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Optics LCHS Notation for Mirrors and Lenses • The object distance is the distance from the object to the mirror or lens • The image distance is the distance from the image to the mirror or lens • The lateral magnification of the mirror or lens is the ratio of the image height to the object height Types of Images • A real image is formed when light rays pass through and diverge from the image point • A virtual image is formed when light rays do not pass through the image point but only appear to diverge from that point Flat Mirrrors Law of Reflection “The angle of incidence equals the angle of reflection.” This is true for both flat mirrors and curved mirrors. Normal Line A Angle of Incidence = Angle of Reflection MIRROR B Diffuse Reflection Locating the Image for Plane Mirrors 1. Draw the image the same distance behind the mirror as the object is in front. 2. Draw a connector line from each object to each image. 3. If the connector line passes through the mirror, the image will be seen. Mirror Images E D C B A These lines are pointed to the only images that will be seen from each of the original locations (A-E) NOTE: No images will be seen from E A B C D E Lateral Magnification Lateral magnification, M, is defined as Im age height h' M Object height h – This is the general magnification for any type of mirror – It is also valid for images formed by lenses – Magnification does not always mean bigger, the size can either increase or decrease Lateral Magnification of a Flat Mirror • The lateral magnification of a flat mirror is 1 • This means that h' = h for all 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 • The reversal is not actually a left-right reversal • The reversal is actually a front-back reversal – It is caused by the light rays going forward toward the mirror and then reflecting back from it Summary • The image is as far behind the mirror as the object is in front – dd = |do| • The image is unmagnified – The image height is the same as the object height • h' = h and M = 1 • The image is virtual • The image is upright – It has the same orientation as the object • There is a front-back reversal in the image The angle of incidence equals the angle of what? a) Dispersion b) Refraction c) Reflection Specular reflections are images seen after __ surface(s). a) rough b) smooth c) no Can you see a reflection of yourself in a diffuse reflection? a) yes b) no Where do you draw the connector lines? a) from the lens to object b) from image to lens c) from object to each image What happens if the connector line passes through the mirror? a) image is invisible b) image is seen Light doesn’t pass through __ images. a) virtual b) real c) large d) small Spherical Mirrors Spherical Mirrors • A spherical mirror has the shape of a segment of a sphere • The mirror focuses incoming parallel rays to a point • A concave spherical mirror has the light reflected from the inner, or concave, side of the curve • A convex spherical mirror has the light reflected from the outer, or convex, side of the curve Concave and Convex Mirrors Concave and convex mirrors are curved mirrors similar to portions of a sphere. light rays Concave mirrors reflect light from their inner surface, like the inside of a spoon. light rays Convex mirrors reflect light from their outer surface, like the outside of a spoon. Concave Mirrors Light from Infinite Distance C F Focuses at the focal point Two Rules for Concave Mirrors • Any incident ray traveling parallel to the principal axis on the way to the mirror will pass through the focal point upon reflection • Any incident ray passing through the focal point on the way to the mirror will travel parallel to the principal axis upon reflection C F C F C F Virtual Image C F Real Image C F Virtual Image C F Convex Mirrors • A convex mirror is sometimes called a diverging mirror – The light reflects from the outer, convex side • The rays from any point on the object diverge after reflection as though they were coming from some point behind the mirror • The image is virtual because the reflected rays only appear to originate at the image point Will an image ever focus at a single point with a convex mirror? F Therefore, the images you see are virtual! Image Formed by a Convex Mirror In general, the image formed by a convex mirror is upright, virtual, and smaller than the object Notes on Images • With a concave mirror, the image may be either real or virtual. When the object is – outside the focal point, the image is real – at the focal point, the image is infinitely far away – inside the focal point, the image is virtual • With a convex mirror, the image is always virtual and upright – As the object distance decreases, the virtual image increases in size Mirror Sign Convention f = focal length 1 1 1 f = di + do di = image distance do = object distance + for real image di - for virtual image + for concave mirrors f - for convex mirrors Magnification hi By definition, m = ho m = magnification hi = image height (negative means inverted) ho = object height AND m = hi / ho = -di / do Magnification is simply the ratio of image height to object height. A positive magnification means an upright image. •F •C Casey decides to join in the fun and she finds a convex mirror to stand in front of. She sees her image reflected 7 feet behind the mirror which has a focal length of 11 feet. Her image is 1 foot tall. Where is she standing and how tall is she? do =19.25 feet ho = 2.75 feet Mirror Equation Sample Problem •C •F Suppose AllStar, who is 3 and a half feet tall, stands 27 feet in front of a concave mirror with a radius of curvature of 20 feet. Where will his image be reflected di = 15.88 feet What will its size be? hi = -2.06 feet Determine the image distance for a 5.00-cm tall object placed 10.0 cm from a concave mirror having a focal length of 15.0 cm. Use 1 / f = 1 / do + 1 / di where f = 15 cm and do = 10.0 cm di = -30.0 cm Determine the image height for a 5.00cm tall object placed 10.0 cm from a concave mirror having a focal length of 15.0 cm. (di = -30.0 cm) Then use hi / ho = -di / do where ho = 5 cm, do = 45 cm, and di = -30.0 cm hi = +15.0 cm A 4.00-cm tall light bulb is placed a distance of 45.7 cm from a concave mirror having a focal length of 15.2 cm. Determine the image distance. 1/f = 1/do + 1/di 1/(15.2 cm) = 1/(45.7 cm) + 1/di 0.0658 cm-1 = 0.0219 cm-1 + 1/di 0.0439 cm-1 = 1/di 22.8 cm = di A 4.00-cm tall light bulb is placed a distance of 45.7 cm from a concave mirror having a focal length of 15.2 cm. Determine the image size. hi/ho = - di/do hi /(4.0 cm) = - (22.8 cm)/(45.7 cm) hi = - (4.0 cm) • (22.8 cm)/(45.7 cm) hi = -1.99 cm Refraction Normal Line Less Dense More Dense Normal Line More Dense Less Dense Normal Line #1 Light Beam Fast AIR Slow WATER AIR Fast Normal Line #2 http://cougar.slvhs.slv.k12.ca.us/~pboomer/physicslectures/secondsemester/light/refraction/refraction.html Snell’s Law Snell’s law states that a ray of light bends in such a way that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant. Mathematically, i r ni nr ni sin i = nr sinr Here ni is the index of refraction in the original medium and nr is the index in the medium the light enters. i and r are the angles of incidence and refraction, respectively. Willebrord Snell Index of Refraction, n The index of refraction of a substance is the ratio of the speed in light in a vacuum to the speed of light in that substance: c n= v n = Index of Refraction c = Speed of light in vacuum v = Speed of light in medium Note that a large index of refraction corresponds to a relatively slow light speed in that medium. Medium n Vacuum 1 Air (STP) 1.00029 Water (20º C) 1.33 Ethanol 1.36 Glass ~1.5 Diamond 2.42 Index of Refraction Equations • n = c/v = speed of light in a vacuum speed of light in medium Index of Refraction Equations • n = c/v = speed of light in a vacuum speed of light in medium • n = sin i/sin r Index of Refraction Equations • n = c/v = speed of light in a vacuum speed of light in medium • n = sin i/sin r • sin A / sin B = nB / nA Index of Refraction Problem A diamond (n = 2.42) is in water (n = 1.33) and a ray of light shines on it making an angle of incidence of 55o. What is the angle of refraction inside the diamond? sin A / sin B = nB / nA sin 55o / sin B = 2.42/1.33 B = 27o Total Internal Reflection... …is the total reflection of light traveling in a medium when it strikes a surface of a less dense medium above the critical angle n r -1 c = sin ni Critical Angle Animation Refraction Critical Angle AIR 49 Total Internal Reflection WATER Light Source A diver basks in the reflection of the Northern Lights underwater by George Karbus Index of Refraction Problem What is the speed of light in water, which has an index of refraction of 1.33? n = c/v v = c/n v = (2.998 x 108 m/s) / 1.33 V = 2.25 x 108 m/s Index of Refraction Problem A ray of light enters a piece of crown glass at an angle of 57o and is refracted to 31o inside the glass. What is the index of refraction? n = sin i/sin r = sin 57o / sin 31o = 1.63 Air – Water Interface sin θ = n2/n1 Air nair = 1 and Water n2 = 1.33 sin θ = 1.00/1.33 = 0.750 sin θ = 0.750 θ = sin-1 0.750 θ = 49o Critical Angle Sample Problem Calculate the critical angle for the diamond (n = 2.42) -air (n = 1) boundary. c = sin-1 (nr / ni) air diamond c = sin-1 (1 / 2.42) = 24.4 Any light shone on this boundary beyond