E X P E R I M E N T 12

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1 E X P E R I M E N T 12 Mirrors and Lenses Produced by the Physics Staff at Collin College Copyright Collin College Physics Department. All Rights Reserved. University Physics II, Exp 12: Mirrors and Lenses Page 1

2 Purpose In this experiment, you will investigate image formation using lenses and mirrors. You will investigate the effects of focal length and object location on image formation and magnification. Equipment Optical bench Lens holders Vernier caliper Two convex lenses Illuminated object Screen w/ holder Concave mirror Theory What do you see when you look at the world around you? Imagine a cone of converging light rays all meeting at your eye. The cone of rays entering each eye contains millions of rays, each coming from a different point in space out in front of you. If effect, there is a 2- dimensional distribution of points, called picture elements (pixels), each providing a light ray into your eye. When you view the scene with both eyes, each eye sees a slightly different twodimensional array of pixels. Your brain creates a three-dimensional image from these two 2-D images. The rays coming from different pixels usually differ in two important properties they can have different intensities (brightness values) and different colors. The complete spatial pattern of these different rays is the picture of the world that comprises the scene in front of you. All light rays originate at luminous objects or substances. If any of the points in your field of view is luminous, the rays coming from those points originate there. Otherwise, all the rays originate somewhere else (at a light source) and are either reflected from opaque objects or transmitted through transparent objects. Your body's mechanism for seeing a scene is an array of tiny nerve cells, collectively called the retina, at the back of your eyeball. When a ray of light strikes one of those cells it sends an electrical impulse to your brain. For you to see a complete scene, all the rays coming from that scene that pass into your eye must strike your retina in the same pattern of intensities and colors as they had when they left the scene. In the study of geometric optics, the pattern of rays leaving the scene is called the object, and the pattern striking the retina is called the image. So the image must contain the same spatial pattern of brightness and color values as the object does. The lens at the front of your eyeball manipulates all the rays entering your eye to make the two patterns match. The lens is said to create an image of the object. University Physics II, Exp 12: Mirrors and Lenses Page 2

3 But exactly how are images created? Consider the light rays coming from any portion of an illuminated scene. If you place a white screen so that it intercepts some of the rays coming from the scene, all you will see on the screen is a fairly uniform distribution of brightness and color no pattern or image. But if you hold a lens between the scene and the screen and move it to a particular distance from the screen, the rays that pass through the lens will strike the screen in the same pattern as the scene itself; i.e., the lens creates an image of the object scene. Many natural and man-made objects can manipulate light rays to create images. Those that create images with reflected light rays are called mirrors. Those that use transmitted rays are called lenses. We are all so accustomed to seeing images created by mirrors and lenses that we rarely stop to consider how they do it. But it is a fascinating process. To examine how a mirror or lens creates an image, it is instructive to: (1) consider one ray at a time; a single ray coming from a single point on the object and creating a single point on the image, and (2) note that for your eye lens to create an image on your retina (for you to see scenes), the rays entering your eye must be diverging like the three rays coming from the tip of the candle flame in Figure Figure 12.1 Of course, the divergence angle of the rays becomes quite small when the object or image is far away; but the angle is zero only when the object is at infinity. In your textbook, you saw how the familiar flat (plane) mirror creates an imaginary (virtual) image. It is imaginary because the rays entering your eyes do not actually pass through the image; they only appear to come from it; the mirror fools your brain. In this experiment, you will examine images created by curved mirrors and lenses. University Physics II, Exp 12: Mirrors and Lenses Page 3

