hapter 13- Refraction and Lenses We have already established that light is an electromagnetic wave, so it does not require a medium to travel through. However, we know from the personal experience of being able to see the world around us that light can travel through a transparent medium, such as air, glass, water, etc. Just as the speed of sound varied depending on the medium it was traveling through, the speed of light will vary with different mediums. Remember that c is technically the speed of light in a vacuum (3 x 10 8 m/s) which is the condition where light will travel the fastest. However, in the presence of medium, light will actually slow down. Note that c remains the maximum speed of light, as light will never travel faster in a medium that will in a vacuum. If light is passing through a single uniform medium, let s say air, thet will always travel in a straight line. However, when the light arrives at the surface of a different medium, it may continue in a different path depending on the characteristics of the second medium. If the medium is opaque, or not transparent, then the light will not be able to travel into that medium very far. An example of an opaque surface could be a glossy tile floor. In this case, some of the light may be absorbed by the second medium (the tiles), but most of the light will be deflected back into the air. This redirection of light that causes it to change direction back into the first medium (the air) is called reflection. If the second medium is also transparent, then some of the light will be transmitted through that medium. As light passes from one medium to another, such as from glass to air, the change in speed from one medium to the other will actually cause the light to bend. The bending of a light wave across the boundary between two mediums is called refraction. In this chapter, we will examine reflection and refraction detail, and then consider some practical applications of these phenomena. Refraction When light reflects off of a surface (like a mirror), it is really reflecting off of the boundary between two mediums. In the case of the wall mirror, the two mediums are air and the mirrored surface. However, sometimes the light does not just reflect off the second medium, but it actually is transmitted through it. Think of looking through a window the light is actually going through two boundaries as it goes from air into the glass, and then back into the air on the inside of the house. In the beginning of this chapter we mentioned that as light travels in different mediums, it will have different speeds. This causes light to bend as it travels from one medium (at one speed) to a second medium (at another speed). This bending of light is called refraction. Let us look at it in more detail. Law of Refraction The index of refraction (represented with the variable n) is the ratio of the speed of light in a vacuum to the speed of light in another medium. We can calculate n for a medium following the equation: n c v where c is the speed of light in a vacuum and v is the speed of light in the other medium. Since n is c divided by another speed, it never have a unit (the two m/s cancel out) and it will always be greater than 1 (since light is fastest in a vacuum). Below you will find tables of the indices of
refraction for multiple substances. Note that the index of refraction for air is very close to the index of refraction for a vacuum. When solving most problems with air, it is acceptable to use n = 1.00. Indices of Refraction for Various Substances Solids at 20 n ubic zirconia 2.20 Diamond 2.419 luorite 1.434 used quartz 1.458 Glass, crown 1.52 Glass, flint 1.66 Ice (at 0) 1.309 Polystyrene 1.49 Sodium chloride 1.544 Zircon 1.923 Liquids at 20 n Benzene 1.501 arbon disulfide 1.628 arbon tetrachloride 1.461 Ethyl Alcohol 1.361 Glycerine 1.473 Water 1.333 Gases at 0, 1 atm n Air 1.000 293 arbon dioxide 1.000 450 The values in the tables above are only valid for visible light with a wavelength of 589 nm in a vacuum. This is because light with different wavelengths will bend differently when entering a new medium. In the visible spectrum, blue light, which has a higher wavelength, bends more than red light, which has a lower wavelength. You may have observed this before without realizing it, if you have ever seen white light separated by a prism into a rainbow. Each color has a different wavelength, and as each wavelength is bent at different angles, you see each color as leaving the prism at a different point, allowing you to see the colors individually, rather than together, which would appear white. Snell s Law In order to measure how much a light wave is bent as it travels from one medium into another, we will not only be concerned with measuring the angle of incidence ( i) as with mirrors, but we will also measure the angle of refraction ( r). onsider the example of light traveling from air into water, as illustrated to the right. The light enters the water at an angle I away from the normal to the surface. However, once the light goes through the surface of the water, its path is changed to follow the straight line at an angle r from the normal. The values of these two angles can be used to compare the indices of refraction for these two substances using Snell s Law: si n r sin r Recall that the frequency of the light does not change as it goes from one medium into another. When a medium has a greater value for n, the light will bend more. It follows that if the speed of light decreases, such as whet travels from air into water, the light will be bent
