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Chapter 23: Geometric Optics © 2013 Pearson Education, Inc. Review Basic Geometry! Ray Approximation • The rays are straight lines perpendicular to the wave fronts • With the ray approximation, we assume that a wave moving through a medium travels in a straight line in the direction of its rays Light Rays: Ignore Diffraction and Interference of waves! Diffraction depends on SLIT WIDTH: the smaller the width, relative to wavelength, the more bending and diffraction. We will assume that λ<<d , where d is the diameter of the opening. This approximation is good for the study of mirrors, lenses, prisms, etc. The Ray Model of Light Let us define a light ray as a line in the direction along which light energy is flowing. Any narrow beam of light, such as a laser beam, is actually a bundle of many parallel light rays. You can think of a single light ray as the limiting case of a laser beam whose diameter approaches zero. © 2013 Pearson Education, Inc. Slide 23-19 The Ray Model of Light Light travels through a transparent material in straight lines called light rays. The speed of light is v c/n, where n is the index of refraction of the material. Light rays do not interact with each other. Two rays can cross without either being affected in any way. © 2013 Pearson Education, Inc. Slide 23-20 The Ray Model of Light An object is a source of light rays. Rays originate from every point on the object, and each point sends rays in all directions. The eye “sees” an object when diverging bundles of rays from each point on the object enter the pupil and are focused to an image on the retina. © 2013 Pearson Education, Inc. Slide 23-22 The Ray Model of Light Light interacts with matter in four different ways: At an interface between two materials, light can be either reflected or refracted. Within a material, light can be either scattered or absorbed. © 2013 Pearson Education, Inc. Slide 23-21 Objects Objects can be either self-luminous, such as the sun, flames, and lightbulbs, or reflective. Most objects are reflective. © 2013 Pearson Education, Inc. Slide 23-23 Objects The diverging rays from a point source are emitted in all directions. Each point on an object is a point source of light rays. A parallel bundle of rays could be a laser beam, or light from a distant object. © 2013 Pearson Education, Inc. Slide 23-24 Ray Diagrams Rays originate from every point on an object and travel outward in all directions, but a diagram trying to show all these rays would be messy and confusing. To simplify the picture, we use a ray diagram showing only a few rays. © 2013 Pearson Education, Inc. Slide 23-25 Reflection & Refraction i r n1 sin 1 n2 sin 2 Law of Reflection • The normal is a line perpendicular to the surface i r – It is at the point where the incident ray strikes the surface • The incident ray makes an angle of θ1 with the normal • The reflected ray makes an angle of θ1’ with the normal Specular Reflection • Specular reflection is reflection from a smooth surface • The reflected rays are parallel to each other • All reflection in this text is assumed to be specular Diffuse Reflection • Diffuse reflection is reflection from a rough surface • The reflected rays travel in a variety of directions • A surface behaves as a smooth surface as long as the surface variations are much smaller than the wavelength of the light Following the Reflected and Refracted Rays •Ray is the incident ray. •Ray is the reflected ray. •Ray is refracted into the lucite. •Ray is internally reflected in the lucite. •Ray is refracted as it enters the air from the lucite. Section 35.5 How many times will the incident beam shown be reflected by each of the parallel mirrors? https://www.youtube.com/watch?v=BQNw8HSixGk Mirror Mirror QuickCheck 23.3 You are looking at the image of a pencil in a mirror. What do you see in the mirror if the top half of the mirror is covered with a piece of dark paper? A. The full image of the pencil. B. The top half only of the pencil. C. The bottom half only of the pencil. D. No pencil, only the paper. © 2013 Pearson Education, Inc. Slide 23-41 QuickCheck 23.3 You are looking at the image of a pencil in a mirror. What do you see in the mirror if the top half of the mirror is covered with a piece of dark paper? A. The full image of the pencil. B. The top half only of the pencil. C. The bottom half only of the pencil. D. No pencil, only the paper. © 2013 Pearson Education, Inc. Slide 23-42 Why are most materials Opaque? (Opaque – Can’t see through) They absorb light without re-emitting it. Vibrations given by the light to their atoms and molecules are turned into random kinetic energy – they become slightly warmer. Opacity: Mirrors Free electrons in opaque reflective surfaces can vibrate, absorb & re-emit at any frequency. Transparency