this angle will be reflected back into the diamond. Lenses Focal Length and Focal Point of a Thin Lens A converging lens has a positive focal length o Therefore, it is sometimes called a positive lens A diverging lens has a negative focal length o It is sometimes called a negative lens Converging or Convex Lens C F F C Converging or Convex Lens Converging or Convex Lens C F F C Converging or Convex Lens C F F C Converging or Convex Lens C F F C Converging or Convex Lens C F F C Converging or Convex Lens C F F C Converging or Convex Lens C F F C Converging or Convex Lens C F F C Converging or Convex Lens C F F C Lens Sign Convention 1 1 1 + = f di do f = focal length di = image distance do = object distance di + for real image - for virtual image + for convex lenses f - for concave lenses Lens/Mirror Sign Convention The general rule for lenses and mirrors is this: di + for real image - for virtual image and if the lens or mirror has the ability to converge light, f is positive. Otherwise, f must be treated as negative for the mirror/lens equation to work correctly. Lens Equation: do> C f = 2 cm, C = 4 cm, ho = 2 cm, do = 5cm, di = ? 1/f = 1/do + 1/di 1/2 = 1/5 + 1/di 1/di = 1/2 - 1/5 = 0.5 – 0.2 = 0.3 di = 3.33 cm M = hi/ho = -di/do (-ho x di )/ do = hi hi = (-2 x 3.3)/5 hi = -1.3 cm Lens Equation: do < f f = 2 cm, C = 4 cm, ho = 2 cm, do = 0.5 cm, di = ? 1/f = 1/do + 1/di 1/2 = 1/1 + 1/di 1/di = 1/2 - 1/0.5 = 0.5 – 2.0 = -1.5 di = -.67 cm M = hi/ho = -di/do (-ho x di )/ do = hi hi = (-2 x -.67)/0.5 hi = +8/3 = +3.67 M = - di / do = +1.33 Lens Sample Problem •2F •F •F •2F Tooter, who stands 4 feet tall (counting his snorkel), finds himself 24 feet in front of a convex lens and he sees his image reflected 35 feet behind the lens. What is the focal length of the lens and how tall is his image? f = 14.24 feet hi = -5.83 feet Diverging or Concave Lens C F F C Concave Lenses 2• F •F •F 2• Rays traveling parallel to the principal axis of a concave lens will refract as if coming from the focus. F Rays traveling toward the focus will refract parallel to the principal axis. •2F •F •F 2• F •2F •F •F 2• F Rays traveling directly through the center of a concave lens will leave the lens traveling in the exact same direction, just as with a convex lens. Concave Lens Diagram object •2F •F image •F •2F No matter where the object is placed, the image will be on the same side as the object. The image is virtual, upright, and smaller than the object with a concave lens. Image Summary • For a converging lens, when the object distance is greater than the focal length (p >ƒ) – The image is real and inverted • For a converging lens, when the object is between the focal point and the lens, (p<ƒ) – The image is virtual and upright • For a diverging lens, the image is always virtual and upright – This is regardless of where the object is placed For a converging lens, the object real and inverted when the object distance is __. a) Greater than the focal length b) Less than the focal length c) Equal to the focal length For a diverging lens, the image is always what? a) virtual b) upright c) real d) a and b A converging lens has a __ focal length and a diverging lens has a __ focal length. a) +, c) +,+ b) -, + d) -,- Fiber Optics spool of optical fiber Fiber optic lines are strands of glass or transparent fibers that allows the transmission of light and digital information over long distances. They are used for the telephone system, the cable TV system, the internet, medical imaging, and mechanical engineering inspection. Optical fibers have many advantages over copper wires. They are less expensive, thinner, lightweight, and more flexible. They aren’t flammable since they use light signals instead of electric signals. Light signals from one fiber do not interfere with signals in nearby fibers, which means clearer TV reception or phone conversations. A fiber optic wire Continued… Fiber Optics Cont. Fiber optics are often long strands of very pure glass. They are very thin, about the size of a human hair. Hundreds to thousands of them are arranged in bundles (optical cables) that can transmit light great distances. There are three main parts to an optical fiber: • Core- the thin glass center where light travels. • Cladding- optical material (with a lower index of refraction than the core) that surrounds the core that reflects light back into the core. • Buffer Coating- plastic coating on the outside of an optical fiber to protect it from damage. Continued… Light travels through the core of a fiber optic by continually reflecting off of the cladding. Due to total internal reflection, the cladding does not absorb any of the light, allowing the light to travel over great