Chapter 29/30 Refraction and Lenses Refraction Refraction the bending of waves as they pass from one medium into another. Caused by a change in the average speed of light. Analogy A car that drives off a road on to the shoulder will pull toward the shoulder. A wave that slows down in the new medium will bend toward the normal. A wave that speeds up in the new medium will bend away from the normal. A wave that enters the new medium along the normal will not be refracted. Refraction of Sound Sound waves are refracted when parts of a wave front travel at different speeds. This can happen in uneven winds Or, when sound is traveling through air of uneven temperature Refraction is not abrupt but gradual Sound travels faster through warm air Sound waves tend to bend away from warm ground Wave Fronts and Rays Wave fronts lines that represent the positions of different crests. The direction a wave travels can also be represented using rays that are drawn perpendicular to the wave fronts. Index of Refraction Index of refraction n = speed of light in a vacuum/speed of light in material Snell s Law: n = sin i/sin r Dispersion in a Prism Light of frequencies closer to the natural frequency of the electron oscillators in a medium travels more slowly in the medium. Due to the process of absorption and reemission. For transparent materials, visible light of higher frequencies travels more lsowly than light of lower frequencies. Light traveling at different speeds will refract at different angles. Dispersion the separation of light into colors arranged according to their frequency. 1
The Rainbow For a rainbow to occur, The sun must be shining in one part of the sky Water droplets in a cloud or in falling rain must be in the opposite part of the sky. If viewed from a high altitude, a rainbow forms a complete circle. Some of the light that strikes a drop is reflected and some is refracted. Note that violet light is bent the most and red the least. The Rainbow Each drop disperses a full spectrum of colors. An observer is in a position to see only a single color from any one drop. If violet light from a single drop enters your eye, red light from the same drop falls below your eye. To see red light, you have to look at a drop higher in the sky. Total Internal Reflection Total internal reflection occurs when a light ray is reflected back into a medium instead of being refracted out of it. Critical angle the angle of incidence that provides an angle of refraction of 90-degrees For the water-air boundary, the critical angle is 48.6-degrees. For the crown glass-water boundary, the critical angle is 61.0- degrees. Total Internal Reflection Prisms are used in place of mirrors in many optical instruments, e.g. binoculars. Total internal reflection results in 100% of incident light being reflected. Many mirrors only reflect 90-95% of incident light. The small critical angle of diamond (24.6 o ) is smaller than any other known substance. Light inside a diamond is more likely to be totally internally reflected than to escape. Only light rays less than 24.6 o from the normal can escape. The speed of light within diamonds is unusually slow. This causes pronounced refraction and wide dispersion Total Internal Reflection Fiber optic cables (light pipes) use total internal reflection to get light to inaccessible places and to send communication signals The index of refraction for the core (n1) is always greater than that of the outer layer (n2) Converging and Diverging Lenses Lens a piece of glass shaped to bend parallel rays of light to form an image. A lens may be thought of as a set of prisms that converge light or diverge light away from a single point. The most net bending of rays occurs at the outermost prisms. 2
Key Features of Converging Lenses Principal axis the line joining the centers of curvature of its surfaces. Focal point the point at which beams parallel to the principal axis converge. Focal plane The possible points above or below the focal point where incident beams not parallel to the principal axis focus. Centers of Curvature Image Formation by a Lens A far away object is viewed through a small angle of view. A close object is seen through a larger angle of view. Magnification occurs when an image is observed through a wider angle with the use of a lens. An object between the focal point and a converging lens will be magnified, virtual, and upright. The rays behave as if they came from the image position. The image cannot be projected on a screen. If the object is beyond the focal point of a converging lens, a real image is formed. The image can be projected on a screen. The real image is inverted. Image Formation by a Lens A lone diverging lens will always produce a virtual image that is upright and reduced. Ray Diagrams To locate an image, you only have to know the paths of any 2 of 3 rays from a point on the object. A ray parallel to the principal axis will be refracted by the lens to the focal point. A ray of light that passes through the center of a lens will not be refracted. A ray of light that passes through the focal point in front of the lens emerges and proceeds parallel to the principal axis. Ray Diagrams & Diverging Lenses A ray parallel to the principal axis will be bent in the same direction as though it had come from the focal point. A ray through the center goes straight through. A ray heading for the focal point on the far side of the lens is bent so that it emerges parallel to the principal axis. The rays appear to come from the virtual the location of the virtual image. Image Formation Summarized For a converging lens An object within one focal length of the lens The image is virtual, magnified, and right-side up. An object beyond one focal length The image is real and inverted. For a diverging lens The image is virtual, reduced, and right-side up. 3
