are the differences between real and virtual images? How can you tell (by looking) whether an image formed by a single lens or mirror is real or virtual?Can you see a virtual image? Explain your response.Can you photograph a virtual image?Can you project a virtual image onto a screen?Is it necessary to project a real image onto a screen to see it?Devise an arrangement of mirrors allowing you to see the back of your head. What is the minimum number of mirrors needed for this task?If you wish to see your entire body in a flat mirror (from head to toe), how tall should the mirror be? Does its size depend upon your distance away from the mirror? Provide a sketch.At what distance is an image always located: at do,di, or f?Under what circumstances will an image be located at the focal point of a spherical lens or mirror?What is meant by a negative magnification? What is meant by a magnification whose absolute value is less than one?Can an image be larger than the object even though its magnification is negative? Explain.Derive the formula for the apparent depth of a fish in a fish tank using Snell’s law.Use a ruler and a protractor to find the image by refraction in the following cases. Assume an air-glass interface. Use a refractive index of 1 for air and of 1.5 for glass. (Hint: Use Snell’s law at the interface.) (a) A point object located on the axis of a concave interface located at a point within the focal length from the vertex. (b) A point object located on the axis of a concave interface located at a point farther than the focal length from the vertex. (c) A point object located on the axis of a convex interface located at a point within the focal length from the vertex. (d) A point object located on the axis of a convex interface located at a point farther than the focal length from the vertex. (e) Repeat (a)—(d) for a point object off the axis.You can argue that a that piece of glass, such as in a window, is like a lens with an infinite focal length. If so, where does it form an image? That is, how are diand dorelated?When you focus a camera, you adjust the distance of the lens from the film. If the camera lens acts like a thin lens, why can it not be a fixed distance from the film for both near and distant objects?A thin lens has two focal points, one on either side of the lens at equal distances from its center, and should behave the same for light entering from either side. Look backward and forward through a pair of eyeglasses and comment on whether they are thin lenses.Will the focal length of a lens change when it is submerged in water? Explain.If the lens of a person’s eye is removed because of cataracts (as has been done since ancient times), why would you expect an eyeglass lens of about 16 D to be prescribed?When laser light is shone into a relaxed normal-vision eye to repair a tear by spot-welding the retina to the back of the eye, the rays entering the eye must be parallel. Why?Why is your vision so blurry when you open your eyes while swimming under water? How does a face mask enable clear vision?It has become common to replace the cataract-clouded lens of the eye with an internal lens. This intraocular lens can be chosen so that the person has perfect distant vision. Will the person be able to read without glasses? If the person was nearsighted, is the power of the intraocular lens greater or less than the removed lens?If the cornea is to be reshaped (this can be done surgically or with contact lenses) to correct myopia, should its curvature be made greater or smaller? Explain.Geometric optics describes the interaction of light with macroscopic objects. Why, then, is it correct to use geometric optics to analyze a microscope’s image?The image produced by the microscope in Figure 2.38 cannot be projected. Could extra lenses or mirrors project it? Explain.If you want your microscope or telescope to project a real image onto a screen, how would you change the placement of the eyepiece relative to the objective?Consider a pair of flat mirrors that are positioned so that they form an angle of 120 . An object is placed on the bisector between the minors. Construct a ray diagram as in Figure 2.4 to show how many images are formed. Figure 2.4 Two minors can produce multiple images. (a) Three images of a plastic lead are visible in the two minors at a right angle. (b) A single object reflecting from two minors at a right angle can produce three images, as shown by the greet, purple, and red images.Consider a pair of flat mirrors that are positioned so that they form an angle of 60 .. An object is placed on the bisector between the mirrors. Construct a ray diagram as in Figure 2.4 to show how many images ale formed. Figure 2.4 Two minors can produce multiple images. (a) Three images of a plastic lead are visible in the two minors at a right angle. (b) A single object reflecting from two minors at a right angle can produce three images, as shown by the greet, purple, and red images.By