distances. Some of the light signal will degrade over time due to impurities in the glass. Fiber Optics (cont.) There are two types of optical fibers: • Single-mode fibers- transmit one signal per fiber (used in cable TV and telephones). • Multi-mode fibers- transmit multiple signals per fiber (used in computer networks). Mirage Pictures Inferior Mirages A person sees a puddle ahead on the hot highway because the road heats the air above it, while the air farther above the road stays cool. Instead of just two layers, hot and cool, there are really many layers, each slightly hotter than the layer above it. The cooler air has a slightly higher index of refraction than the warm air beneath it. Rays of light coming toward the road gradually refract further from the normal, more parallel to the road. (Imagine the wheels and axle: on a light ray coming from the sky, the left wheel is always in slightly warmer air than the right wheel, so the left wheel continually moves faster, bending the axle more and more toward the observer.) When a ray is bent enough, it surpasses the critical angle and reflects. The ray continues to refract as it heads toward the observer. The “puddle” is really just an inverted image of the sky above. This is an example of an inferior mirage, since the cool are is above the hot air. Sunlight after Sunset Lingering daylight after the sun Apparent is below the horizon is another effect of refraction. Light travels position Observer of sun at a slightly slower speed in Earth’s atmosphere than in space. As a result, sunlight is Actual refracted by the atmosphere. In Earth position the morning, this refraction of sun causes sunlight to reach us before the sun is actually above Atmosphere the horizon. In the evening, the sunlight is bent above the horizon after the sun has actually set. So daylight is extended in the morning and evening because of the refraction of light. Note: the picture greatly exaggerates this effect as well as the thickness of the atmosphere. Rainbows A rainbow is a spectrum formed when sunlight is dispersed by water droplets in the atmosphere. Sunlight incident on a water droplet is refracted. Because of dispersion, each color is refracted at a slightly different angle. At the back surface of the droplet, the light undergoes total internal reflection. On the way out of the droplet, the light is once more refracted and dispersed. Although each droplet produces a complete spectrum, an observer will only see a certain wavelength of light from each droplet. (The wavelength depends on the relative positions of the sun, droplet, and observer.) Because there are millions of droplets in the sky, a complete spectrum is seen. The droplets reflecting red light make an angle of 42o with respect to the direction of the sun’s rays; the droplets reflecting violet light make an angle of 40o. Primary Rainbow Secondary Rainbow Secondary Primary Alexander’s dark region The secondary rainbow is a rainbow of radius 51, occasionally visible outside the primary rainbow. It is produced when the light entering a cloud droplet is reflected twice internally and then exits the droplet. The color spectrum is reversed in respect to the primary rainbow, with red appearing on its inner edge. Dispersion is the ___ of white light into pure colors. a) combination c) absorption b) separation d) decomposition What color can bend the most? a) violet c) red b) cyan d) magenta Which of these can raindrops not do to sunlight? a) Refract c) Reflect b) Absorb d) Disperse Credits Snork pics: http://www.geocities.com/EnchantedForest/Cottage/7352/indosnor.html Snorks icons: http://www.iconarchive.com/icon/cartoon/snorks_by_pino/ Snork seahorse pic: http://members.aol.com/discopanth/private/snork.jpg Mirror, Lens, and Eye pics: http://www.physicsclassroom.com/ Refracting Telescope pic: http://csep10.phys.utk.edu/astr162/lect/light/refracting.html Reflecting Telescope pic: http://csep10.phys.utk.edu/astr162/lect/light/reflecting.html Fiber Optics: http://www.howstuffworks.com/fiber-optic.htm Willebrord Snell and Christiaan Huygens pics: http://micro.magnet.fsu.edu/optics/timeline/people/snell.html Chromatic Aberrations: http://www.dpreview.com/learn/Glossary/Optical/Chromatic_Aberrations_01.htm Mirage Diagrams: http://www.islandnet.com/~see/weather/elements/mirage1.htm Sir David Brewster pic: http://www.brewstersociety.com/brewster_bio.html Mirage pics: http://www.polarimage.fi/ http://www.greatestplaces.org/mirage/desert1.html http://www.ac-grenoble.fr/college.ugine/physique/les%20mirages.html Diffuse reflection: http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l1d.html Diffraction: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/grating.html