Chapter 25: Geometric Optics PDF

Chapter 25: Geometric Optics Mrs. Pooja Brahmaiahchari Introduction Through vision, the most valued of our senses, light can evoke spiritual emotions, such as when we view a magnificent sunset or glimpse a rainbow breaking through the clouds. Optics is the branch of physics that deals w...

Chapter 25: Geometric Optics Mrs. Pooja Brahmaiahchari Introduction Through vision, the most valued of our senses, light can evoke spiritual emotions, such as when we view a magnificent sunset or glimpse a rainbow breaking through the clouds. Optics is the branch of physics that deals with the behavior of visible light and other electromagnetic waves. In particular, optics is concerned with the generation and propagation of light and its interaction with matter. Geometric Optics Light from this page or screen is formed into an image by the lens of your eye, much as the lens of the camera that made this photograph.  Mirrors, like lenses, can also form images that in turn are captured by your eye. It is convenient to divide optics into two major parts based on the size of objects that light encounters. When light interacts with an object that is several times as large as the light’s wavelength, its observable behavior is like that of a ray; it does not prominently display its wave characteristics. We call this part of optics “geometric optics.” The Ray Aspect of Light There are three ways in which light can travel from a source to another location. It can come directly from the source through empty space, such as from the Sun to Earth. Or light can travel through various media, such as air and glass, to the person. Light can also arrive after being reflected, such as by a mirror. Light very often travels in straight lines. We represent light using rays, which are straight lines emanating from an object.  In all of these cases, light is modeled as traveling in straight lines called rays. The word “ray” comes from mathematics and here means a straight line that originates at some point. Part of optics, where the ray aspect of light dominates, is therefore called geometric optics. There are two laws that govern how light changes direction when it interacts with matter. These are the law of reflection, for situations in which light bounces off matter, and the law of refraction, for situations in which light passes through matter. The Law of Reflection The law of reflection states that angle of incidence is equal to angle of reflection. The angles are measured relative to the perpendicular to the surface at the point where the ray strikes the surface. A mirror illuminated by many parallel rays reflects them in only one direction, since its surface is very smooth. Only the observer at a particular angle will see the reflected light.  When we see ourselves in a mirror, it appears that our image is actually behind the mirror. This is illustrated in Figure. We see the light coming from a direction determined by the law of reflection. The angles are such that our image is exactly the same distance behind the mirror as we stand away from the mirror. If the mirror is on the wall of a room, the images in it are all behind the mirror, which can make the room seem bigger. The Law of Refraction The changing of a light ray’s direction (loosely called bending) when it passes through variations in matter is called refraction. Refraction is responsible for a tremendous range of optical phenomena, from the action of lenses to voice transmission through optical fibers. The speed of light not only affects refraction, it is one of the central concepts of Einstein’s theory of relativity. As the accuracy of the measurements of the speed of light were improved, was found not to depend on the velocity of the source or the observer.  The speed of light is now known to great precision. In fact, the speed of light in a vacuum is so important that it is accepted as one of the basic physical quantities and has the fixed value. 8 𝑚 𝑐 = 2.99792458 𝑥 10 ≈ 3.00 𝑥 108 𝑚/𝑠 𝑠 The speed of light through matter is less than it is in a vacuum, because light interacts with atoms in a material. The speed of light depends strongly on the type of material, since its interaction with different atoms, crystal lattices, and other substructures varies.  We define the index of refraction of a material to be 𝑐 𝑛= 𝑣 where c is the observed speed of light in the material. Since the speed of light is always less than in matter and equals only in a vacuum, the index of refraction is always greater than or equal to one. The change in direction of the light ray depends on how the speed of light changes.  