towards the normal. If the speed of light increases, such as from water into air, the light will bend away from the normal. Let s look at an example. Example: Red light with a wavelength of 690 nm travels from air into crown glass (n = 1.52) at an angle of 30 degrees. ind the angle of refraction. Solution: si n r sin r 1 sin30 1.52sin r sin r 1 sin30 r 19.2 1.52 How to catch a fish: We now know that light will bend at a greater angle from the normal in air than water. Imagine a fish that is swimming below the water. If you are spear fishing, do you want to aim directly at the place where you see the fish? N! As the light reflected from the body of the fish crosses the water-air boundary, the light bends away from the normal. When that light reaches your eye, your brain does not account for the fact that the light is bent differently in the water that is in the air, so you will see the fish as appearing on the line that is made by connecting the point of your eye to the point where the light left the water. See the apparent fish in the image above. ollowing the law of refraction, we can see that the light from the fish originated from a position closer to the normal. In reality, the fish would be closer to the normal that appears, and you would need to aim below and in front of it to catch it. ish are not the only objects that appear to be in a different location due to the refraction. All objects beneath the surface of the water will seem to be farther in front of you and fairly close to the surface of the water, when reality they are closer and deeper. The result may cause streams and ponds to appear to be much more shallow than they actually are. Atmospheric Refraction It is very obvious that water and air are two different mediums. However, we should realize that a single medium, such as air, may not be completely uniform. Recall that temperature effects the density of air. As the density of air changes, light will bend at different amounts as it travels through air pockets with varying densities. Pressure (altitude), humidity, and other factors may also play a role in causing light to bend differently. This non-uniform bending of light is known as atmospheric refraction. In this section, we will examine three consequences of atmospheric refraction. 1. Light After Sunset
If you have ever observed a sunset from the beach, you may have noticed that the sun appears to waver and stretch as it falls below the horizon. As the light from the sun travels towards your eye, it travels a longer length through the atmosphere that would whet is directly overhead. What you may not expect is that at the point that you see the Sun disappear, it has actually already fallen below the horizon! See in the diagram to the right, that if the light traveled in a straight line from the setting sun, it would actually not reach your eye. However, due to atmospheric refraction, the light rays actually bend around the surface of the earth toward your eye, allowing you to see a prolonged sunset. Note that S is the position where the setting sun actually is, but due to the bending light, you see the sun as still being at position S. 2. Dispersion The process of separating white light into the separate colors that it is composed of is known as dispersion. When Isaac Newton first separated white light into colors using a prism, he demonstrated this principle of dispersion and proved that the prism was not creating colors from white light, but rather was separating what was already there. Dispersios most commonly seen in nature in rainbows, which are created through a combination of refraction and reflection of sunlight in water droplets. When light from the sun enters a water droplet it is refracted, with the violet light bending the most, and red bending the least. Since different colors bend different amounts, they will hit the back of the water droplet at slightly different points. As the light reaches the backside of the water droplet, some of it is reflected back into the water, rather than being transmitted out into the air. This reflected light is then refracted even more as it travels back to the front of the water drop. By the time the light leaves the water droplet, the frequencies of light will be traveling along different paths, with violet light at about 40 below its original path, and red at about 42. The rainbow that you see is actually formed by many water droplets, each separating the light in this same way. A droplet that is higher in the sky will allow you to see red light, which leaves the droplet at greater angle than violet light. The violet light that leaves this droplet will pass overhead unnoticed because it does not reach your eye. However, a water droplet that is lower in the sky will reflect violet light towards your eye, while the red light from that droplet will hit the ground. This allows you to see the violet light, but not the red light from that second drop. The result is the image of a rainbow that you are familiar with, with red light coming from the top, violet from the bottom, and the rest of the spectrum in between. Since a light is reflected of the back of the water droplets to make a rainbow, you will most often see rainbows when the Sus behind you and there is a rain cloud in front