using more than one flat mirror, construct a ray diagram showing how to create an inverted image.The following figure shows a light bulb between two spherical mirrors. One mirror produces a beam of light with parallel rays; the other keeps light from escaping without being put into the beam. Where is the filament of the light in relation to the focal point or radius of curvature of each mirror?Why are diverging mirrors often used for rearview mirrors in vehicles? What is the main disadvantage of using such a mirror compared with a flat one?Some telephoto cameras use a mirror rather than a ens. What radius of curvature mirror is needed to replace a 800 mm-focal length telephoto lens?Calculate the focal length of a mirror formed by the shiny back of a spoon that has a 3.00 cm radius of curvature.Electric room heaters use a concave mirror to reflect infrared (IR) radiation from hot coils. Note that IR radiation follows the same law of reflection as visible light. Given that the mirror has a radius of curvature of 50.0 cm and produces an image of the coils 3.00 m away from the mirror, where are the coils?Find the magnification of the heater element in the previous problem. Note that its large magnitude helps spread out the reflected energy.What is the focal length of a makeup mirror that produces a magnification of 1.50 when a person’s face is 12.0 cm away? Explicitly show how you follow the steps in the Problem-Solving Strategy: Spherical Mirrors.A shopper standing 3.00 m from a convex security mirror sees his image with a magnification of 0.250. (a) Were is his image? (b) What is the focal length of the minor? (c) What is its radius of curvature?An object 1.50 cm high is held 3.00 cm from a person’s cornea, and its reflected image is measured to be 0.167 cm high. (a) What is the magnification? (b) Where is the image? (c) Find the radius of curvature of the convex mirror formed by the cornea. (Note that this technique is used by optometrists to measure the curvature of the cornea for contact lens fitting. The instrument used is called a keratometer, or curve measurer.)Ray tracing for a flat mirror shows that the image is located a distance behind the mirror equal to the distance of the object from the mirror. This is stated as di=do, since this is a negative image distance (it is a virtual image). What is the focal length of a flat mirror?Show that, for a flat mirror, hi=ho, given that the image is the same distance behind the mirror as the distance of the object from the mirror.Use the law of reflection to prove that the focal length of a mirror is half its radius of curvature. That is, prove that f = R/2. Note this is true for a spherical mirror only if its diameter is small compared with its radius of curvature.Referring to the electric room heater considered in problem 5, calculate the intensity of IR radiation in W/m2 projected by the concave mirror on a person 3.00 m away. Assume that the heating element radiates 1500 W and has an area of 100 cm2 , and that half of the radiated power is reflected and focused by the mirror.Two mirrors are inclined at an angle of 60 ° and an object is placed at a point that is equidistant from the two mirrors. Use a protractor to draw rays accurately and locate all images. You may have to draw several figures so that that rays for different images do not clutter your drawing.Two parallel mirrors are facing each other and are separated by a distance of 3 cm. A point object is placed between the mirrors 1 cm from one of the mirrors. Find the coordinates of all the images.An object is located in air 30 cm from the vertex of a concave surface made of glass with a radius of curvature 10 cm. Where does the image by refraction form and what is its magnification? Use nair=1and nglass=1.5. .An object is located in air 30 cm from the vertex of a convex surface made of glass with a radius of curvature 80 cm. Where does the image by refraction form and what is its magnification?An object is located in water 15 cm from the vertex of a concave surface made of glass with a radius of curvature 10 cm. Where does the image by refraction form and what is its magnification? Use nwater=4/3and nglass=1.5.An object is located in water 30 cm from the vertex of a convex surface made of Plexiglas with a radius of curvature of 80 cm. Where does the image form by refraction and what is its magnification? nwater=4/3 and nPlexiglas=1.65.An object is located in air 5 cm from the vertex of a concave surface made of glass with a radius of curvature 20 cm. Where does the image form by refraction and what is its magnification? Use nair=1 and nglass=1.5 .Derive the spherical interface equation for refraction at a concave surface. (Hint: Follow the derivation in the text for the convex surface.)How far from the lens must the film in a camera be, if the lens has a 35.0-mm focal length and is being used to photograph a flower 75.0 cm away? Explicitly show how you follow the steps in the Problem-solving