10.2 Images Formed by Lenses SUMMARY. Refraction in Lenses. Section 10.1 Questions

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Dominick Price
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1 10.2 SUMMARY Refraction in Lenses Converging lenses bring parallel rays together after they are refracted. Diverging lenses cause parallel rays to move apart after they are refracted. Rays are refracted at the surfaces of lenses according to Snell s law. Section 10.1 Questions Understanding Concepts 1. igure 3 shows light rays in air approaching three glass lenses. Draw diagrams of the same shape, but larger in size, in your notebook and draw the approximate direction of each light ray as it travels into the glass and back into the air. (Hint: Draw the appropriate normals wherever a ray strikes a surface.) 2. Draw a large double concave lens, then draw three parallel rays on one side of the lens. Apply the method you used in question 1 to show that light rays emerging from the lens diverge. Extend the rays straight back until they meet, or almost meet. Label this point appropriately. (a) (b) (c) igure 3 or question 1 Applying Inquiry Skills 3. Pretend that the fingers of your wide-open hand, when spread out flat on a piece of paper, represent diverging light rays that originate at a single virtual focal point. Devise a way to measure the distance from the focal point to the tip of your middle finger. Making Connections 4. The lenses in your science classroom are likely double convex and double concave lenses. These are not suited for eyeglasses. Why? Which lenses in igure 2 are best suited for eyeglasses? Why? 10.2 Images ormed by Lenses In all lenses, the geometric centre is called the optical centre (O), as shown in igure 1. A vertical line drawn through the optical centre is called the optical axis (OA) of the lens. A horizontal line drawn through the optical centre is optical centre: (O) the geometric centre of all lenses optical axis: (OA) a vertical line through the optical centre principal focus f focal length optical axis (OA) focal plane igure 1 Lenses and the Eye 357

2 principal axis: () a horizontal line drawn through the optical centre principal focus: ( ) the point on the principal axis through which a group of rays parallel to the principal axis is refracted focal length: (f ) the distance between the principal focus and the optical centre, measured along the principal axis focal plane: the plane, perpendicular to the principal axis, on which all focal points lie light rays from distant igure 2 f focal plane l called the principal axis (). If a lens is thin, a group of rays parallel to the principal axis is refracted through a point on the principal axis called the principal focus ( ). The focal length (f ) is the distance between the principal focus and the optical centre, measured along the principal axis. A beam of parallel rays that is not aligned with the principal axis converges at a focal point that is not on the principal axis (igure 1). All focal points, including the principal focus, lie on the focal plane, perpendicular to the principal axis. When a converging lens refracts light from a distant (igure 2), the rays arriving at the lens are nearly parallel; thus, a real is formed at a distance close to one focal length from the lens. Since light can enter a lens from either side, there are two principal foci.the focal length is the same on both sides of the lens, even if the curvature on each side is different. The notation is always given to the primary principal focus, the point at which the rays converge or from which they appear to diverge; the secondary principal focus is usually expressed as. Characteristics of an Image Once an has been observed or located in a ray diagram, we can use four characteristics to describe it relative to the : magnification, attitude, location, and type of the. Magnification of the Image The height of an is written as h o ; the height of the is written as h i. To compare their heights, the magnification (M) of the is found by calculating the ratio of the height to the height: hi M = h o Note that magnification has no units because it is a ratio of heights. Attitude of the Image The attitude of the refers to its orientation relative to the. or example, when an forms on film in a camera, the is inverted relative to the that was photographed. An is either upright or inverted, relative to the. Location of the Image The distance between the subject of a photograph and the lens of a camera is the distance, designated by d o. An of the is formed on the film inside the camera. The distance between the on the film and the lens is the distance, d i. The is located either on the side of the lens, or on the opposite side of the lens. When discussing and distances, we refer to the location as between and the lens, between and 2, or beyond 2. point: point at which light from an point converges Type of the Image An can be either real or virtual. A real can be placed onto a screen; a virtual cannot. Recall that light diverges from a real point (igure 3(a)). An optical device can converge light from an point to a point called the point. A diverging beam from this point must enter your eye in order for you to see the point. Such an point is called a real point. By using a screen to scatter the light from the real point, the is visible from many angles. An optical device can also change the direction of a diverging beam from an point so that the rays appear to diverge from behind the point 358 Chapter 10