of you. This allows the light to come from overhead, refract in the water droplets, and reflect back to your eyes. It is possible for double rainbows to occur when there is a double reflection the water droplets. The second rainbow will occur on top of the first rainbow, and will be inverted, with red appearing on the bottom, and violet on top. Some people have reported seeing triple and even quadruple rainbows. 3. Mirages Mirages occur due to a transfer of heat, generally in conditions where the ground is significantly warmer than the air above it, such as in the desert or on a highway on a hot sunny day. Some light rays (A) will travel directly from an object to the observer parallel to the ground, while others travel towards the ground (B). The light rays along this path are refracted as they go from cooler to warmer air, which bends those rays back up to the observer as well. These refracted rays are inverted, which means the observer sees an upright image from the (A) rays, and anverted image from the (B) rays. Where else do we see this in nature? As a reflection off of a pool of water! Since our experience tells us that seeing this double image indicates there is a reflection from water, we can be tricked into thinking that we see water off on the horizon. While you may not have personally experienced the mirages of a desert, think about the wet spots that appear to be visible on the road during a hot sunny day. As you approach them, and the angle that you are observing the light changes, you can see that there really is not a puddle, but you had been seeing the blue rays refracted from the sky. Total Internal Reflection When a light ray moves from a medium with a higher index of refraction to a lower index of refraction, the path that the light will take after encountering the boundary between the two mediums will depend on the angle of incidence. onsider the example of light that is coming from water and going into air. If the light hits the boundary head on, (parallel to the normal), it will be transmitted through the boundary in a straight line. As the angle of incidence increases, the light will be refracted more and more. At some angle, called the critical angle, the refracted light will move parallel to the boundary between the two mediums. or angles of incidence greater than the critical angle, the light ray will actually reflect back into the first medium, rather that transmitting into the second, similar to if it had hit a perfectly reflective surface. At angles above the critical angle, the law of reflection (from the mirror section) applies. This means that the angle of reflectios equal to the angle of incidence. In the image above, you can see the reflection of the turtle on the boundary between the water and air. The water has a higher
index of refraction than the air, and the photo is taken at an angle greater than the critical angle. Snell s Law can be used to determine the critical angle if we use 90 for the angle of reflection: sin c n r sin90 where c represents the critical angle. This equation can be simplified: sin c n r 1 sin c n r Remember that this only applies when > n r. Total internal reflection occurs in prisms when light reaches the boundary between the prism and the air around it. When arranged at the proper angle, prisms can be used in place of mirrors in optical equipment, which is a desirable application, as they reflect light more efficiently and are often more durable than mirrors. In the image to the right, you can see the light reflecting multiple times inside of the prism before exiting out the end. Diamonds are particularly valued, in part, because of their sparkle. Diamond has a high index of refraction and a critical angle of about 24. Since this critical angle is so small, most of the light that enters into a cut diamond is totally internally reflected. The light will bounce off multiple surfaces before eventually exiting the diamond through the visible faces. Diamonds are engineered so that the maximum amount of light entering the diamond is reflected back to the face of the diamond, giving the observer the brightest sparkle possible. Lenses As light enters a pane of glass, it is bent toward the normal. As the light leaves the glass, it is bent again, this time away from the normal. Since the amount of refractios the same for the light entering the glass as it is leaving the glass, the light will leave the glass pane parallel to its original path before it entered the glass, but shifted slightly sideways. The size of this shift will vary with the thickness of the glass and the indices of refraction for the two mediums (in this case the glass and air). By curving the surface of the glass pane, we can alter where the outgoing light rays will be directed. bjects that use this principle to change where light will be refracted are called lenses. Simple lenses can be categorized into two groupsconverging and diverging lenses. You can see a concave lens and a convex lens in the image to the right.