strategy: Lenses.A certain slide projector has a 100 mm-focal length lens. (a) How far away is the screen if a slide is placed 103 mm from the lens and produces a sharp image? (b) If the slide is 24.0 by 36.0 mm, what are the dimensions of the image? Explicitly show how you follow the steps in the Problem-solving strategy: Lenses.A doctor examines a mole with a 15.0-cm focal length magnifying glass held 13.5 cm from the mole. (a) Where is the image? (b) What is its magnification? (c) How big is the image of a 5.00 mm diameter mole?A camera with a 50.0-mm focal length lens is being used to photograph a person standing 3.00 m away. (a) How far from the lens must the film be? (b) If the film is 36.0 mm high, what fraction of a 1.75-m-tall person will fit on it? (c) Discuss how reasonable this seems, based on your experience in taking or posing for photographs.A camera lens used for taking close-up photographs has a focal length of 22.0 mm. The farthest it can be placed from the film is 33.0 mm. (a) What is the closest object that can be photographed? (b) What is the magnification of this closest object?Suppose your 50.0 mm-focal length camera lens is 51.0 mm away from the film in the camera. (a) How far away is an object that is in focus? (b) What is the height of the object if its image is 2.00 cm high?What is the focal length of a magnifying glass that produces a magnification of 3.00 when held 5.00 cm from an object, such as a rare coin?The magnification of a book held 7.50 cm from a 10.0 cm-focal length lens is 3.00. (a) Find the magnification for the book when it is held 8.50 cm from the magnifier. (b) Repeat for the book held 9.50 cm from the magnifier. (c) Comment on how magnification changes as the object distance increases as in these two calculations.Suppose a 200 mm-focal length telephoto lens is being used to photograph mountains 10.0 km away. (a) Where is the image? (b) What is the height of the image of a 1000 m high cliff on one of the mountains?A camera with a 100 mm-focal length lens is used to photograph the sun. What is the height of the image of the sun on the film, given the sun is 1.40106 km in diameter and is 1.50108km away?Use the thin—lens equation to show that the magnification for a thin lens is determined by its focal length and the object distance and is given by m=f/(fd0).An object of height 3.0 cm is placed 5.0 cm in front of a converging lens of focal length 20 cm and observed from the other side. Where and how large is the image?An object of height 3.0 cm is placed at 5.0 cm in front of a diverging lens of focal length 20 cm and observed from the other side. Where and how large is the image?Au object of height 3.0 cm is placed at 25 cm in front of a diverging lens of focal length 20 cm. Behind the diverging lens, there is a converging lens of focal length 20 cm. The distance between the lenses is 5.0 cm. Fluid the location and size of the final image.Two convex lenses of focal lengths 20 cm and 10 cm are placed 30 cm apart, with the lens with the longer focal length on the right. An object of height 2.0 cm is placed midway between them and observed through each lens from the left and from the right. Describe what you will see, such as where the image(s) will appear, whether they will be upright or inverted and their magnifications.What is the power of the eye when viewing an object 50.0 cm away?Calculate the power of the eye when viewing an object 3.00 m away.The print in many books averages 3.50 mm in height. How high is the image of the print on the retina when the book is held 30.0 cm from the eye?Suppose a certain person’s visual acuity is such that he can see objects clearly that form an image 4.00mhigh on his retina. What is the maximum distance at which he can read the 75.0-cm-high letters on the side of an airplane?People who do very detailed work close up, such as jewelers, often can see objects clearly at much closer distance than the normal 25 cm. (a) What is the power of the eyes of a woman who can see an object clearly at a distance of only 8.00 cm? (b) What is the image size of a 1.00-mm object, such as lettering inside a ring, held at this distance? (c) What would the size of the image be if the object were held at the normal 25.0 cm distance?What is the far point of a person whose eyes have a relaxed power of 50.5 D?What is the near point of a person whose eyes have an accommodated power of 53.5 D?