11.3. Lenses. Seeing in the Dark

Transcription

1 .3 Lenses Here is a summary o what you will learn in this section: Lenses reract light in useul ways to orm s. Concave lenses, which cause light to diverge, are usen multi-lens systems to help produce s. Convex lenses cause light to converge and can be usen magniying glasses or to project s on a screen. When the object is arther away rom a convex lens than the ocal point, the is real annverted. When the object is closer to the lens than the ocal point, the is virtual and upright. igure.45 Police oicers using night vision goggles while patrolling along the St. Lawrence River Seeing in the Dark Imagine taking to the skies in a helicopter over the orests o northern Ontario. It is the middle o the night, and all you see out the window is total blackness. Your task is to ly to a remote orest location and rescue a team o ireighters needing emergency evacuation. Or picture yoursel on night patrol watching along the shores o the St. Lawrence River (igure.45). Would you be ready or such a mission? In addition to excellent training, it helps to have good equipment, including radar, radio, lights, and night vision goggles (igure.46). Modern night vision goggles are so sensitive that the tiny amounts o starlight relecting o orests can be ampliied to levels visible to pilots and rescue sta to give a clear view o the countryside. With these ultra-sensitive devices, you can literally ly and search by starlight. igure.46 Night vision goggles 448 UNIT D Light and Geometric Optics

2 Night vision goggles use lenses to ocus light onto a device called an intensiier. Inside the intensiier, the light energy releases a stream o particles. These particles then hit a phosphor-coated screen. The phosphors glow when the particles strike them. The person wearing the goggles sees a glowing green (igure.47). igure.47 The intensiier o night goggles ampliies the particles beore they hit the screen. The appears as shades o green. D2 Quick Lab Observing Lenses Purpose To observe how concave and convex lenses aect light Materials & Equipment convex lens (bulges out) concave lens (middle is thinner than the edges) light source, such as a candle screen, such as a piece o paper candle holder, such as sand and a metal tray CAUTION: I an open lame is used, it must be secured so that it cannot all over. Keep all combustible materials away rom open lames. Tie back long hair beore using an open lame. Procedure. Look though each lens at the printed text in this student book. Record your observations. 2. Look through both lenses at some printed text. Record your observations. 3. Try to use each o the lenses to project a candle lame or light onto a screen or piece o paper. Record your observations. Questions 4. Which single lens would be most useul as a magniying glass? 5. How should the convex and concave lenses be arranged to make a distant object appear closer? 6. What arrangement o lenses is most eective in projecting the o a light source onto a piece o paper? Ray diagrams model the behaviour o light in mirrors and lenses. 449

3 WORDS MATTER The word lens is derived rom the Latin word lenticula, which means lentil. A lens is in the shape o a lentil like those shown here. Types o Lenses I you have ever used a microscope, telescope, binoculars, or a camera, you have worked with one or more lenses (igure.48). A lens is a curved transparent material that is smooth and regularly shaped so that when light strikes it, the light reracts in a predictable and useul way. Most lenses are made o transparent glass or very hard plastic. These materials have several useul properties. or example, they are strong and hard. They can also be shaped and polished. By shaping both sides o the lens, it is possible to make light rays diverge or converge as they pass through the lens. The most important aspect o lenses is that the light rays that reract through them can be used to magniy s or to project s onto a screen. Relative to the object, the produced by a thin lens can be real or virtual, inverted or upright, larger or smaller. converging lenses (thickest in the middle) diverging lenses (thinnest in the middle) igure.49 Lenses can be groupento two types, converging and diverging, depending on how they reract the light that enters them. (a) (b) axis o symmetry axis o symmetry igure.50 (a) Converging lens and (b) diverging lens Lens Terminology igure.50 illustrates some o the terms associated with both converging and diverging lenses: principal axis principal axis The principal axis is an imaginary line drawn through the optical centre perpendicular to both suraces. The axis o symmetry is an imaginary vertical line drawn through the optical centre o a lens. Both kinds o lenses have two principal ocuses. The ocal point where the light either comes to a ocus or appears to diverge rom a ocus is given the symbol, while that on the opposite side o the lens is represented by. The ocal length,, is the distance rom the axis o symmetry to the principal ocus measured along the principal axis. Since light behaves the same way travelling in either direction through a lens, both types o thin lenses have two equal ocal lengths. 450 UNIT D Light and Geometric Optics