The change in the speed of light is related to the indices of refraction of the media involved. The change in direction of a light ray depends on how the speed of light changes when it crosses from one medium to another. The speed of light is greater in medium 1 than in medium 2 in the situations shown here. You need to know what direction the refracted ray will go whether it is towards or away from the normal This is know, if it goes from lower to higher refractive index it will go towards the normal If it goes from higher to lower it will go away from the normal  The amount that a light ray changes its direction depends both on the incident angle and the amount that the speed changes. For a ray at a given incident angle, a large change in speed causes a large change in direction, and thus a large change in angle. The exact mathematical relationship is the law of refraction, or “Snell’s Law,” which is stated in equation form as 𝑛1 𝑠𝑖𝑛𝜃1 = 𝑛2 sin 𝜃2 Here 𝑛1 and 𝑛2 are the indices of refraction for medium 1 and 2, and 𝜃1 and 𝜃2 are the angles between the rays and the perpendicular in medium 1 and 2,  The incoming ray is called the incident ray and the outgoing ray the refracted ray, and the associated angles the incident angle and the refracted angle. The law of refraction is also called Snell’s law after the Dutch mathematician Willebrord Snell (1591–1626), who discovered it in 1621. Snell’s experiments showed that the law of refraction was obeyed and that a characteristic index of refraction could be assigned to a given medium. Snell was not aware that the speed of light varied in different media, but through experiments he was able to determine indices of refraction from the way light rays changed direction. Total Internal Reflection A ray of light strikes the surface between two materials, such as is shown in Figure. Part of the light crosses the boundary and is refracted; the rest is reflected. The index of refraction for the second medium is less than for the first, the ray bends away from the perpendicular. Now imagine what happens as the incident angle is increased. This causes θ2 to increase also. The largest the angle of refraction θ2 can be is 90o, as shown in Figure  The critical angle θc for a combination of materials is defined to be the incident angle θ1 that produces an angle of refraction of 90o. That is, θc is the incident angle for which θ2 = 90o. If the incident angle is greater than the critical angle, as shown in Figure. Then all of the light is reflected back into medium 1, a condition called total internal reflection. Know the relation between the angle of total internal reflection and the critical angle Total internal reflection is just when the angle of incidence is the same as the angle of reflection  Snell’s law states the relationship between angles and indices of refraction. It is given by 𝑛1 sin 𝜃1 = 𝑛2 sin 𝜃2 When the incident angle equals the critical angle (𝜃1 = 𝜃2 ), the angle of refraction is 90o(𝜃2 = 90𝑜 ). Noting that ,Snell’s law in this case becomes 𝑛1 sin 𝜃1 = 𝑛2 The critical angle θc for a given combination of materials is thus 𝜃𝑐 = sin−1 𝑛1 Τ𝑛2 for 𝑛1 > 𝑛2 Total internal reflection occurs for any incident angle greater than the critical angle and it can only occur when the second medium has an index of refraction less than the first. Fiber Optics: Endoscopes to Telephones Fiber optics employs the transmission of light down fibers of plastic or glass. Because the fibers are thin, light entering one is likely to strike the inside surface at an angle greater than the critical angle and, thus, be totally reflected. The index of refraction outside the fiber must be smaller than inside, a condition that is easily satisfied by coating the outside of the fiber with a material having an appropriate refractive index. In fact, most fibers have a varying refractive index to allow more light to be guided along the fiber through total internal refraction. Rays are reflected around corners as shown, making the fibers into tiny light pipes.  Bundles of fibers can be used to transmit an image without a lens, as illustrated in Figure (a). The output of a device called an endoscope is shown in Figure (b) Endoscopes are used to explore the body through various orifices or minor incisions. Light is transmitted down one fiber bundle to illuminate internal parts, and the reflected light is transmitted back out through another to be observed. Surgery can be performed, such as arthroscopic surgery on the knee joint, employing cutting tools attached to and observed with the endoscope.  Fiber optics has revolutionized surgical techniques and observations within the body. There are a host of medical diagnostic and therapeutic uses. The flexibility of the fiber optic bundle allows it to navigate around difficult and small regions in the body, such as the intestines, the heart, blood vessels, and joints. Optical fibers have in fact enabled microsurgery and remote surgery where the incisions are small, and the surgeon’s fingers do not need to touch the diseased tissue.  Fibers in bundles are surrounded by a cladding material that has a lower index of refraction than the core. The cladding prevents light from escaping out of the fiber; Instead, most of the light is propagated along the length of the fiber, minimizing the loss of signal and ensuring that a quality image is formed at the other end. The cladding and an additional protective layer make optical fibers flexible and durable. Most telephone conversations and Internet communications are now carried by laser signals along optical fibers.  The fibers can be made so transparent that light can travel many kilometers before it becomes dim enough to require amplification— much superior to copper conductors. This property of optical fibers is called low loss. Lasers emit light with characteristics that allow far more conversations in one fiber than are possible with electric signals on a single conductor. This property of optical fibers is called high bandwidth. Optical signals in one fiber do not produce undesirable effects in other adjacent fibers. Know the qualites of fiber optics This property of optical fibers is called reduced crosstalk. 