(a) A laser reshaping the cornea of a myopic patient reduces the power of his eye by 9.00 D, with a ±5.0 % uncertainty in the final correction. What is the range of diopters for eyeglass lenses that this person might need after this procedure? (b) Was the person nearsighted or farsighted before the procedure? How do you know?The power for normal close vision is 54.0 D. In a vision-correction procedure, the power of a patient’s eye is increased by 3.00 D. Assuming that this produces normal close vision, what was the patient’s near point before the procedure?For normal distant vision, the eye has a power of 50.0 D. What was the previous far point of a patient who had laser vision correction that reduced the power of her eye by 7.00 D, producing normal distant vision?The power for normal distant vision is 50.0 D. A severely myopic patient has a far point of 5.00 cm. By how many diopters should the power of his eye be reduced in laser vision correction to obtain normal distant vision for him?A student’s eyes, while reading the blackboard, have a power of 51.0 D. How far is the board from his eyes?The power of a physician’s eyes is 53.0 D while examining a patient. How far from her eyes is the object that is being examined?The normal power for distant vision is 50.0 D. A young woman with normal distant vision has a 10.0% ability to accommodate (that is, increase) the power of her eyes. What is the closest object she can see clearly?The far point of a myopic administrator is 50.0 cm. (a) What is the relaxed power of his eyes? (b) If he has the normal 8.00% ability to accommodate, what is the closest object lie can see clearly?A very myopic man has a far point of 20.0 cm. What power contact lens (when on the eye) will correct his distant vision?Repeat the previous problem for eyeglasses held 1.50 cm from the eyes.A myopic person sees that her contact lens prescription is —4.00 D. What is her far point?Repeat the previous problem for glasses that are 1.75 cm from the eyes.The contact lens prescription for a mildly farsighted person is 0.750 D, and the person has a near point of 29.0 cm. What is the power of the tear layer between the cornea and the lens if the correction is ideal, taking the tear layer into account?If the image formed on the retina subtends an angle of 30and the object subtends an angle of 5, what is the magnification of the image?What is the magnification of a magnifying lens with a focal length of 10 cm if it is held 3.0 cm from the eye and the object is 12 cm from the eye?How far should you hold a 2.1 cm-focal length magnifying glass from an object to obtain a magnification of 10 x ? Assume you place your eye 5.0 cm from the magnifying glass.You hold a 5.0 cm-focal length magnifying glass as close as possible to your eye. If you have a normal near point, what is the magnification?You view a mountain with a magnifying glass of focal length f = 10 cm . What is the magnification?You view an object by holding a 2.5 cm-focal length magnifying glass 10 cm away from it. How far from your eye should you hold the magnifying glass to obtain a magnification of 10 ?A magnifying glass forms an image 10 cm on the opposite side of the lens from the object, which is 10 cm away. What is the magnification of this lens for a person with a normal near point if their eye 12 cm from the object?An object viewed with the naked eye subtends a 2° angle. If you view the object through a 10 x magnifying glass, what angle is subtended by the image formed on your retina?For a normal, relaxed eye, a magnifying glass produces an angular magnification of 4.0. What is the largest magnification possible with this magnifying glass?What range of magnification is possible with a 7.0 cm-focal length converging lens?A magnifying glass produces an angular magnification of 4.5 when used by a young person with a near point of 18 cm. What is the maximum angular magnification obtained by an older person with a near point of 45 cm?A microscope with an overall magnification of 800 has an objective that magnifies by 200. (a) What is the angular magnification of the eyepiece? (b) If there are two other objectives that can be used, having magnifications of 100 and 400, what other total magnifications are possible?(a) What magnification is produced by a 0.150 cm-focal length microscope objective that is 0.155 cm from the object being viewed? (b) What is the overall magnification if an 8 x eyepiece (one that produces an angular magnification of 8.00) is used?Where does an object need to be placed relative to a microscope for its 0.50 cm-focal length objective to produce a magnification of —400?An amoeba is 0.305 cm away from the 0.300 cm- focal length objective lens of a microscope. (a) Where is the image formed by the objective lens? (b) What is this image’s magnification? (C) An eyepiece with a 2.00-cm focal length is placed 20.0 cm from the objective. Where is the final image? (d) What angular magnification is produced by the eyepiece? (e) What is the overall magnification? (See Figure 2.39.)Unreasonable Results Your friends show you an image through a microscope. They tell you that the microscope has an objective with a 0.500-cm focal length and an eyepiece with a 5.00-cm focal length. The resulting overall magnification is 250,000. Are these viable values for a microscope? Unless otherwise stated, the lens-to-retina distance is 2.00 cm.What is the angular magnification of a telescope that has a 100 cm-focal length objective and a 2.50 cm-focal length eyepiece?Find the distance