Some Common Optical Instruments The Camera Consists of a lens and sensitive film The lens can be moved back and forth to adjust the distance between lens and film. The lens forms a real, inverted image on the film. The amount of light that gets to the film is regulated by Shutter controls the length of time that the film is exposed to light Diaphragm controls the opening that light passes through to reach the film. Varying the size of the opening (aperture) varies the amount of light that reaches the film. Some Common Optical Instruments Simple Telescope Uses a lens to form a real image of a distant object. A second lens (eyepiece) is positioned so that the image produced by the first lens is within one focal length of the eyepiece. The eyepiece forms an enlarged virtual image of the real image. Terrestrial Telescope (also binoculars) A third lens or a pair of reflecting prisms produce an image that is right side up (see page 472). A pair of such telescopes are binoculars. Some Common Optical Instruments Compound Microscope Uses two converging lenses of short focal length (see page 472). The first is the objective lens, which produces an enlarged real image of a close object. The second the eye piece forms a virtual image that is further enlarged. Projector A concave mirror reflects light form an intense source onto a pair of condenser lenses (see page 473). The condenser lenses direct light through the slide or movie frame to a projection lens. The projection lens can be moved back and forth to focus the image on the screen. The Eye The eye is similar to a camera Iris controls the amount of light that enters the eye. Pupil the opening of the eye that allows light to enter. Cornea the transparent covering of the eye. Retina light sensitive layer of tissue at the back of the eye on which light is focused. Fovea a small region in the center of our field of view where we have the most distinct vision. Blind spot A spot in the retina where the nerves from the retina leave the eye and form the optic nerve. Accommodation Focusing occurs by changing the thickness and shape of the lens. This is called accommodation Brought about by the ciliary muscle that surrounds the lens. Some Defects in Vision A normal eye can accommodate to clearly see objects from infinity (the far point) to 25 cm (the near point). Farsighted the eyes form images behind the retina. The eyeball is too short. Farsighted people must hold things farther than 25 cm away Remedy: wearing eyeglasses or contact lenses that are made from converging lenses. 4
Some Defects in Vision Nearsighted the eyes focus too near the lens in front of the retina. The eyeball is too long. Remedy: wearing eyeglasses or contact lenses that diverge rays from distant object Astigmatism a defect that results when the cornea is curved more in one direction than another. The eye does not form sharp images. Remedy: cylindrical corrective lenses that have more curvature in one direction than another. Some Defects of Lenses Aberrations distortions in an object. Aberrations can be minimized by using compound lenses (several simple lenses together). Spherical aberration light passing through the edges of a lens focuses at a slightly different place from light passing through the center of the lens. Can be corrected by covering the edges of the lens or by using compound lenses. Chromatic aberration the result of different colors of light undergoing different refractions because of their different speeds through a lens. Achromatic lenses combine simple lenses of different kinds of glass to correct this defect. Vision is sharpest when the pupil is smallest. The light only passes through the center of the lens. Spherical and chromatic aberrations are minimal. Light bends the least through the center minimal focusing required. End of Notes Graphics Follow This Slide 1. Once these incident rays strike the lens, refract Pick a point on the top of the object and draw three incident rays traveling towards the lens. Using a straight edge, accurately draw one ray so that it passes exactly through the focal point on the way to the lens. Draw the second ray such that it travels exactly parallel to the principal axis. Place arrowheads upon the rays to indicate their direction of travel. Draw the third incident ray such that it travels directly to the exact center of the lens. them according to the three rules of refraction for converging lenses. 2.The ray that passes through the focal point on the way to the lens will refract and travel parallel to the principal axis. Use a straight edge to accurately draw its path. The ray which traveled parallel to the principal axis on the way to the lens will refract and travel through the focal point. And the ray which traveled to the exact center of the lens will continue in the same direction. Place arrowheads upon the rays to indicate their direction of travel. Extend the rays past their point of intersection. 5