Selective Absorption Glass resonates strongly with UV and infrared, absorbing those frequencies while transmitting visible frequencies. Refraction: Bending Light into Focus Refraction: Bending of Light Transmitted through Materials Light Bends because it Slows Down. Atoms are Optical Tuning Forks Light slows down as it travels through glass because it takes time to be absorbed and re-emitted. Light Slows Down in Materials Light Bends Toward the Normal when going from a medium of lower refractive index to one that has a higher refractive index and visa versa. lower n higher n Index of Refraction c n v n 1 Vacuum: 1 Water: 1.33 Glass: 1.46 Diamond: 2.4 The Index of Refraction • Refraction: Light Bends in Transmission • The speed of light in any material is less than its speed in vacuum • The index of refraction, n, of a medium can be defined as • For a vacuum, n = 1 – We assume n = 1 for air speed of light in a vacuum c λ n also speed of light in a m edium v λn • For other media, n > 1 λ λ in vacuum • n is a dimensionless number greater than unity, not n λn λ in a medium necessarily an integer Some Indices of Refraction Variation of Index of Refraction with Wavelength sp e e d o f lig h t in a v a cu u m c λ n sp e e d o f lig h t in a m e d iu m v λ n • This dependence of n on λ is called dispersion • The index of refraction for a material generally decreases with increasing wavelength • Violet light bends more than red light when passing into a refracting material Frequency Doesn’t Change! • As light travels from one medium to another, its frequency does not change – Both the wave speed and the wavelength do change – The wavefronts do not pile up, nor are created or destroyed at the boundary, so ƒ must stay the same Snell’s Law of Refraction Angles are always measured from the normal. n1 sin 1 n2 sin 2 Snell’s Law Light is refracted into a crown glass slab. n1 = 1.00 and n2 = 1.52 If θ1 = 30.0o, θ2 = ? θ2 = sin-1(n1 / n2) sin θ1 = 19.2o The ray bends toward the normal, as expected because n2 > n1 Snell’s Law of Refraction In general: n1 sin 1 n2 sin 2 n11 n22 n1 2 1 n2 If n2 n1 , then 1 2 measured from the normal! QuickCheck 23.4 A laser beam passing from medium 1 to medium 2 is refracted as shown. Which is true? A. n1 < n2. B. n1 > n2. C. There’s not enough information to compare n1 and n2. © 2013 Pearson Education, Inc. Slide 23-53 QuickCheck 23.4 A laser beam passing from medium 1 to medium 2 is refracted as shown. Which is true? A. n1 < n2. B. n1 > n2. C. There’s not enough information to compare n1 and n2. © 2013 Pearson Education, Inc. Slide 23-54 Prelab Emerging Beam is Parallel to Incident Beam but offset distance d, called the Lateral Shift and is the subject of this week’s lab! Fig. 35-15, p. 989 Prelab for Next Lab: https://www.youtube.com/watch?v=1-0eQGeWLfY&t=2s Real and Apparent Depth Real and apparent depth. The refraction of light at the surface of water makes ponds and swimming pools appear shallower than they really are. A 1m deep pond would only appear to be 0.75 m deep when viewed from directly above. When light emerges from glass or water into air it speeds up again. © 2013 Pearson Education, Inc. Apparent Depth © 2013 Pearson Education, Inc. QuickCheck 23.6 A fish in an aquarium with flat sides looks out at a hungry cat. To the fish, the distance to the cat appears to be A. Less than the actual distance. B. Equal to the actual distance. C. More than the actual distance. © 2013 Pearson Education, Inc. Slide 23-71 QuickCheck 23.6 A fish in an aquarium with flat sides looks out at a hungry cat. To the fish, the distance to the cat appears to be A. Less than the actual distance. B. Equal to the actual distance. C. More than the actual distance. © 2013 Pearson Education, Inc. Slide 23-72 Beam & Refraction Directions • Possible directions of the beam are indicated by rays numbered 1 through 5 • The refracted rays are bent away from the normal since n 1 > n2 • An application of internal reflection • Plastic or glass rods are used to “pipe” light from one place to another • Applications include: – medical use of fiber optic cables for diagnosis and correction of medical problems – Telecommunications • A flexible light pipe is called an optical fiber • A bundle of parallel fibers (shown) can be used to construct an optical transmission line Fiber Optics Construction of an Optical Fiber • The transparent core is surrounded by cladding – The cladding has a lower n than the core – This allows the light in the core to experience total internal reflection • The combination is surrounded by the jacket Total Internal Reflection 2 90 n1 sin 1 n2 sin 2 n2 sin C n1 The Critical Angle Critical Angle • There is a particular angle of incidence that will result in an angle of refraction of 90° – This angle of incidence is called the critical angle, θC n2 sin θ C (for n1 n 