between the objective and eyepiece lenses in the telescope in the above problem needed to produce a final image very far from the observer, where vision is most relaxed. Note that a telescope is normally used to view very distant objects.A large reflecting telescope has an objective mirror with a 10.0-rn radius of curvature. What angular magnification does it produce when a 3.00 m-focal length eyepiece is used?A small telescope has a concave mirror with a 2.00-rn radius of curvature for its objective. Its eyepiece is a 4.00 cm-focal length lens. (a) What is the telescope’s angular magnification? (b) What angle is subtended by a 25,000 km-diameter sunspot? (c) What is the angle of its telescopic image?A 7.5 binocular produces an angular magnification of —7.50, acting like a telescope. (Mirrors are used to make the image upright.) If the binoculars have objective lenses with a 75.0-cm focal length, what is the focal length of the eyepiece lenses?Construct Your Own Problem Consider a telescope of the type used by Galileo, having a convex objective and a concave eyepiece as illustrated in part (a) of Figure 2.40. Construct a problem in which you calculate the location and size of the image produced. Among the things to be considered are the focal lengths of the lenses and their relative placements as well as the size and location of the object. Verify that the angular magnification is greater than one. That is, the angle subtended at the eye by the image is greater than the angle subtended by the object.Trace rays to find which way the given ray will emerge after refraction through the thin lens in the following figure. Assume thin-lens approximation. (Hint: Pick a point P on the given ray in each case. Treat that point as an object. Now, find its image Q. Use the rule: All rays on the other side of the lens will either go through Q or appear to be coming from Q.)Copy and draw rays to find the final image in the following diagram. (Hint: Find the intermediate image through lens alone. Use the intermediate image as the object for the mirror and work with the mirror alone to find the final image.)A concave mirror of radius of curvature 10 cm is placed 30 cm from a thin convex lens of focal length 15 cm. Find the location and magnification of a small bulb sitting 50 cm from the lens by using the algebraic method.An object of height 3 cm is placed at 25 cm in front of a converging lens of focal length 20 cm. Behind the lens there is a concave mirror of focal length 20 cm. The distance between the lens and the mirror is 5 cm. Find the location, orientation and size of the final image.An object of height 3 cm is placed at a distance of 25 cm in front of a converging lens of focal length 20 cm, to be referred to as the first lens. Behind the lens there is another converging lens of focal length 20 cm placed 10 cm from the first lens. There is a concave mirror of focal length 15 cm placed 50 cm from the second lens. Find the location, orientation, and size of the final image.An object of height 2 cm is placed at 50 cm in front of a diverging lens of focal length 40 cm. Behind the lens, there is a convex mirror of focal length 15 cm placed 30 cm from the converging lens. Find the location, orientation, and size of the final image.Two concave mirrors are placed facing each other. One of them has a small hole in the middle. A penny is placed on the bottom mirror (see the following figure). When you look from the side, a real image of the penny is observed above the hole. Explain how that could happen.A lamp of height S cm is placed 40 cm in front of a converging lens of focal length 20 cm. There is a plane mirror 15 cm behind the lens. Where would you find the image when you look in the mirror?Parallel rays from a faraway source strike a converging lens of focal length 20 cm at an angle of 15 degrees with the horizontal direction. Find the vertical position of the real image observed on a screen in the focal plane.Parallel rays from a faraway source strike a diverging lens of focal length 20 cm at an angle of 10 degrees with the horizontal direction. As you look through the lens, where in the vertical plane the image would appear?A light bulb is placed 10 cm from a plane mirror, which faces a convex mirror of radius of curvature 8 cm. The plane mirror is located at a distance of 30 cm from the vertex of the convex mirror. Find the location of two images in the convex mirror. Are there other images? If so, where are they located?A point source of light is 50 cm in front of a converging lens of focal length 30 cm. A concave mirror with a focal length of 20 cm is placed 25 cm behind the lens. Where does the final image form, and what are its orientation and magnification?Copy and trace to find how a horizontal ray from S comes out after the lens. Use nglass=1.5for the prism material.Copy and trace how a horizontal ray from S comes out after the lens. Use n=1.55 for the glass.Copy and draw rays to figure out the