1. Repeat the process for the bottom of the object. 1. Mark the image of the top of the object. 2.The image point of the top of the object is the point where the two refracted rays intersect. All three rays should intersect at exactly the same point. This point is merely the point where all light from the top of the object would intersect upon refracting through the lens. Of course, the rest of the object has an image as well and it can be found by applying the same three steps to another chosen point. (See note below.) 2.The goal of a ray diagram is to determine the location, size, orientation, and type of image which is formed by the double convex lens. Typically, this requires determining where the image of the upper and lower extreme of the object is located and then tracing the entire image. After completing the first three steps, only the image location of the top extreme of the object has been found. Thus, the process must be repeated for the point on the bottom of the object. If the bottom of the object lies upon the principal axis (as it does in this example), then the image of this point will also lie upon the principal axis and be the same distance from the mirror as the image of the top of the object. At this point the entire image can be filled in. Refraction Rules for a Diverging Lens Any incident ray traveling parallel to the principal axis of a diverging lens will refract through the lens and travel in line with the focal point (i.e., in a direction such that its extension will pass through the focal point). Any incident ray traveling towards the focal point on the way to the lens will refract through the lens and travel parallel to the principal axis. An incident ray which passes through the center of the lens will in effect continue in the same direction that it had when it entered the lens. Refraction Rules for a Converging Lens Any incident ray traveling parallel to the principal axis of a converging lens will refract through the lens and travel through the focal point on the opposite side of the lens. Any incident ray traveling through the focal point on the way to the lens will refract through the lens and travel parallel to the principal axis. An incident ray which passes through the center of the lens will in effect continue in the same direction that it had when it entered the lens. 6
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Case 1: The object is located beyond 2F Case 1: the object is located beyond the 2F point Case 2: the object is located at the 2F point Case 3: the object is located between the 2F point and the focal point (F) Case 4: the object is located at the focal point (F) Case 5: the object is located in front of the focal point (F) When the object is located at a location beyond the 2F point, the image will always be located somewhere in between the 2F point and the focal point (F) on the other side of the lens. Regardless of exactly where the object is located, the image will be located in this specified region. In this case, the image will be an inverted image. That is to say, if the object is right-side up, then the image is upside down. In this case, the image is reduced in size; in other words, the image dimensions are smaller than the object dimensions. If the object is a six-foot tall person, then the image is less than six feet tall. Earlier in Unit 13, the term magnification was introduced; the magnification is the ratio of the height of the object to the height of the image. In this case, the magnification is a number less than 1. Finally, the image is a real image. Light rays actually converge at the image location. If a sheet of paper was placed at the image location (as was done in the Converging Lens Lab), the actual replica of the object would appear projected upon the sheet of paper. Case 2: The object is located at 2F When the object is located at the 2F point, the image will also be located at the 2F point on the other side of the lens. In this case, the image will be inverted (i.e., a right-side-up object results in an upside-down image). The image dimensions are equal to the object dimensions. A six-foot tall person would have an image which is six feet tall; the magnification is exactly 1. Finally, the image is a real image. Light rays actually converge at the image location. As such, the image of the object could be projected upon a sheet of paper. Case 3: The object is located between 2F and F When the object is located in front of the 2F point, the image will be located beyond the 2F point on the other side of the lens. Regardless of exactly where the object is located between C and F, the image will be located in the specified region. In this