3 10.2 (a) point diverging beam converging beam diverging beam real point 2' ' 2 rom here, the real point can be viewed in space. (b) rays traced backwards virtual point diverging beam from lens point 2' ' 2 diverging beam from point Virtual point viewed from here. igure 3 (a) Real point (b) Virtual point (igure 3(b)). Such an point is called a virtual point. The virtual point is located by extending the rays backward until they intersect. Since the light itself does not intersect at the virtual point, a virtual cannot be formed or captured on a screen. Whenever extending rays backward to locate virtual points, use dashed lines on your ray diagrams. virtual point: point from which rays from an point appear to diverge Practice Understanding Concepts 1. Create a chart, table, or detailed diagram to summarize the symbols and meanings of these properties or variables of lenses: optical centre, optical axis, principal axis, principal focus, focal length, distance, height, distance, height, magnification. 2. The word MALT can be used as a memory aid for the four characteristics of s. Print the word vertically. Add the complete word that corresponds to each letter. In each case, list the possible choices used to describe an. Images ormed by Converging Lenses An gives off light rays in all directions, but, for the purpose of locating its, we are only interested in those rays that pass through the lens (igure 4). 2' ' igure 4 Three rays are particularly convenient for locating points, since they either pass through the lens reference positions and or are parallel to the principal axis. Once points are located, we can predict the four characteristics of the. Lenses and the Eye 359

4 (a) (b) The following are the three rays and their rules that can be used: igure 5 (a) (b) lateral displacement OA actual path of light ray PHY11_U4_10.2.4a OA ray diagram Rules for Rays in a Converging Lens 1. A light ray travelling parallel to the principal axis refracts through the principal focus (). 2. A light ray that passes through the secondary principal focus ( ) refracts parallel to the principal axis. 3. A light ray that passes through the optical centre goes straight through, without refracting. Note: Only two rays are needed to locate an point. The third ray can be used as a check of accuracy. It may seem strange that the ray that passes through the optical centre in igure 4 is not refracted, since most rays passing through the optical centre are laterally displaced. However, in thin lenses, the lateral displacement of the ray is so small that we can assume that the ray is not refracted (igure 5). In both diagrams in igure 6, a construction line has been drawn through the optical centre perpendicular to the principal axis. The actual path of the light ray is indicated by a solid line. An can form either in front of a lens or behind it, and measurements are made either above or below the principal axis. Therefore, we need a sign convention to distinguish between real and virtual s and to interpret magnification calculations. Sign Convention 1. Object and distances are measured from the optical centre of the lens. 2. Object distances are positive if they are on the side of the lens from which light is coming; otherwise they are negative. 3. Image distances are positive if they are on the opposite side of the lens from which light is coming; if they are on the same side, the distance is negative. (Image distance is positive for real s, negative for virtual s.) 4. Object heights and heights are positive when measured upward and negative when measured downward from the principal axis. Using this convention, a converging lens has a real principal focus and a positive focal length. A diverging lens has a virtual principal focus and a negative focal length. The orientation of the is predicted using the sign convention. Magnification is positive for an upright and negative for an inverted. igure 6 or simplicity, when drawing ray diagrams in lenses, we can represent all the refraction of light as occurring just once along the optical axis instead of twice at the curved surfaces of the lens. This results in the same location. Sample Problem 1 A 1.5-cm-high is 8.0 cm from a converging lens of focal length 2.5 cm. (a) Draw a ray diagram to locate the of the and state its attitude, location, and type. (b) Measure the height from your diagram, and calculate the magnification. Solution (a) igure 7 shows an arrow resting on the principal axis. Three incident rays and three refracted rays are drawn, according to the rules for converging lenses. The resulting, seen in the ray diagram, is real and inverted. The is located on the opposite side of the lens between and Chapter 10

5 10.2 ' 2 (b) h o = 1.5 cm h i = 0.70 cm hi M = h o = cm 1. 5 cm M = 0.47 (negative because it was measured below the principal axis) The magnification of the is Note that a negative magnification is consistent with both the sign convention and the ray diagram. igure 7 The position of an relative to a lens affects how the is formed. There are five cases for a converging lens (igure 8). (a) beyond 2' (d) at ' 2' ' The is: between and 2, real, inverted, smaller than the. (b) at 2' 2' ' The is: at 2, real, inverted, the same size as the. (c) between ' and 2' 2' ' The is: beyond 2, real, inverted, larger than the ' No is formed because the refracted rays are parallel and never meet. (e) between lens and ' ' The is: behind the, virtual, erect, larger than the. igure 8 Lenses and the Eye 361