final image.By ray tracing or by calculation, find the place inside the glass where rays from S converge as a result of refraction through the lens and the convex air-glass interface. Use a ruler to estimate the radius of curvature.A diverging lens has a focal length of 20 cm. What is the power of the lens in diopters?Two lenses of focal lengths of f1and f2are glued together with transparent material of negligible thickness. Show that the total power of the two lenses simply add.What will be the angular magnification of a convex lens with the focal length 2.5 cm?What will be the formula for the angular magnification of a convex lens of focal length f if the eye is very close to the lens and the near point is located a distance D from the eye?Use a ruler and a protractor to draw rays to find images in the following cases. (a) A point object located on the axis of a concave minor located at a point within the focal length from the vertex. (b) A point object located on the axis of a concave mirror located at a point farther than the focal length from the vertex. (c) A point object located on the axis of a convex mirror located at a point within the focal length from the vertex. (d) A point object located on the axis of a convex mirror located at a point farther than the focal length from the vertex. (e) Repeat (a)—(d) for a point object off the axis.Where should a 3 cm tall object be placed in front of a concave minor of radius 20 cm so that its image is real and 2 cm tall?A 3 cm tall object is placed 5 cm in front of a convex mirror of radius of curvature 20 cm. Where is the image formed? How tall is the image? What is the orientation of the image?You are looking for a mirror so that you can see a four- fold magnified virtual image of an object when the object is placed 5 cm from the vertex of the mirror. What kind of mirror you will need? What should be the radius of curvature of the mirror?Derive the following equation for a convex mirror: 1VO1VI=1VF where VO is the distance to the object O from vertex V, VI the distance to the image I from V, and VF is the distance to the focal point F from V. (Hint: use two sets of similar triangles.)(a) Draw rays to form the image of a vertical object on the optical axis and farther than the focal point from a converging lens. (b) Use plane geometry in your figure and prove that the magnification m is given by m=hiho=didoUse another ray-tracing diagram for the same situation as given in the previous problem to derive the thin-lens equation, 1d0+1di=1f.You photograph a 2.0-m-tall person with a camera that has a 5.0 cm-focal length lens. The image on the film must be no mote than 2.0 cm high. (a) What is the closest distance the person can stand to the lens? (b) For this distance, what should be the distance from the lens to the film?Find the focal length of a thin plano-convex lens. The front surface of this lens is flat, and the rear surface has a radius of curvature of R2=35cm . Assume that the index of refraction of the lens is 1.5.Find the focal length of a meniscus lens with R1=20cmand R2=15cm . Assume that the index of refraction of the lens is 1.5.A nearsighted man cannot see objects clearly beyond 20 cm from his eyes. How close must he stand to a mirror in order to see what he is doing when he shaves?A mother sees that her child’s contact lens prescription is 0.750 D. What is the child’s near point?Repeat the previous problem for glasses that are 2.20 cm from the eyes.The contact-lens prescription for a nearsighted person is —4.00 D and the person has a far point of 22.5 cm. What is the power of the tear layer between the cornea and the lens if the correction is ideal, taking the tear layer into account?Unreasonable Results A boy has a near point of 50 cm and a far point of 500 cm. Will a —4.00 D lens correct his far point to infinity?Find the angular magnification of an image by a magnifying glass of f = 5.0 cm if the object is placed do=4.0cmfrom the lens and the lens is close to the eye.Let objective and eyepiece of a compound microscope have focal lengths of 2.5 cm and 10 cm, respectively and be separated by 12 cm. A 70- mobject is placed 6.0 cm from the objective. How large is the virtual image formed by the objective-eyepiece system?Draw rays to scale to locate the image at the retina if the eye lens has a focal length 2.5 cm and the near point is 24 cm. (Hint: Place an object at the near point.)The objective and the eyepiece of a microscope have the focal lengths 3 cm and 10 cm respectively. Decide about the distance between the objective and the eyepiece if we need a 10 x magnification from the objective/eyepiece compound system.A far-sighted person has a near point of 100 cm. How far in front or behind the retina does the image of an object placed 25 cm from the eye form? Use the cornea to retina distance of 25 cm.A near-sighted person has afar point of 80 cm. (a) What kind of corrective lens the person will need if the lens is to be placed 