2 ) n1 Critical Angle Sample Problem A ray of light, emitted by a laser located beneath the surface of an unknown liquid with air above it, undergoes total internal refection as shown. What is the index of refraction for the liquid? What is its likely identification? QuickCheck 23.5 A laser beam undergoes two refractions plus total internal reflection at the interface between medium 2 and medium 3. Which is true? A. n1 < n3. B. n1 > n3. C. There’s not enough information to compare n1 and n3. © 2013 Pearson Education, Inc. Slide 23-62 QuickCheck 23.5 A laser beam undergoes two refractions plus total internal reflection at the interface between medium 2 and medium 3. Which is true? A. n1 < n3. B. n1 > n3. C. There’s not enough information to compare n1 and n3. © 2013 Pearson Education, Inc. Slide 23-63 If you pass white light through a prism, it separates into its component colors. long wavelengths short wavelengths R.O.Y. G. B.I.V The index of refraction depends on WAVELENGTH. long wavelengths short wavelengths R.O.Y. G. B.I.V The speed and wavelength change but the FREQUENCY does NOT. Fr Frequency depends on the oscillating source! long wavelengths short wavelengths R.O.Y. G. B.I.V Why does Violet Light bend more than Red Light? Violet light slows down more because the atoms in the material are tuned to higher frequencies. As the violet light travels through glass it takes more time to be absorbed and re-emitted. Variation of Index of Refraction with Wavelength sp e e d o f lig h t in a v a cu u m c λ n sp e e d o f lig h t in a m e d iu m v λ n • This dependence of n on λ is called dispersion • The index of refraction for a material generally decreases with increasing wavelength • Violet light bends more than red light when passing into a refracting material Refraction in a Prism •Since all the colors have different angles of deviation, white light will spread out into a spectrum. – Violet deviates the most. – Red deviates the least. – The remaining colors are in between. Section 35.7 Compare: Dispersion via Diffraction and Refraction Why does Violet Light bend more than Red Light? Violet light slows down more because the atoms in the material are tuned to higher frequencies. As the violet light travels through glass it takes more time to be absorbed and re-emitted. Angle of Deviation • Since all the colors have different angles of deviation, white light will spread out into a spectrum – Violet deviates the most – Red deviates the least – The remaining colors are in between Dispersion Sample Problem The index of refraction for violet light in silica flint glass is 1.66, and that for red light is 1.62. What is the angular dispersion of visible light passing through a prism of apex angle 60.0° if the angle of incidence is 50.0°? red (660 nm) violet (410 nm) Use Snell’s Law twice and some geometry! Angles are always measured from the normal. n1 sin 1 n2 sin 2 How are Rainbows Formed? Dispersion: Raindrops Act like Prisms • A ray of light strikes a drop of water in the atmosphere • It undergoes both reflection and refraction – First refraction at the front of the drop • Violet light will deviate the most • Red light will deviate the least The Rainbow • At the back surface the light is reflected • It is refracted again as it returns to the front surface and moves into the air • The rays leave the drop at various angles – The angle between the white light and the most intense violet ray is 40° – The angle between the white light and the most intense red ray is 42° Observing the Rainbow • If a raindrop high in the sky is observed, the red ray is seen • A drop lower in the sky would direct violet light to the observer • The other colors of the spectra lie in between the red and the violet The droplets form a circular arc, with each droplet within the arc dispersing light and reflecting it back towards the observer with the greatest concentration of outgoing rays found at these 40-42 degree angles of deviation. Every droplet within the arc is refracting and dispersing the entire visible light spectrum (ROYGBIV). Rainbow facts • an observer is in a position to see only a single color from any one droplet of water. • your rainbow is slightly different from the rainbow seen by others • your rainbow moves with you • disk within the bow is brighter because of overlapping of multiple refractions (which don’t occur outside the disk) A line drawn from your eye to the top of the rainbow forms a 42degree angle with the imaginary line from the sun through your eye. (If there is a secondary rainbow, it forms an angle of 51degrees). Because these angles determine the position of the rainbow in the sky, it will sink as the sun rises and rise as the sun sinks. At some points, the entire rainbow, not just the bottom half, will be below the horizon where you can't see it. That's why you'll