case, the image will be inverted (i.e., a right-side-up object results in an upside-down image). The image dimensions are larger than the object dimensions. A six-foot tall person would have an image which is larger than six feet tall; the magnification is greater than 1. Finally, the image is a real image. Light rays actually converge at the image location. As such, the image of the object could be projected upon a sheet of paper. Case 5: The object is located in front of F Case 4: The object is located at F When the object is located at the focal point, no image is formed. As discussed earlier in Lesson 5, the refracted rays neither converge or diverge. After refracting, the light rays are traveling parallel to each other and When the object is located at a location beyond the focal point, the image will always be located somewhere on the same side of the lens as the object. Regardless of exactly where in front of F the object is located, the image will always be located on the object's side of the lens and somewhere further from the lens. In this case, the image will be an upright image. That is to say, if the object is right-side up, then the image will also be right-side up. In this case, the image is enlarged; in other words, the image dimensions are greater than the object dimensions. A six-foot tall person would have an image which is larger than six feet tall; the magnification is greater than 1. Finally, the image is a virtual image. Light rays diverge upon refraction; for this reason, the image location can only be found by extending the refracted rays backwards beyond the mirror. The point of their intersection is the virtual image location. It would appear to any observer as though light from the object were diverging from this location. Any attempt to project such an image upon a sheet of paper would fail since light does not actually pass through the image location. 8
The method of drawing ray diagrams for a double concave lens is described below. 1. Pick a point on the top of the object and draw three incident rays traveling towards the lens. 2.Using a straight edge, accurately draw one ray so that it travels towards the focal point on the opposite side of the lens; this ray will strike the lens before reaching the focal point; stop the ray at the point of incidence with the lens. Draw the second ray such that it travels exactly parallel to the principal axis. Draw the third ray to the exact center of the lens. Place arrowheads upon the rays to indicate their direction of travel. 3. 1. Once these incident rays strike the lens, refract them according to the three rules of refraction for double concave lenses. 2.The ray that travels towards the focal point will refract through the lens and travel parallel to the principal axis. Use a straight edge to accurately draw its path. The ray which traveled parallel to the principal axis on the way to the lens will refract and travel in a direction such that its extension passes through the focal point. Align a straight edge with the point of incidence and the focal point, and draw the second refracted ray. The ray which traveled to the exact center of the lens will continue to travel in the same direction. Place arrowheads upon the rays to indicate their direction of travel. The three rays should be diverging upon refraction. 1. Locate and mark the image of the top of the object. 2.The image point of the top of the object is the point where the three refracted rays intersect. Since the three refracted rays are diverging, they must be extended behind the lens in order to intersect. Using a straight edge, extend each of the rays using dashed lines. Draw the extensions until they intersect. All three extensions should intersect in the same location. The point of intersection is the image point of the top of the object. The three refracted rays would appear to diverge from this point. This is merely the point where all light from the top of the object would appears to diverge from after refracting through the double concave lens. Of course, the rest of the object has an image as well and it can be found by applying the same three steps to another chosen point. See note below. 1. Repeat the process for the bottom of the object. 2.The goal of a ray diagram is to determine the location, size, orientation, and type of image which is formed by the double concave lens. Typically, this requires determining where the image of the upper and lower extreme of the object is located and then tracing the entire image. After completing the first three steps, only the image location of the top extreme of the object has been found. Thus, the process must be repeated for the point on the bottom of the object. If the bottom of the object lies upon the principal axis (as it does in this example), then the image of this point will also lie upon the principal axis and be the same distance from the lens as the image of the top of the object. At this point the complete image can be filled in. 9
The diagrams above shows that in each case, the image is located behind the lens a virtual image an upright image reduced in size (i.e., smaller than the object) 10