1.5 cm from the eye? (b) What would be the power of the contact lens needed? Assume distance to contact lens from the eye to be zero.In a reflecting telescope the objective is a concave mirror of radius of curvature 2m and an eyepiece is a convex lens of focal length 5 cm. Find the apparent size of a 25-m tree at a distance of 10 km that you would perceive when looking through the telescope.Two stars that are 109km apart are viewed by a telescope and found to be separated by an angle of 105 radians. If the eyepiece of the telescope has a focal length of 1.5 cm and the objective has a focal length of 3 meters, how far away are the stars from the observer?What is the angular size of the Moon if viewed from a binocular that has a focal length of 1.2 cm for the eyepiece and a focal length of 8 cm for the objective? Use the radius of the moon 1.74106 m and the distance of the moon from the observer to be 3.8108m .An unknown planet at a distance of 1012 m from Earth is observed by a telescope that has a focal length of the eyepiece of 1 cm and a focal length of the objective of I m. If the far away planet is seen to subtend an angle of 105 radian at the eyepiece, what is the size of the planet?Check Your Understanding In the system used in the preceding examples, at what angles are the first and the second bright fringes formed?Check Your Understanding Going further with Example 3.4, what are the next two thicknesses of soap bubble that would lead to (a) constructive interference, and (b) destructive interference?Check Your Understanding Although m, the number of fringes observed, is an integer, which is often regarded as having zero uncertainty, in practical terms, it is all too easy to lose track when counting fringes. In Example 3.6, if you estimate that you might have missed as many as five fringes when you reported m=122 fringes, (a) is the value for the index of refraction worked out in Example 3.6 too large or too small? (b) By how much?Young’s double-slit experiment breaks a single light beam into two sources. Would the same pattern be obtained for two independent sources of light, such as the headlights of a distant car? Explain.Is it possible to create a experimental setup in which there is only destructive interference? Explain.Why won’t two small sodium lamps, held close together, produce an interference pattern on a distant screen? What if the sodium lamps were replaced by two laser pointers held close together?Suppose you use the same double slit to perform Young’s double-slit experiment in air and then repeat the experiment in water. Do the angles to the same parts of the interference pattern get larger or smaller? Does the color of the light change? Explain.Why is monochromatic light used in the double slit experiment? What would happen if white light were used?What effect does increasing the wedge angle have on the spacing of interference fringes? If the wedge angle is too large, fringes are not observed. Why?How is the difference in paths taken by two originally in-phase light waves related to whether they interfere constructively or destructively? How can this be affected by reflection? By refraction?Is there a phase change in the light reflected from either surface of a contact lens floating on a person’s tear layer? The index of refraction of the lens is about 1.5, and its top surface is dry.In placing a sample on a microscope slide, a glass cover is placed over a water drop on the glass slide. Light incident from above can reflect from the top and bottom of the glass cover and from the glass slide below the water drop. At which surfaces will there be a phase change in the reflected light?Answer the above question if the fluid between the two pieces of crown glass is carbon disulfide.While contemplating the food value of a slice of ham, you notice a rainbow of color reflected from its moist surface. Explain its origin.An inventor notices that a soap bubble is dark at its thinnest and realizes that destructive interference is taking place for all wavelengths. How could she use this knowledge to make a nonreflective coating for lenses that is effective at all wavelengths? That is, what limits would there be on the index of refraction and thickness of the coating? How might this be impractical?A nonreflective coating like the one described in Example 3.3 works ideally for a single wavelength and for perpendicular incidence. What happens for other wavelengths and other incident directions? Be specific.Why is it much more difficult to see interference fringes for light reflected from a thick piece of glass than from a thin film? Would it be easier if monochromatic light were used?Describe how a Michelson interferometer can be used to measure the index of refraction of a gas (including air).At what angle is the first-order maximum for 450-nm wavelength blue light falling on double slits separated by 0.0500 mm?Calculate the angle for the third-order maximum of 580-nm wavelength yellow