1. Calculate the index of refraction for a medium in which the speed of light is 2.012 x 108 m/s. n= C/V n = 3.0x10^8 / 2.012 x10^8 n=1.49 2. Light traveling from water to a gemstone strikes the surface at an angle of 80o and has an angle of refraction of 15.2o.What is the speed of light in the gemstone? 1.33 sin 80 = n2 sin 15.2 This number for refractive index makes sense as we know that if the light goes from a lower to higher refractive index it will go towards the normal and therefore decreasing the angle! If the light went from the water to a vacuum n2 = (1.33 sin 80) / sin15.2 n2 = 5 3. Calculate the critical angle between glass and water? (given n = 1.5 for glass and n = 1.33 for water) 1.3 / 1.5 = sin n1 sin 1 = n2 critical angle = 60 1.5 sin 1 = 1.3 4. The critical angle for the interface between a solid and air is 40o. What is the index of refraction of solid? Dont know the answer to 4. n1 sin 40 = n2 n1 sin 40 = 1 sin 40 = n2 n1 = 1.55 n2 = 0.64 can double check for question three by 1.5 sin 60 = 1.3 sin x 1.5 sin 60 / 1.3 = sin x sin x = 1 so x = 90 degrees which is the critical angle  Corner Reflectors and Diamonds A light ray that strikes an object consisting of two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came. This is true whenever the reflecting surfaces are perpendicular, and it is independent of the angle of incidence. Such an object, shown in Figure, is called a corner reflector, since the light bounces from its inside corner. Many inexpensive reflector buttons on bicycles, cars, and warning signs have corner reflectors designed to return light in the direction from which it originated. It was more expensive for astronauts to place one on the moon.  Total internal reflection, coupled with a large index of refraction, explains why diamonds sparkle more than other materials. The critical angle for a diamond-to-air surface is only 24.4o, and so when light enters a diamond, it has trouble getting back out. Good diamonds are very clear, so that the light makes many internal reflections and is concentrated at the few places it can exit—hence the sparkle.  The colors you see emerging from a sparkling diamond are not due to the diamond’s color, which is usually nearly colorless. Those colors result from dispersion. Colored diamonds get their color from structural defects of the crystal lattice and the inclusion of minute quantities of graphite and other materials. The Argyle Mine in Western Australia produces around 90% of the world’s pink, red, champagne, and cognac diamonds. Around 50% of the world’s clear diamonds come from central and southern Africa. Dispersion: The Rainbow and Prisms Everyone enjoys the spectacle of a rainbow glimmering against a dark stormy sky. We see about six colors in a rainbow—red, orange, yellow, green, blue, and violet; sometimes indigo is listed, too. Those colors are associated with different wavelengths of light. When our eye receives pure-wavelength light, we tend to see only one of the six colors, depending on wavelength. The thousands of other hues we can sense in other situations are our eye’s response to various mixtures of wavelengths.  White light, in particular, is a fairly uniform mixture of all visible wavelengths. Sunlight, considered to be white, actually appears to be a bit yellow because of its mixture of wavelengths, but it does contain all visible wavelengths. Dispersion is defined as the spreading of white light into its full spectrum of wavelengths. More technically, dispersion occurs whenever there is a process that changes the direction of light in a manner that depends on wavelength. Dispersion, as a general phenomenon, can occur for any type of wave and always involves wavelength-dependent processes. Refraction is responsible for dispersion in rainbows and many other situations.  (a) A pure wavelength of light falls onto a prism and is refracted at both surfaces. (b) White light is dispersed by the prism (shown exaggerated). Since the index of refraction varies with wavelength, the angles of refraction vary with wavelength. A sequence of red to violet is produced, because the index of refraction increases steadily with decreasing wavelength.  Rainbows are produced by a combination of refraction and reflection. You may have noticed that you see a rainbow only when you look away from the sun. Light enters a drop of water and is reflected from the back of the drop, The actual rainbow of colors seen by an observer depends on the myriad of rays being refracted and reflected toward the observer’s eyes from numerous drops of water. The effect is most spectacular when the background is dark, as in stormy weather, but can also be observed in waterfalls and lawn sprinklers. 5. A beam of white light goes from air into water at an incident angle of 75o. At what angles are the red (660 nm) and violet (410 nm) parts of the light refracted? Image Formation by Lenses The word lens derives from the Latin word for a lentil bean, the shape of which is similar to the convex lens. The convex lens shown has been shaped so that all light rays that enter it parallel to its axis cross one another at a single point on the opposite side of the lens. Such a lens is called a converging (or convex) lens for the converging effect it has on light rays.  