never see a summer rainbow at midday. Double Rainbow • The secondary rainbow is fainter than the primary • The secondary rainbow arises from light that makes two reflections from the interior surface before exiting the raindrop • Higher-order rainbows are possible, but their intensity is low • • Halos are caused by the light of the sun or moon passing through a very thin layer of cirruform (ice-crystal) clouds in the upper atmosphere. The ice crystals refract the light of the moon, similar to the way water droplets in the lower atmosphere can refract sunlight to produce a rainbow. Just like a rainbow, strong halos can have bands of color in them, due to slightly different refractive properties of the ice crystals for different colors. Essentially, halos ARE rainbows caused by primary refraction in ice crystals. Some interesting facts about halos: Halos always occur exactly 22 degrees away from the sun or moon. Occasionally, intense halos can be double halos, just as intense rainbows can be doubled. Intense halos can also produce "moondogs" or "sundogs," very bright regions on the halo evenly spaced at 90 degree intervals around the halo. Colored Filters and Colored Objects Green glass is green because it absorbs any light that is “not green.” If a green filter and a red filter are overlapped, no light gets through. The green filter transmits only green light, which is then absorbed by the red filter because it is “not red.” © 2013 Pearson Education, Inc. Slide 23-83 Colored Filters and Colored Objects The figure below shows the absorption curve of chlorophyll, which is essential for photosynthesis in green plants. The chemical reactions of photosynthesis absorb red light and blue/violet light from sunlight and puts it to use. When you look at the green leaves on a tree, you’re seeing the light that was reflected because it wasn’t needed for photosynthesis. © 2013 Pearson Education, Inc. Slide 23-84 Light Scattering: Blue Skies and Red Sunsets Rayleigh scattering © 2013 Pearson Education, Inc. Light Scattering: Blue Skies and Red Sunsets Light can scatter from small particles that are suspended in a medium. Rayleigh scattering from atoms and molecules depends inversely on the fourth power of the wavelength: Iscattered 4 © 2013 Pearson Education, Inc. Slide 23-85 Light Scattering: Blue Skies and Red Sunsets Sunsets are red because all the blue light has scattered as the sunlight passes through the atmosphere. © 2013 Pearson Education, Inc. Slide 23-86 Galileo In the early 17th century, many scientists believed that there was no such thing as the "speed of light"; they thought light could travel any distance in no time at all. Galileo disagreed, and he came up with an experiment to measure light's velocity: he and his assistant each took a shuttered lantern, and they stood on hilltops one mile apart. Galileo flashed his lantern, and the assistant was supposed to open the shutter to his own lantern as soon as he saw Galileo's light. Galileo would then time how long it took before he saw the light from the other hilltop. The problem was that the speed of light is simply too fast to be measured this way; light takes such a short time (about 0.000005 seconds, in fact) to travel one mile that there's no way the interval could have been measured using the tools Galileo had. The Speed of Light? • • • • 186,000 miles per second 300,000 kilometers per second 3 x 108 m/s first successfully determined by Danish astronomer Ole Roemer in 1675: 2.3 x 108 m/s • First Terrestrial Measurement by Fizeau in 1849: 2.9979 x 108 m/s • In 1926, Michelson used a rotating prism to measure the time it took light to make a round trip from Mount Wilson to Mount San Antonio in California, a distance of about 22 miles (36 km). The precise measurements yielded a speed of 186,285 miles per second (299,796 kilometres per second). Lenses and Mirrors Image Formation Why aren’t images of objects produced on the wall without a lens or hole? Why aren’t images of objects produced on the wall without a lens or hole? Images Image formation is a consequence of light traveling in straight lines The first camera—the pinhole camera— illustrates this fact. Apertures A camera obscura is a darkened room with a single, small hole, called an aperture. The geometry of the rays causes the image to be upside down. The object and image heights are related by: © 2013 Pearson Education, Inc. Slide 23-26 Apertures We can apply the ray model to more complex apertures, such as the L-shaped aperture below. © 2013 Pearson Education, Inc. Slide 23-27 QuickCheck 23.1 The dark screen has a small hole, 2 mm in diameter. The lightbulb is the only source of light. What do you see on the viewing screen? © 2013 Pearson Education, Inc. Slide 23-28 QuickCheck 23.1 The dark screen has a small hole, 2 mm in diameter. The lightbulb is the only