6 Practice Answers 3. (a) M = 1.0 (b) M = 1.6 (c) M = 2.0 Understanding Concepts 3. A 15-mm-high is viewed with a converging lens of focal length 32 mm. or each distance listed below, draw a ray diagram using all the rules to locate the of the. State the characteristics of each, including the magnification for (a) d o = 64 mm (b) d o = 52 mm (c) d o = 16 mm Making Connections 4. Below is a list of optical devices that match the arrangements of lenses and s in igure 8. or each of the following cases, explain where the lens must be placed relative to the : (a) a copy camera produces an that is real and the same size (b) a hand magnifier produces an that is virtual and larger (c) a slide projector produces an that is real and larger (d) a 35-mm camera produces an that is real and smaller (e) a spotlight produces parallel light; there is no (f) a photographic enlarger produces an that is real and larger Images ormed by Diverging Lenses In a diverging lens, parallel rays are refracted so that they radiate outward from the principal focus ( ) as shown in igure 9. focal plane principal focus (virtual) diverging lens igure 9 A diverging lens f (focal length) The rays used to locate the position of the in a diverging lens are similar to those used with converging lenses (igure 10). As a result, one set of rules is used for all lenses. The important difference is that the principal focus in the converging lens is real, whereas in the diverging lens it is virtual. igure 10 Rays refracting through a diverging lens As with converging lenses, we assume with ray diagrams in diverging lenses that all refraction occurs at the optical axis of the lens. This makes the ray diagram easier to draw. ' 362 Chapter 10

7 10.2 Rules for Rays in a Diverging Lens 1. A light ray travelling parallel to the principal axis refracts in line with the principal focus ( ). 2. A light ray that is aimed toward the secondary principal focus ( ) refracts parallel to the principal axis. 3. A light ray that passes through the optical centre goes straight through, without refracting. igure 11 illustrates the formation of an by a diverging lens. or all positions of the, the is virtual, upright, and smaller. The is always located between the principal focus and the optical centre. igure 11 Image formation in a diverging lens Sample Problem 2 A 1.5-cm-high is located 8.0 cm from a diverging lens of focal length 2.5 cm. (a) Draw a ray diagram to locate the of the and state its attitude, location, and type. (b) Measure the height from your ray diagram, and calculate the magnification. Solution (a) The refracted rays must be extended straight back to where they meet on the side of the lens where the is located, as in igure 12.The resulting is virtual, upright, and located between and the lens. ' igure 12 Lenses and the Eye 363

8 (b) h o = 1.5 cm h i = 0.40 cm hi M = h o = 0.40 cm 1.5 cm M = 0.27 (positive because it was measured above the principal axis) The magnification of the is Practice Answers 5. (a) M = 0.33 (b) M = 0.50 Understanding Concepts 5. A 12-mm-high is viewed using a diverging lens of focal length 32 mm. Using all the rules, draw a ray diagram to locate the of the for each situation listed below. State the characteristics of each, and write a conclusion describing what happens to the magnification of the viewed in a diverging lens as the distance decreases for (a) d o = 64 mm (b) d o = 32 mm Questioning Hypothesizing Predicting Planning Conducting INQUIRY SKILLS Recording Analyzing Evaluating Communicating Investigation Images of a Pinhole Camera, Converging Lens, and Diverging Lens The principle of the pinhole camera was described early in the 10th century by the Egyptian scholar Alhazen, who used it to indirectly view a solar eclipse. The purpose of this Investigation is to study the s formed by a pinhole camera and compare them with s produced by lenses. Question What are the differences among s formed by a pinhole camera, a converging lens, and a diverging lens? Materials opaque screen with a pinhole (pinhole camera) small light source (miniature light bulb) translucent screen (white paper) converging lens (f = 20 cm) diverging lens (f = 20 cm) metre stick optical bench Prediction (a) Predict the differences among the s formed by the pinhole camera, the converging lens, and the diverging lens. 364 Chapter 10