light falling on double slits separated by 0.100 mm.What is the separation between two slits for which 610-nm orange light has its first maximum at an angle of 30.0°?Find the distance between two slits that produces the first minimum for 410-nm violet light at an angle of 45.0°.Calculate the wavelength of light that has its third minimum at an angle of 30.0° when falling on double slits separated by 3.00 m. Explicitly show how you follow the steps from the Problem-Solving Strategy: Wave Optics, located at the end of the chapter.What is the wavelength of light falling on double slits separated by 2.00 m if the third-order maximum is at an angle of 60.0° ?At what angle is the fourth-order maximum for the situation in the preceding problem?What is the highest-order maximum for 400-nm light falling on double slits separated by 25.0 m?Find the largest wavelength of light falling on double slits separated by 1.20 m for which there is a first-order maximum. Is this in the visible part of the spectrum?What is the smallest separation between two slits that will produce a second-order maximum for 720-nm red light?(a) What is the smallest separation between two slits that will produce a second-order maximum for any visible light? (b) For all visible light?(a) If the first-order maximum for monochromatic light falling on a double slit is at an angle of 10.0° , at what angle is the second-order maximum? (b) What is the angle of the first minimum? (c) What is the highest-order maximum possible here?Shown below is a double slit located a distance x from a screen, with the distance from the center of the screen given by y. When the distance d between the slits is relatively large, numerous bright spots appear, called fringes. Show that, for small angles (where sin , with in radians), the distance between fringes is given by y=x/dUsing the result of the preceding problem, (a) calculate the distance between fringes for 633-nm light falling on double slits separated by 0.0800 mm, located 3.00 m from a screen. (b) What would be the distance between fringes if the entire apparatus were submersed in water, whose index of refraction is 1.33?Using the result of the problem two problems prior, find the wavelength of light that produces fringes 7.50 mm apart on a screen 2.00 m from double slits separated by 0.120 mm.In a double-slit experiment, the fifth maximum is 2.8 cm from the central maximum on a screen that is 1.5 m away from the slits. If the slits are 0.15 mm apart, what is the wavelength of the light being used?The source in Young’s experiment emits at two wavelengths. On the viewing screen, the fourth maximum for one wavelength is located at the same spot as the fifth maximum for the other wavelength. What is the ratio of the two wavelengths?If 500-nm and 650-nm light illuminates two slits that are separated by 0.50 mm, how far apart are the second-order maxima for these two wavelengths on a screen 2.0 m away?Red light of wavelength of 700 nm falls on a double slit separated by 400 nm. (a) At what angle is the first-order maximum in the diffraction pattern? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?Ten narrow slits are equally spaced 0.25 mm apart and illuminated with yellow light of wavelength 580 nm. (a) What are the angular positions of the third and fourth principal maxima? (b) What is the separation of these maxima on a screen 2.0 m from the slits?The width of bright fringes can be calculated as the separation between the two adjacent dark fringes on either side. Find the angular widths of the third- and fourth-order bright fringes from the preceding problem.For a three-slit interference pattern, find the ratio of the peak intensities of a secondary maximum to a principal maximum.What is the angular width of the central fringe of the interference pattern of (a) 20 slits separated by d=2.0103 mm? (b) 50 slits with the same separation? Assume that =600 nm.A soap bubble is 100 nm thick and illuminated by white light incident perpendicular to its surface. What wavelength and color of visible light is most constructively reflected, assuming the same index of refraction as water?An oil slick on water is 120 nm thick and illuminated by white light incident perpendicular to its surface. What color does the oil appear (what is the most constructively reflected wavelength), given its index of refraction is 1.40?Calculate the minimum thickness of an oil slick on water that appears blue when illuminated by white light perpendicular to its surface. Take the blue wavelength to be 470 nm and the index of refraction of oil to be 1.40.Find the minimum thickness of a soap bubble that appears red when illuminated by white light perpendicular to its surface. Take the wavelength to be 680 nm, and assume the same index of refraction as water.A film of soapy water (n=1.33) on top of a plastic cutting board has a thickness of 233 nm. What color is most strongly reflected if it is illuminated perpendicular to its surface?