source of light. What do you see on the viewing screen? © 2013 Pearson Education, Inc. Slide 23-29 QuickCheck 23.2 Two point sources of light illuminate a narrow vertical aperture in a dark screen. What do you see on the viewing screen? © 2013 Pearson Education, Inc. Slide 23-30 QuickCheck 23.2 Two point sources of light illuminate a narrow vertical aperture in a dark screen. What do you see on the viewing screen? © 2013 Pearson Education, Inc. Slide 23-31 Lenses A lens nicely bends the straight-line paths of light. Real and Virtual Images Real images can be displayed on screens Virtual Images can not be displayed onto screens. Lenses Lenses two common types converging (convex) lens thicker at the center than edges converges light diverging (concave) lens thinner at the center than edges diverges light Lens Refraction Converging & Diverging Converging Lens Diverging Lens Concave vs Convex Focal Length: Where Parallel rays come to a focus Lenses The photos below show parallel light rays entering two different lenses. The left lens, called a converging lens, causes the rays to refract toward the optical axis. The right lens, called a diverging lens, refracts parallel rays away from the optical axis. Slide 23-87 Focal Length: Converging Lens Focal Length: Diverging Lens QuickCheck 23.8 You can use the sun’s rays and a lens to start a fire. To do so, you should use A. A converging lens. B. A diverging lens. C. Either a converging or a diverging lens will work if you use it correctly. Slide 23-90 QuickCheck 23.8 You can use the sun’s rays and a lens to start a fire. To do so, you should use A. A converging lens. B. A diverging lens. C. Either a converging or a diverging lens will work if you use it correctly. Slide 23-91 Thin Lenses: Ray Tracing Three situations form the basis for ray tracing through a thin converging lens. Situation 1: A ray initially parallel to the optic axis will go through the far focal point after passing through the lens. Slide 23-92 Thin Lenses: Ray Tracing Three situations form the basis for ray tracing through a thin diverging lens. Situation 2: A ray directed along a line toward the far focal point becomes parallel to the optic axis after passing through the lens. Slide 23-116 Thin Lenses: Ray Tracing Three situations form the basis for ray tracing through a thin converging lens. Situation 3: A ray through the center of a thin lens is neither bent nor displaced but travels in a straight line. Slide 23-94 Thin Lenses: Ray Tracing Rays from an object point P are refracted by the lens and converge to a real image at point P. Slide 23-95 Lenses In an actual lens, rays refract twice, at spherical surfaces having radii of curvature R1 and R2. Slide 23-130 Lenses Key features of lenses principal axis focal point line joining the centers of curvature of the two lens surfaces point at which all the light rays come together focal length distance between the center of the lens and either focal point Notation for Mirrors and Lenses The object distance is the distance from the object to the mirror or lens: Denoted by p The image distance is the distance from the image to the mirror or lens: Denoted by q The lateral magnification of the mirror or lens is the ratio of the image height to the object height Denoted by M The focal point: f The radius of curvature: R = 2f M Image height h' Object height h 1 1 2 1 p q R f FRONT is on the same side as the object and BACK is the other side! The Principal axis goes through the focal point and the center of curvature of the lens or mirror!! Signs for Thin Lenses 1 1 2 1 p q R f h' q M h p https://phet.colorado.edu/en/simulation/legacy/geometricoptics The Thin Lens Equation The object distance s is related to the image distance s by: where f is the focal length of the lens, which can be found from: where R1 is the radius of curvature of the first surface, and R2 is the radius of curvature of the second surface, and the material surrounding the lens has n = 1. © 2013 Pearson Education, Inc. Slide 23-131 Thin Lenses: Refraction Theory If an object is located at distance s from a spherical refracting surface, an image will be formed at distance s given by: Slide 23-125 Tactics: Ray Tracing for a Converging Lens © 2013 Pearson Education, Inc. Slide 23-103 © 2013 Pearson Education, Inc. Ray Diagram for Converging Lens, p > f The image is real (q>0) The image is inverted (M<0) The image is on the back side of the lens (q>0) Ray Diagram for Converging Lens, p < f The image is virtual (q < 0) The image is upright (M>0) The image is larger than the object (M>1) The image is on the front side of the lens (q<0) Magnifying Glass: Virtual Images You can “see” a virtual image by looking through the lens. This is exactly what you do with a magnifying glass, microscope or binoculars. Slide 23-111 QuickCheck 23.15 A lens creates an image