50 #22 Lenses and Mirrors Objective To study characteristics of lenses and mirrors and the images they produce. Introduction and Theory Thin Lenses The formation of images by lenses is one of the most important studies in the field of optics. The purpose of this exercise is to observe the real images formed by various lenses. When a beam of rays parallel to the principal axis of a converging lens impinges up- on the lens, it is brought together at a point called the principal focus of the lens. The dis- tance from the principal focus to the center of the lens is called the focal length of the lens, f. The focal length is positive for a converging (convex) lens and negative for a diverging (concave) lens. The relation between the object distance, o, the image distance, i, and the focal length, f, of a thin lens is given by the lens equation 1 1 1 + O i f 6-7 (22.1) and is shown in Fig. 22.1. The magnification, M, produced by a lens (i.e., the linear magnification) is defined as the ratio of the size of the image, I, to the size of the object, O. This can be shown to be equal to the ratio of the image distance to the object distance. Therefore image size image distance Magnification = object size object distance or > i | M1 = = = (22.2) Images will be positive if they are upright and negative if inverted. Therefore, a positive magnification indicates an upright image while a negative magnification indicates an inverted image. The principal focal length of a converging lens may be determined by forming an image of a very distant object on a screen and measuring the distance from the screen to the lens. This distance will be the focal length, since the rays of light from a very distant object are very nearly parallel. A more accurate method of determining the focal length of a converging lens is to measure the image distance corresponding to a suitable and known object distance, and to calculate the focal length from the lens equation above. 51 Object Focal Point Image 0 Fig. 22.1. Schematics for the optical bench, showing the relation between o, i, and f for a real image. When two thin lenses are in contact, the equivalent focal length of the combination may be measured experimentally by one of the above methods. It may also be calculated in terms of the individual focal lengths. Thus 1 1 1 + (22.3) f fi f2 }, where fis the equivalent focal length of the lens combination, f, is the focal length of the first lens, and f2 is the focal length of the second lens. A concave lens by itself cannot form a real image since it is a diverging lens. Hence, a different method must be used for measuring its focal length. This is done by placing the negative lens in contact with a positive lens of shorter and known focal length, measuring the equivalent focal length of the combination experimentally, and then using Eq. 22.2 to solve for the focal length of the negative lens. Mirrors no A spherical mirror is a small section of a spherical shell. If the mirror surface is on the same side as the center of curvature or the sphere, it is called a concave mirror. If, however, the mirror surface is on the outside of the spherical shell, it is called a convex mirror. When a beam of light from an infinitely distant object is incident upon a spherical mirror, it is either converged to a real focus in front of the mirror (concave mirror) or di- verged to that it appears to come from a virtual focus behind the mirror (convex mirror). The principal focus of a mirror is defined as the point through which a bundle of rays par- allel to the axis of the mirror pass, or appear to pass, after reflection. The distance from the reflecting surface to the principal focus is called the focal length. A real focus is one through which the beam of light actually passes, while a virtual focus is one to which the light only appears to go. The corresponding images are likewise referred to as real and virtual. Eqs. 22.1 and 22.2 hold for mirrors as well as for lenses. When a thin lens and a mir- ror are in contact, the equivalent focal length of the combination may be measured exper- imentally by one of the above methods. It may also be calculated in terms of the individu- al focal lengths. Thus - 1 2 1 + f fi f2 (22.4) where fis the equivalent focal length of the lens/mirror combination, fı is the focal length of the lens, and f2 is the focal length of the mirror. Experimental Procedure 1. Measure the focal length of lens A directly by obtaining the image of a very distant object on the screen, and measuring the image distance. The object may be a tree or building more than a block away. Repeat for lens B, and for the lens combination BC. 2. Repeat step 1 for concave mirror D, placing the concave surface of the mirror toward the infinitely distant source and the "L"-shaped screen at a slight angle to the rays com- ing from the source so that the reflected rays can focus on the screen. Repeat for the lens/mirror combination BE. 3. Place the illuminated object at one end of the optical bench. Place the screen at a dis- tance from the object of about four or five times the focal length of the lens. With the object and screen fixed, find the position of lens A for which a sharp, enlarged image is produced on the screen. Make sure that the object, lens, and screen all lie along the same straight line, the principal axis of the lens, and that they are all perpendicular to the axis. Record the positions of the object, the lens, and the screen to an accuracy of one millimeter. Measure the size of the object and the size of the image and record these measurements. 4. Switch the positions of the object and image leaving the lens in place. Do you notice any differences? If so, record them. 5. Repeat steps 3 and 4 for lens B and for the lens combination BC. 6. Place the object behind the “L”-shaped screen. Starting with the mirror very close to the source of light, slide the mirror along the bench away from the source until a sharp im- age of the arrow is formed on the opaque part of the screen. (It may be necessary to tilt the mirror slightly so that the image is focused on the screen.) Note and record the dis- tance between the source and the mirror (object distance) and the distance between the screen and the mirror (image distance) as well as the sizes of the object and image. 7. Next move the mirror as far away from the source as possible. Approach the position of sharp definition of the image from the opposite direction and again record the data as in step 6. 8. The convex mirror E cannot be used to form a real image so step 1 can not be used to find its focal length directly. Observe the image of yourself in the convex mirror and record your observations of the reflections for different distances to the mirror. 9. Repeat step 6 for the lens/mirror combination BE. 53 Analysis of Data 1. Use Eq. 22.1 to calculate the focal length of lenses A, B, and combination BC from the data taken in steps 3-5. Compare these values with the value obtained in step 1. 2. Calculate the magnification for each of the lenses from the measured image and object distances. Calculate the magnification for each of the lenses from the measured image and object sizes. Compare the results. 3. Comment on the differences observed when o and i are reversed. 4. Calculate the focal length of diverging lens C from Eq. 22.3 using the average focal length of lens B and the average focal length of the lens combination BC. 5. Use Eq. 22.1 to calculate the focal length of the concave mirror D and the combination BE from the data taken in steps 6 and 9. Compare these values with those obtained in step 2. 6. Calculate the magnification for the concave mirror and the combination BE from the measured image and object distances. Calculate the magnifications from the measured image and object sizes. Compare the magnifications obtained using both methods. 7. How does the data for mirror D from Procedure step 7 compare with that of step 6? 8. Calculate the focal length of the convex mirror E from Eq. 22.3 using the average focal length of lens B and the average focal length of the lens/mirror combination B/E. Be- fore you begin substituting numbers, carefully picture the experimental procedure so that you substitute the correct values. Explain why the lens must be accounted for twice. 9. Comment on the image formed with the convex mirror E. 10. Summarize your results in a table as shown on the next page. Conclusion Comment on the agreement of the calculated focal lengths of lenses A and B and mirror D with those measured directly. Why is it not possible to measure the focal lengths of lens C and mirror E directly? How did reversing the source and screen for the lenses affect the image size? Also comment on the agreement of the magnifications calculated by two different methods. What are the principal sources of error in this experiment?
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