Due to the lens’s shape, light is thus bent toward the axis at both surfaces. The point at which the rays cross is defined to be the focal point F of the lens. The distance from the center of the lens to its focal point is defined to be the focal length of the lens. The greater effect a lens has on light rays, the more powerful it is said to be. For example, a powerful converging lens will focus parallel light rays closer to itself and will have a smaller focal length than a weak lens. The power P of a lens is defined to be the inverse of its focal length. 1 𝑃= 𝑓  The concave lens is a diverging lens, because it causes the light rays to bend away (diverge) from its axis. In this case, the lens has been shaped so that all light rays entering it parallel to its axis appear to originate from the same point, F , defined to be the focal point of a diverging lens.  Ray Tracing and Thin Lenses Ray tracing is the technique of determining or following (tracing) the paths that light rays take. For rays passing through matter, the law of refraction is used to trace the paths. A thin lens is defined to be one whose thickness allows rays to refract but does not allow properties such as dispersion and aberrations. An ideal thin lens has two refracting surfaces, but the lens is thin enough to assume that light rays bend only once. A thin symmetrical lens has two focal points, one on either side and both at the same distance from the lens. Another important characteristic of a thin lens is that light rays through its center are deflected by a negligible amount, The rules for ray tracing for thin lenses A ray entering a converging lens parallel to its axis passes through the focal point F of the lens on the other side. A ray entering a diverging lens parallel to its axis seems to come from the focal point F. A ray passing through the center of either a converging or a diverging lens does not change direction. A ray entering a converging lens through its focal point exits parallel to its axis. A ray that enters a diverging lens by heading toward the focal point on the opposite side exits parallel to the axis. Image formation Consider an object some distance away from a converging lens, Ray tracing is used to locate the image formed by a lens. Rays originating from the same point on the object are traced—the three chosen rays each follow one of the rules for ray tracing, so that their paths are easy to determine. The image is located at the point where the rays cross. In this case, a real image—one that can be projected on a screen—is formed.  The image in which light rays from one point on the object actually cross at the location of the image and can be projected onto a screen, a piece of film, or the retina of an eye is called a real image. We define do to be the object distance, the distance of an object from the center of a lens. Image distance di is defined to be the distance of the image from the center of a lens. The height of the object and height of the image are given the symbols h0 and hi, respectively. Images that appear upright relative to the object have heights that are positive and those that are inverted have negative heights.  The thin lens equations are, 1 1 1 + = 𝑑0 𝑑𝑖 𝑓 and ℎ𝑖 𝑑𝑖 =− =𝑚 ℎ𝑜 𝑑𝑜 ℎ𝑖 We define the ratio of image height to object height ( ) to be the ℎ𝑜 magnification. A converging lens (convex lens) produces a virtual image when the object is placed closer to the lens than its focal point (within the focal length). In this case: Position: The virtual image will appear on the same side of the lens as the object. Characteristics: This virtual image is upright and magnified compared to the object. Distance: The virtual image appears to be located at a distance beyond the lens from where the rays appear to diverge when traced backward.  Ray tracing predicts the image location and size for an object held closer to a converging lens than its focal length. Ray 1 enters parallel to the axis and exits through the focal point on the opposite side, while ray 2 passes through the center of the lens without changing path. The two rays continue to diverge on the other side of the lens, but both appear to come from a common point, locating the upright, magnified, virtual image. For a diverging lens (concave lens), a virtual image is always produced, regardless of the position of the object. Here’s how it works: Position: The virtual image forms on the same side of the lens as the object. Characteristics: The virtual image is upright and smaller (reduced in size) than the object. Distance: The image appears to be located at a point where the diverging rays from the lens seem to originate when traced backward. Points to remember: 1. If di = +ve, then it is a Real image and the image will form in the right side. 2. If di = -ve, then it is a Virtual image and image is formed in left side of lens. Summary: Images form on a point where the light rays either come together or appear to come together. This differentiates a concave lens and a convex 3. Magnification m: lens. the convex (converging) lens will have a real image that is on the opposite site and is inverted. it is real because the light rays truely do ▪ m = +ve, Upright image converge at