4 Spherical and Cylindrical Mirrors The images created by a curved mirror depend on whether the observer and object are outside or inside the curvature. If you are outside the curved surface the mirror is called convex; if you are inside the curvature it is called concave. In most curved mirrors the radius of curvature is constant, i.e., the curve is a section of a circle. If the curvature is in only one dimension, the mirror surface is cylindrical. If the curvature is in two dimensions, the mirror surface is spherical. In either case, the center of curvature is the key to analyzing the image created by the mirror. To analyze the reflection of rays from a curved mirror, refer to Figure First draw a 2-D projection of the curved surface of the mirror (a portion of a circle). Next, draw a diameter extending on both sides of the circle (this is called the optic axis of the mirror). Figure 12.2 Now, mark the position on the axis of the center of curvature C on both sides of the mirror. Finally, mark two more points on the axis midway between the mirror surface and each C. These last two points are called the mirror's focal points F. The focal length of the mirror f is half its radius of curvature: f R 2 Equation 12.1 Now consider an object located a distance d o from the mirror, beyond the center C. The object in Figure 12.3 is an arrow extending up from the axis. Follow three single rays coming from the arrow's tip (object point) as they reflect from the mirror: 1. Ray (1) passes through the center of curvature C of the concave mirror, or is on a path toward the center of curvature of the convex mirror. Because its path goes through C, it will strike the mirror at a right angle. The reflected ray (1) will therefore follow the same path back. 2. Ray (2) is parallel to the axis. It will reflect from either mirror along the path (2) that passes through the focal point F of the concave mirror, or appears to pass through the University Physics II, Exp 12: Mirrors and Lenses Page 4

5 focal point of the convex mirror. This is because the incident and reflected rays make equal angles with the radius of curvature at the point of incidence. 3. Ray (3) passes through the focal point F of the concave mirror, or is on a path toward the focal point of the convex mirror. It will reflect from either mirror along the path (3) that is parallel to the axis. Again, the incident and reflected rays make equal angles with the radius of curvature at the point of incidence. Figure 12.3(a) All three of these reflected rays intersect at a single point. This is the image point (the tip of the arrow). The three rays diverge as they continue beyond this image point. If you place your eye in that region (beyond the image), the diverging rays will create another image of the arrow s tip on your retina. All other rays coming from the same object point (the arrow s tip) and striking the mirror will intersect at that same image point. So the mirror has created an image of the arrow s tip. In a like manner, the rays coming from any other object point (any other point on the arrow) that strike the mirror will reflect and intersect at image points between the axis and the image of the arrow tip. So every point on the object has a corresponding image point. The mirror has created a complete image of the object. Several characteristics of this reflection imaging process are worth special notice: 1. All incident rays parallel to the axis that reflect from a concave mirror, as in Figure 12.3(a), will converge and intersect at the focal point. A concave mirror is therefore called a converging mirror. 2. All incident rays parallel to the axis that reflect from a convex mirror, as in Figure 12.3(b), will diverge as if they were coming from its focal point. A convex mirror is therefore called a diverging mirror. 3. All incident rays passing through (or on a path toward) a focal point are reflected from either mirror in a direction parallel to the axis. 4. All incident rays passing through (or on a path toward) a center of curvature are reflected from either mirror back through (or on a path toward) the center point. University Physics II, Exp 12: Mirrors and Lenses Page 5

6 5. If the rays that eventually enter your eye actually diverge from the image, it is called a real image. If they only appear to be diverging from the image, it is called a virtual image. 6. If the image has the same orientation as the object, the image is said to be erect. If it has the opposite orientation, it is said to be inverted. 7. If the image is larger than the object, it is said to be magnified. If it is smaller, it is reduced. Figure 12.3(b) By using similar triangles formed by rays, you can show that the image distance from the mirror d i is related to the object distance d o by Equation 12.2 di do f R where f is the focal length and R is the radius of curvature. So if you know the focal length and the object distance, you can solve Equation 12.2 for the image distance: do f di Equation 12.3 d f o By the same similar triangle analysis, the magnification M is given by di M Equation 12.4 do The sign conventions for these quantities in mirrors and lenses are summarized below: University Physics II, Exp 12: Mirrors and Lenses Page 6