as shown. In this situation, A. B. C. D. s < f. f < s < 2f. s > 2f. There’s not enough information to compare s to f. © 2013 Pearson Education, Inc. Slide 23-132 QuickCheck 23.15 A lens creates an image as shown. In this situation, A. B. C. D. s < f. f < s < 2f. s > 2f. There’s not enough information to compare s to f. The image is real, which requires s > f. The image is taller than the object, and s’ > s requires s < 2f. © 2013 Pearson Education, Inc. Slide 23-133 Tactics: Ray Tracing for a Diverging Lens © 2013 Pearson Education, Inc. Slide 23-120 Ray Diagram for Diverging Lens For a diverging lens, the image is always virtual and upright (M>0) This is regardless of where the object is placed The image is on the front side of the lens (q<0) Example 23.10 Demagnifying a Flower © 2013 Pearson Education, Inc. Slide 23-122 Example 23.10 Demagnifying a Flower © 2013 Pearson Education, Inc. Slide 23-123 Example 23.17 Analyzing a Concave Mirror © 2013 Pearson Education, Inc. Slide 23-150 Example 23.17 Analyzing a Concave Mirror © 2013 Pearson Education, Inc. Slide 23-151 Example 23.17 Analyzing a Concave Mirror © 2013 Pearson Education, Inc. Slide 23-152 QuickCheck 23.14 Light rays are converging to point 1. The lens is inserted into the rays with its focal point at point 1. Which picture shows the rays leaving the lens? © 2013 Pearson Education, Inc. Slide 23-118 QuickCheck 23.14 Light rays are converging to point 1. The lens is inserted into the rays with its focal point at point 1. Which picture shows the rays leaving the lens? © 2013 Pearson Education, Inc. Slide 23-119 Flat Mirrors Make Virtual Images Virtual Image: An image that cannot be projected onto a surface. A virtual image only appears like light rays came from the location of the image, they are not really there. Flat mirrors make Virtual Images. Images Formed by Flat Mirrors Light rays leave the source and are reflected from the mirror Images are always located by extending diverging rays back to a point at which they intersect One ray starts at point P, travels to Q and reflects back on itself Another ray follows the path PR and reflects according to the law of reflection h’ = h for all images Flat mirrors make virtual images Reversals in a Flat Mirror A flat mirror produces an image that has an apparent left-right reversal For example, if you raise your right hand the image you see raises its left hand Properties of the Image Formed by a Flat Mirror – Summary The image is as far behind the mirror as the object is in front |p| = |q| The image is unmagnified The image height is the same as the object height The image is virtual The image is upright h’ = h and M = 1 It has the same orientation as the object There is a front-back reversal in the image Mirror Reflection Convex & Concave “Object” on the left, image on the right. Convex Mirror Convave Mirror Focal Length& Radius of Curvature When the object is very far away, then p → ∞ and the incoming rays are essentially parallel In this special case, the image point is called the focal point The distance from the mirror to the focal point is called the focal length The focal length is ½ the radius of curvature R = 2f Lateral Magnification Image height h' M Object height h Magnification does not always mean bigger, the size can either increase or decrease. M>1: Increase Positive: Upright M<1: Decrease Negative: Inverted Ray Diagrams:Concave Mirrors Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected through the focal point, F Ray 2 is drawn from the top of the object through the focal point and is reflected parallel to the principal axis Ray 3 is drawn through the center of curvature, C, and is reflected back on itself The intersection of any two of the rays at a point locates the image. The third ray serves as a check of the construction Concave Mirror, p > R The center of curvature is between the object and the concave mirror surface (f >0) The image is real (q>0) The image is inverted (M<0) The image is smaller than the object (absM<1) 1 1 2 1 p q R f A Real Image Formed by a Concave Mirror Slide 23-140 Concave Mirror, p < f The object is between the mirror surface and the focal point (p>0) The image is virtual (q<0) The image is upright (M>0) The image is larger than the object (M>1) Ray Diagrams:Convex Mirrors Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected away from the focal point, F Ray 2 is drawn from the top of the object toward the focal point and is reflected parallel to the principal axis Ray 3 is drawn through the center of curvature, C, on the back side of the mirror and is reflected back on itself Convex Mirror The object is in front of a convex mirror (p>0) The focal point distance q is negative (q <0) The image is always virtual and upright (M>0) As the object distance decreases, the virtual image size increases The image is