the point where the image is produced this is why it is called real because you could put this image on a photograph. ▪ m = -ve, inverted image The concave lens however will always produce a virtual image that appears behind the object. This is because in a concave lens the light rays on the ▪ |m| >1, then its an enlarged image opposite side of the lens will never converge as they are going away from each other. However if you were to ray trace the light rays you they would ▪ |m|< 1, then its an reduced image converge at a point on the same side as the object. It is virtual in this case because the light rays do not acctually converge at this point. ▪ |m| = 1, Neither enlarged nor reduced. Concave (diverging) lenses: Always produce a virtual, upright, and smaller image on the same side of the lens as the object. Convex (converging) lenses: Produce a virtual, upright, and enlarged image only when the object is within the focal length of the lens. 6. What is the power in diopters of a camera lens that has a 50.0 mm focal length? convert focal length to meters 1/0.05 = power P = 20 7. What is the focal length of 1.75 D reading glasses found on the rack in a pharmacy? P = 1.75 P = 1/ f f = 0.57 meters or 570 mm 8. 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)What is its magnification? 1/do +1/di = 1/f 1/3 + 1/di = 20 Real image because it is on the opposite side of 1/di = 20-1/3 the lens and is possible to put onto film di = 0.05 m Virtual images would be on the same side as the object and impossible to put onto film Magnification = - di/ do m= -0.0166 Will be inverted and smaller Image Formation by Mirrors Like lenses, mirrors can form a variety of images. For example, dental mirrors may produce a magnified image, just as makeup mirrors do. Security mirrors in shops, on the other hand, form images that are smaller than the object. We will use the law of reflection to understand how mirrors form images.  Two rays are shown emerging from the same point, striking the mirror, and being reflected into the observer’s eye. The rays can diverge slightly, and both still get into the eye. If the rays are extrapolated backward, they seem to originate from a common point behind the mirror, locating the image.  Now let us consider the focal length of a mirror—for example, the concave spherical mirrors in Figure. For a mirror that is large compared with its radius of curvature, as in Figure(a), we see that the reflected rays do not cross at the same point, and the mirror does not have a well-defined focal point. If the mirror had the shape of a parabola, the rays would all cross at a single point, and the mirror would have a well-defined focal point. For a concave mirror, a virtual image is produced only when the object is placed between the mirror and its focal point. Here’s how it behaves in this case: Position: The virtual image appears behind the mirror (on the same side as the object). Characteristics: This virtual image is upright and enlarged compared to the object. Distance: The virtual image is located at a distance that appears further behind the mirror than the object.  But parabolic mirrors are much more expensive to make than spherical mirrors. The solution is to use a mirror that is small compared with its radius of curvature, as shown in Figure(b). To a very good approximation, this mirror has a well-defined focal point at F that is the focal distance from the center of the mirror. The focal length of a concave mirror is positive, since it is a converging mirror. A more strongly curved mirror has a shorter focal length and a greater power. Using the law of reflection and some simple trigonometry, it can be shown that the focal length is half the radius of curvature, 𝑅 𝑓= 2  Convex mirrors: Always produce a virtual, upright, and smaller image. The smaller the radius of curvature, the smaller the focal length and, thus, the more powerful the mirror. Parallel rays of light reflected from a convex spherical mirror seem to originate from a well- defined focal point at the focal distance behind the mirror. Convex mirrors diverge light rays and, thus, have a negative focal length. For a convex mirror, the image formed is always virtual, regardless of the object’s position. Here’s how it behaves: Position: The virtual image appears behind the mirror. Characteristics: This virtual image is upright and smaller (diminished) than the object. Distance: The image appears closer to the mirror than the actual object distance. Points to remember 1. If di = +ve, then it is a Real image and the image will form in front of the mirror. 2. If di = -ve, then it is a Virtual image and image is formed behind the mirror. 3. Magnification m: m = +ve, Upright image m = -ve, inverted image |m| >1, then its an enlarged image(tall) |m|< 1, then its an reduced image(short) |m| = 1, Neither enlarged nor reduced. Focal length is positive for concave mirror. Focal length is negative for convex mirror. 9. Consider a concave mirror of f = 8cm, do = 24cm, ho = 4cm. Where is the image located and what is the magnification? 1/do +1/di = 1/f 1/24 + 1/di = 1/8 1/di = 1/8 - 1/24 di = 12 cm Inverted, smaller and real m = -di/do m = 12/ 24 m= -0.5 https://collegephysicsanswers.com/openstax-solutions/camera-500-mm-focal-length-lens-being-used-photograph-person-standing-300-m- away THANK YOU