Ray Diagrams for Lenses Ray diagrams for lenses are quite similar to those of mirrors, in that you will draw three rays to predict where the image will appear. Remember that all rays are drawn from the top of the object to the respective points. The point where at least two rays intersect will be where the top of the image forms. or a onverging Lens, you will draw the following three rays: 1. A ray parallel to principle axis into the lens, then through the focal point on the other side of the lens (called a Parallel Ray) 2. A ray through the focal point into the lens, then out of the lens parallel to principle axis (called a ocal Ray) 3. A ray through the center of the lens that continues in a straight path (called the entral Ray) onverging Lenses Similar to mirrors, the images produced by lenses will vary depending on the position of the object relative to the lens. A lens can produce either a real or a virtual image, depending on the location of the object. Below you will find ray diagrams for five different situations. 1) An object that is beyond the center of curvature of the lens will produce anverted real, smaller image between and on the opposite side of the lens. This is the situation that you would find in a camera lens or the lens of the human eye. utside I 2) An object at the center of curvature () will produce anverted, real image of the same size at the center of curvature on the other side of the lens. This arrangement is used in some telescopes, where the image from another lens acts as the object for this lens. At I 3) An object between and produces anverted, real magnified image beyond the center of curvature on the other side of the lens. This can be seen movie projectors, which require the image to be formed on a screen a great distance away from the lens.
Between and I 4) An object at produces amage at infinity, similar to the situation of a concave mirror with an object at the same distance. Since the rays never intersect, no image will be formed. Lighthouses and searchlights are able to project light great distances using this arrangement, where the light bulb is used as the object. At 5) An object at a point closer to the lens than will produce a magnified virtual image on the same side of the lens. Binoculars and microscopes use this arrangement to allow you to see magnified views of an object, at great distance or small scales, respectively. loser than I 6) A sixth situation could occur if the object were an infinite distance from the lens. In this case, there would be a point image at on the other side of the lens. We see this application with a magnifying glass, which when held up on a sunny day, will form a point image of the Sun. The opposite is also true if the magnifying glass is held so that a small object is located at the focal point, then you will see an enlarged version of that image, where the light rays spread out towards your eye following the same pattern as the light rays were focused from the Sun to a point. Never try to combine these two applications! Looking at the Sun with any kind of optical magnification can be dangerous! Diverging Lenses Unlike the many situations possible with converging lenses, a diverging lens will always produce a virtual, upright, and smaller image. The most common uses for diverging lenses are correcting vision with eyeglasses or contacts, certain telescopes, telescopic camera lenses, and projectors. Equations for Lenses
The equations for lenses are the same as those for mirrors, but we will review them and look at the applicable conventions. The Thin Lens Equation: 1 d o 1 d i 1 f The sign conventions that we use for lenses are as follows: Values in front of the lens are positive Values behind the lens are negative f is positive for a converging lens and negative for a diverging lens The Magnification Equation: M d i d o A positive value for M indicates an upright and virtual image A negative value for M indicates anverted and real image If M > 1, the image is enlarged If M < 1, the image is smaller orrecting Eyesight Using Lenses The cornea of the human eye is actually a converging lens that allows light to be focused on the retina at the back of the eye. When a persos able to see things clearly from far away, but is unable to see objects up close, we say that they are farsighted. arsightedness is caused by an abnormality in the eye called hyperopia, where the eyeball is either too short, or the muscles aren t able to properly focus the image from a nearby object, causing the image to form behind the retina. While the light does not actually go behind the retina, when the rays hit the retina, they have not yet intersected to form a clear image. Hyperopia can be corrected using a converging lens, which further refracts the light allowing the image to focus directly on the retina. A person who can see an object up-close but has difficulty focusing on an object far away is said to be nearsighted. This is caused by the abnormality myopia, which is when the eyeball is too long or the maximum focal length of the lens is too small to produce a clear image on the retina. In this case, the image forms in front of the retina, so it can be corrected by using a diverging lens. If a person uses glasses or contacts, converging or diverging lenses will be used to correct either nearsightedness or farsightedness. Whether you wear glasses, contacts, or have 20-20 vision, remember that whenever you see something, it is the result of lenses in action!