9 10.2 Procedure 1. Position the light source at the end of the optical bench (igure 13). viewing direction 1 50 cm opaque screen with pinhole or a lens (at midpoint) 35 cm (translucent screen) viewing direction 2 2. Place the opaque screen 50 cm away from the light source. 3. In a dark part of the room, place the translucent screen about 35 cm from the opaque screen. Slowly move the translucent screen forward and backward. Make observations from both viewing directions at different positions along the principal axis. Record what happens to the. 4. Remove the translucent screen and make observations from both viewing directions, as shown in igure Replace the opaque screen with a converging lens. 6. (a) Replace the translucent screen and move it until you get a relatively sharp on the screen. It may help to angle the converging lens up slightly. Slowly move the screen forward and backward along the principal axis. Record what happens to the. (b) Cover the upper half of the lens with a piece of paper. Observe and record the effect on the. Cover the left half of the lens. Observe and record the effect on the. 7. Repeat step Replace the converging lens with a diverging lens. 9. Repeat step Place the converging lens between the opaque screen and the light source. Observe the resulting with a screen. Move the screen back and forth along the principal axis. Change the relative positions of the lens, pinhole, and screen to obtain the sharpest possible. Record the positions of the pinhole, lens, and screen. Analysis (b) Using the evidence you have collected, answer the Question. Your explanation should include neatly drawn ray diagrams and a discussion of linear propagation, formation of real and virtual s, and scattering of a real by a screen. (c) Which arrangement of pinhole camera, lens, and screen produced the sharpest? Evaluation (d) Evaluate your predictions. (e) Describe the sources of error in the investigation and evaluate their effect on the results. Suggest one or two improvements to the experimental design. igure 13 Setup for Investigation Lenses and the Eye 365

10 SUMMARY Images ormed by Lenses An can be described by four characteristics: magnification, attitude, location, and type. The magnification of an is the ratio of the height to the height. Images are real or virtual; real s can be scattered by a screen and viewed from many angles. Section 10.2 Questions Understanding Concepts 1. When a converging lens is used as an ordinary magnifying glass, is the it produces real or virtual? Include a sketch with your explanation. 2. Copy igure 14 into your notebook and draw the path of each ray after it has been refracted igure Copy igure 15 into your notebook and draw the path of each ray after it has been refracted igure Images in a diverging lens are always virtual. When an is placed between the focus and optical centre of a converging lens, the is also virtual. What are the differences between these virtual s? Explain by studying the four characteristics of each. 5. Use ray diagrams to explain why a half-covered lens still produces a complete of an. 6. or each situation described below, draw a ray diagram to locate the of the, then describe the four characteristics of each. (In each case, try to use all three rules for drawing ray diagrams.) (a) f = 3.0 cm; d o = 7.5 cm; h o = 2.0 cm (b) f = 4.4 cm; d o = 2.2 cm; h o = 1.2 cm (c) f = 2.8 cm; d o = 5.0 cm; h o = 2.2 cm 366 Chapter 10

11 10.3 Applying Inquiry Skills 7. (a) Draw a ray diagram of your own design of a variable-length pinhole camera made of common household materials, such as cardboard tubes or shoe boxes. (b) How would you use your design to determine the relationship between the magnification of the and the distance between the pinhole and the? 10.3 Mathematical Relationships for Thin Lenses You have studied how a converging lens produces s for light sources placed at different positions along the principal axis. Now you will study the quantitative relationship between and positions. Investigation Predicting the Location of Images Produced by a Converging Lens The purpose of this investigation is to determine the relationship among the focal length, the distance, and the distance of a converging lens. Questioning Hypothesizing Predicting Planning Conducting INQUIRY SKILLS Recording Analyzing Evaluating Communicating Question What is the relationship among the focal length, the distance, and the distance of a converging lens? Materials converging lens (f =10 cm 25 cm) small light source (miniature light bulb) translucent screen (white paper) optical bench Prediction (a) Predict what will happen to the size and distance of an as an is moved closer to a converging lens. Procedure Part 1: Real Image 1. Create a table similar to Table In a dark part of the room, hold the lens so that light from a distant passes through it and onto the screen, as in igure 1. Move the screen back and forth until the is clearly focused. Measure the focal length, f, of the lens, the distance between the lens and the screen. Lenses and the Eye 367

Lab 11: Lenses and Ray Tracing

Name: Lab 11: Lenses and Ray Tracing Group Members: Date: TA s Name: Materials: Ray box, two different converging lenses, one diverging lens, screen, lighted object, three stands, meter stick, two letter