7 Quantity Conditions Sign Focal length f Concave mirror + Convex mirror Concave lens Convex lens + Object distance d o All objects + Image distance d i Real image + Virtual image Magnification M Erect image + Inverted image Magnified image > 1 Minified image < 1 Spherical and Cylindrical Lenses All lenses (both convex and concave) have two radii of curvature (one for each surface) called R 1 and R 2. The two radii may be equal to or different from each other. Each surface has a center of curvature but these points are not as important in locating lens images graphically as they were with mirrors. Every lens has a focal length f, the same distance from the center plane of the lens on each side even if the radii are different. As it does in mirrors, the focal length of a lens depends on the values of the radius. But unlike mirrors, it also depends on the index of refraction of the lens material n (since rays pass through the lens material). The relationship is 1 f n 1 1 R 1 1 R 2 Equation 12.5 where R is positive for convex surfaces and negative for concave surfaces. For most lens materials, the value of n is between 1.4 and 1.8. For a symmetrical lens (R 1 = R 2 = R) with n = 1.5, Equation 12.5 yields f = R. The focal length of other symmetrical lenses may be less or greater than R (depending on the value of n). A convex lens is called converging because rays parallel to the axis will converge at the focal point on the image side, as shown in Figure 12.4(a). A concave lens is called diverging because these parallel rays appear to diverge from the focal point on the object side. Figure 12.4(a) University Physics II, Exp 12: Mirrors and Lenses Page 7

8 Figure 12.4(b) To use ray tracing techniques to analyze lens images, draw a figure analogous to Figure 12.2 with a lens instead of a mirror. Then consider an object located a distance d o from the central plane of the lens, beyond the focal point F as in Figure The object is again an arrow extending up from the axis. Again follow three single rays (slightly different from the mirror s rays) coming from the arrow s tip as they refract through the lens: Ray (1) passes through the geometric center of the lens so that its total refraction angle is zero for either type of lens. It will continue along the same path projected through the lens. Ray (2) is again parallel to the axis. It will refract through the lens along the path (2) that passes through the opposite focal point F of the convex lens, or appears to pass through the object-side focal point of the concave lens. This essentially defines the positions of the focal points. Ray (3) passes through the focal point of the convex lens, or on a path toward the opposite focal point of the concave lens. It will refract through the lens along the path (3) that is parallel to the axis. This is again based on the definition of the focal points. All three of these refracted rays intersect (or appear to intersect) at a single point the image point of the object point (the tip of the arrow). As they continue beyond (or appear to emanate from) this image point, the three rays continue to diverge. If you place your eye in that region, these diverging rays would create another image of the arrow's tip on your retina. All other rays coming from that same object point and passing through the lens will intersect at that same image point. So the lens has created an image of the arrow s tip. In a like manner, the rays coming from all other object points (any other points on the arrow) that transit the lens will refract and intersect at corresponding image points between the axis and the arrow tip image. So every point on the object has a corresponding image point. The lens has created a complete image of the object. With obvious differences, the same characteristics of the mirror-imaging process apply to the lens imaging process: 1. All incident rays parallel to the axis that refract through a convex lens, as in Figure 12.4(a), will converge and intersect at the opposite focal point. Thus, this lens is called converging. 2. All incident rays parallel to the axis that refract through a concave lens, as in Figure 12.4(b), will diverge as if they were coming from its object-side focal point. Thus, this lens is called diverging. University Physics II, Exp 12: Mirrors and Lenses Page 8