smaller than the object (0<M<1) Image Formation with Spherical Mirrors A city skyline is reflected in this polished sphere. Slide 23-143 Sign Conventions: Mirrors 1 1 2 1 p q R f The Mirror Equation For a spherical mirror with negligible thickness, the object and image distances are related by: where the focal length f is related to the mirror’s radius of curvature by: © 2013 Pearson Education, Inc. Slide 23-146 Compare Signs for Mirrors and Thin Lenses Thin Lenses QuickCheck 23.9 A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if the lens is removed? A. An inverted but blurry image. B. An image that is dimmer but otherwise unchanged. C. A sharp, upright image. D. A blurry, upright image. E. No image at all. Slide 23-96 QuickCheck 23.9 A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if the lens is removed? A. An inverted but blurry image. B. An image that is dimmer but otherwise unchanged. C. A sharp, upright image. D. A blurry, upright image. E. No image at all. Slide 23-97 QuickCheck 23.10 A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if a piece of dark paper is lowered to cover the top half of the lens? A. An inverted but blurry image. B. An image that is dimmer but otherwise unchanged. C. Only the top half of the image. D. Only the bottom half of the image. E. No image at all. Slide 23-98 QuickCheck 23.10 A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if a piece of dark paper is lowered to cover the top half of the lens? A. An inverted but blurry image. B. An image that is dimmer but otherwise unchanged. C. Only the top half of the image. D. Only the bottom half of the image. E. No image at all. Slide 23-99 QuickCheck 23.11 A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if the lens is covered by a dark mask having only a small hole in the center? A. B. C. D. E. An inverted but blurry image. An image that is dimmer but otherwise unchanged. Only the middle piece of the image. A circular diffraction pattern. No image at all. Slide 23-100 QuickCheck 23.11 A lens produces a sharply focused, inverted image on a screen. What will you see on the screen if the lens is covered by a dark mask having only a small hole in the center? A. B. C. D. E. An inverted but blurry image. An image that is dimmer but otherwise unchanged. Only the middle piece of the image. A circular diffraction pattern. No image at all. Slide 23-101 Image Formation The figure is a close-up view of the rays very near the image plane. To focus an image, you must either move the screen to coincide with the image plane or move the lens or object to make the image plane coincide with the screen. Slide 23-102 Example 23.12 A Goldfish in a Bowl Slide 23-126 Example 23.12 A Goldfish in a Bowl s 8.3 cm Slide 23-128 Tactics: Ray Tracing for a Converging Lens Slide 23-103 Tactics: Ray Tracing for a Converging Lens Slide 23-104 QuickCheck 23.12 A lens creates an image as shown. In this situation, the object distance s is A. Larger than the focal length f. B. Equal to the focal length f. C. Smaller than focal length f. Slide 23-105 QuickCheck 23.12 A lens creates an image as shown. In this situation, the object distance s is A. Larger than the focal length f. B. Equal to the focal length f. C. Smaller than focal length f. Slide 23-106 QuickCheck 23.13 A lens creates an image as shown. In this situation, the image distance s is A. Larger than the focal length f. B. Equal to the focal length f. C. Smaller than focal length f. Slide 23-107 QuickCheck 23.13 A lens creates an image as shown. In this situation, the image distance s is A. Larger than the focal length f. B. Equal to the focal length f. C. Smaller than focal length f. Slide 23-108 Lateral Magnification The image can be either larger or smaller than the object, depending on the location and focal length of the lens. The lateral magnification m is defined as: A positive value of m indicates that the image is upright relative to the object. A negative value of m indicates that the image is inverted relative to the object. The absolute value of m gives the size ratio of the image and object: h/h |m|. Slide 23-109 Virtual Images Consider a converging lens for which the object is inside the focal point, at distance s < f. You can see all three rays appear to diverge from point P. Point P is an upright, virtual image of the object point P. Slide 23-110 Virtual Images You can “see” a virtual image by looking through the lens. This is exactly what you do with a magnifying glass, microscope or binoculars. Slide 23-111 Example 23.9 Magnifying a Flower Slide 23-112 Example 23.9 Magnifying a Flower Slide 23-113 Example 23.9 Magnifying a Flower Slide 23-114 Thin Lenses: Ray Tracing Three situations form the basis for ray tracing through a thin diverging lens. Situation 1: A ray initially parallel to the optic axis will appear to diverge from the near focal point after passing through the lens. Slide 23-115 Thin Lenses: Ray Tracing Three situations form the basis for ray tracing through a thin diverging lens. Situation 2: A ray directed along a line toward the far focal point becomes parallel to the optic axis after passing through the lens. Slide 23-116 Thin Lenses: Ray Tracing Three situations form the basis for ray tracing through a thin diverging lens. Situation 3: A ray through the center of a thin lens is neither bent nor displaced but travels in a straight line. Slide 23-117 QuickCheck 23.14 Light rays are converging to point 1. The lens is inserted into the rays with its focal point at point 1. Which picture shows the rays leaving the lens? Slide 23-118 QuickCheck 23.14 Light rays are converging to point 1. The lens is inserted into the rays with its focal point at point 1. Which picture shows the rays leaving the lens? Slide 23-119 Tactics: Ray Tracing for a Diverging Lens Slide 23-120 Example 23.10 Demagnifying a Flower Slide 23-121 Example 23.10 Demagnifying a Flower Slide 23-122 Example 23.10 Demagnifying a Flower Slide 23-123 Thin Lenses: Refraction Theory Consider a spherical boundary between two transparent media with indices of refraction n1 and n2. The sphere has radius of curvature R and is centered at point C. Slide 23-124 Thin Lenses: Refraction Theory If an object is located at distance s from a spherical refracting surface, an image will be formed at distance s given by: Slide 23-125 Example 23.12 A Goldfish in a Bowl Slide 23-126 Example 23.12 A Goldfish in a Bowl Slide 23-127 Example 23.12 A Goldfish in a Bowl Slide 23-128 Example 23.12 A Goldfish in a Bowl s 8.3 cm Slide 129 Lenses In an actual lens, rays refract twice, at spherical surfaces having radii of curvature R1 and R2. Slide 23-130 The Thin Lens Equation The object distance s is related to the image distance s by: where f is the focal length of the lens, which can be found from: where R1 is the radius of curvature of the first surface, and R2 is the radius of curvature of the second surface, and the material surrounding the lens has n = 1. Slide 23-131 QuickCheck 23.15 A lens creates an image as shown. In this situation, A. B. C. D. s < f. f < s < 2f. s > 2f. There’s not enough information to compare s to f. Slide 23-132 QuickCheck 23.15 A lens creates an image as shown. In this situation, A. B. C. D. s < f. f < s < 2f. s > 2f. There’s not enough information to compare s to f. The image is real, which requires s > f. The image is taller than the object, and s’ > s requires s < 2f. Slide 23-133 Example 23.13 Focal Length of a Meniscus Lens Slide 23-134 Example 23.13 Focal Length of a Meniscus Lens Slide 23-135 Example 23.15 A Magnifying Lens Slide 23-136 Example 23.15 A Magnifying Lens Slide 23-137 Example 23.15 A Magnifying Lens Slide 23-138 Image Formation with Concave Spherical Mirrors The figure shows a concave mirror, a mirror in which the edges curve toward the light source. Rays parallel to the optical axis reflect and pass through the focal point of the mirror. Slide 23-139 A Real Image Formed by a Concave Mirror Slide 23-140 Image Formation with Convex Spherical Mirrors The figure shows parallel light rays approaching a mirror in which the edges curve away from the light source. This is called a convex mirror. The reflected rays appear to come from a point behind the mirror. Slide 23-141 A Real Image Formed by a Convex Mirror Slide 23-142 Image Formation with Spherical Mirrors A city skyline is reflected in this polished sphere. Slide 23-143 Tactics: Ray Tracing for a Spherical Mirror Slide 23-144 Tactics: Ray Tracing for a Spherical Mirror Slide 23-145 The Mirror Equation For a spherical mirror with negligible thickness, the object and image distances are related by: where the focal length f is related to the mirror’s radius of curvature by: Slide 23-146 QuickCheck 23.16 You see an upright, magnified image of your face when you look into magnifying “cosmetic mirror.” The image is located A. B. C. D. In front of the mirror’s surface. On the mirror’s surface. Behind the mirror’s surface. Only in your mind because it’s a virtual image. Slide 23-147 QuickCheck 23.16 You see an upright, magnified image of your face when you look into magnifying “cosmetic mirror.” The image is located A. B. C. D. In front of the mirror’s surface. On the mirror’s surface. Behind the mirror’s surface. Only in your mind because it’s a virtual image. Slide 23-148 Example 23.17 Analyzing a Concave Mirror Slide 23-149 Example 23.17 Analyzing a Concave Mirror Slide 23-150 Example 23.17 Analyzing a Concave Mirror Slide 23-151 Example 23.17 Analyzing a Concave Mirror Slide 23-152 Chapter 23 Summary Slides Slide 23-153 General Principles Slide 23-154 General Principles Slide 23-155 Important Concepts Slide 23-156 Important Concepts Slide 23-157