9 3. All incident rays passing through the near focal point of a convex lens, or on a path to pass through the opposite focal point of a concave lens, are refracted through either lens in a direction parallel to the axis. 4. All incident rays passing through the geometric center of either lens are not refracted and continue along the same path. 5. If the rays that eventually enter your eye actually diverge from the image, it is called a real image. If they only appear to be diverging from the image, it is called a virtual image. 6. If the image has the same orientation as the object, the image is said to be erect. If it has the opposite orientation, it is said to be inverted. 7. If the image is larger than the object, it is said to be magnified. If it is smaller, it is reduced. As for spherical mirrors, the object and image distances for a lens, d o and d i, are related by Equation 12.6 di do f So if you know the focal length and the object distance, you can find the image distance with Equation And the magnification M is still given by Equation Procedure Equipment Setup To determine the characteristics of mirrors and lenses in the laboratory, you use an optical bench a long, flat bed with a magnetic surface (it is magnetic in order to secure the lens/mirror holders firmly). The lens/mirror holders can carry a mirror or lens, a lighted object, or a screen on which you can project and view the image. A metric scale is attached to the side of the optical bench to help you measure object and image distances. You will first find the focal length of a concave spherical mirror. Once you know the focal length, you place the mirror at one end of the bench and the object at the other end. You then move the screen until an image is formed on it. A. Focal Length of a Concave Spherical Mirror 1. Place the spherical mirror in the mirror holder and clamp it at one end of the optical bench, recording its position in your lab notebook. Make sure that the concave surface faces the other end of the bench. 2. Place the lighted object on the other end of the optical bench, making sure that the axes of the object and the mirror are aligned. 3. Hold the white screen above the lighted object such that the screen and object are in the same plane. 4. Move the mirror toward the object/screen until you see a sharply focused image on the screen. 5. Record the object distance d o and image distance d i (in centimeters) in Table University Physics II, Exp 12: Mirrors and Lenses Page 9

10 6. Calculate and record the radius of curvature R and focal length f of the mirror. Calculate and record the percent error between your measured focal length and the actual focal length of the mirror (this should be written on the mirror). B. Image Formed by Concave Mirror 7. Return the concave mirror to the end of the optical bench. 8. Place the lighted object at the other end of the bench. Measure and record the object distance in Table Place the screen on the bench facing the mirror (you may need to hold the mirror to the side of the bench to avoid blocking the lighted object). 10. Move the screen toward the mirror until you see a sharp image on the screen. Record this experimental image distance d i in Table Calculate and record the theoretical image distance d i and the % error. 12. Using a vernier caliper, measure and record the object height h o and image height h i. Calculate and record the experimental and theoretical values of the magnification M and the % error in M. 13. Reduce the object distance by 20 cm, and repeat steps C. Focal Length of Converging Lens 14. Position the optical bench so it is pointing at a bright object on the other side of the lab room. 15. Clamp the screen at the end of the optical bench away from the object. Record its position in Table Place the 15 cm focal length convex lens in the lens holder and set the lens holder on the bench in front of the screen. 17. While looking at the screen, move the lens toward or away from it until you see a sharp image of the distant object on the screen. Read and record this position of the lens on the bench. The distance between the lens and the screen is the experimental focal length of the lens. 18. Record the values of both the actual (written on the lens) and experimental focal lengths, and calculate and record the % error. 19. Repeat steps for the 7.5 cm focal length convex lens. D. Converging Lens Image for Various Object Distances 20. Position the optical bench parallel to the length of the lab table. 21. Place the 7.5 cm focal length convex lens in the lens holder and clamp the lens holder in the middle of the optical bench. 22. Place the lighted object at one end of the bench. Adjust the vertical position of the object and/or lens so that they all have the same optic axis. Record the object distance d o in Table University Physics II, Exp 12: Mirrors and Lenses Page 10

11 23. Place the screen at the other end of the bench. Move it until you see a sharp image of the object on it. Clamp the screen in this position. Record the experimental image distance d i. 24. Calculate and record the theoretical value of the image distance. Then calculate and record the % error. 25. Using a vernier caliper, carefully measure and record the size of the object h o and of the image h i. Calculate and record the theoretical and experimental values of magnification M and the percent error in M. 26. Move the lens 10 cm toward the lighted object. Repeat steps Repeat steps with the 15 cm focal length convex lens. University Physics II, Exp 12: Mirrors and Lenses Page 11