4 Concave Lenses A diverging lens is sometimes called a concave lens because it is thinner in the centre than at the edges. As parallel light rays pass through a concave lens, they are reracted away rom the principal axis. This means the light rays diverge and they will never meet on the other side o the lens (igure.5). The ormes always upright and smaller than the object (igure.52 and Table.6). igure.5 A concave lens causes light rays to diverge. igure.52 A diverging lens orms an upright, smaller. Table.6 Images ormed by Concave Lenses Distance o Object rom Lens Type o Image ormed How the Image Is Used Ray Diagram All distances Smaller, upright Some types o eyeglasses and telescopes make use o the diverging properties o concave lenses. These lenses are oten usen combination with converging lenses. object ray ray 2 Drawing a Ray Diagram or a Lens A ray diagram is a useul tool or predicting and understanding how s orm as a result o light rays emerging rom a lens. The index o reraction o a lens is greater than the index o reraction o air. This means that when a light ray passes rom air into the lens, the light ray bends, or reracts, away rom the lens surace and toward the normal. When the light passes out o the lens at an angle, the light rays reract again, this time bending away rom the normal. In other words, light rays undergo two reractions, the irst on entering the lens and the second on leaving the lens (igure.53). (a) (b) igure.53 (a) Concave lens (b) and convex lens Ray diagrams model the behaviour o light in mirrors and lenses. 45

5 In your ray diagrams in this unit, assume you are working with a thin lens. A thin lens is a lens that has a thickness that is slight compared to its ocal length. An example o a thin lens is an eyeglass lens. You can simpliy drawing a ray diagram o a thin lens without aecting its accuracy by assuming that all the reraction takes place at the axis o symmetry. object ray ray 2 igure.54 Concave lens ray diagram Drawing a Concave Lens Ray Diagram Ray diagrams or lenses are similar to ray diagrams or curved mirrors. You need to use two rays to predict location. You can ollow the steps in igure.54 to draw a ray diagram o a concave lens.. The irst ray o a concave lens ray diagram travels rom the tip o the object parallel to the principal axis (ray ). When it emerges rom the lens, it appears to come rom the principal ocus. 2. The second ray travels rom the tip o the object through the optical centre o the lens ans not reracted (ray 2). 3. Draw the virtual where the rays appear to intersect. Convex Lenses A converging lens is also called a convex lens because it is thicker at the centre than at the edges. As parallel light rays travel through a convex lens, they are reracted toward the principal axis. This causes the rays to move toward each other. The light rays cross at the ocal point o the lens. Converging lenses are oten used as magniying glasses (igure.56). igure.55 A convex lens causes light rays to converge. igure.56 A converging lens can be used as a magniying glass. 452 UNIT D Light and Geometric Optics

6 orming a Real Image Convex lenses are useul because they can orm a real on a screen. or example, the light rays coming rom one point on the lame in igure.57 diverge and strike the lens at dierent places. However, the lens redirects all those rays so that they converge at a single point. The screen must be placed so that the light rays strike it exactly as they converge. This way, when the light rays relect o the screen, they are coming rom a single point, just like when they originally let a single point on the candle. At the same time, the lens must also redirect all light rays that come rom a point at the base o the candle and send them to a single point on the screen. The rays then relect o the screen in all directions, just like when the light rays rom the base o the candle let the candle. When the rays rom every point on the candle are sent to the screen, a complete is ormed. You can compare the type o ormed at dierent distances as well as some o the uses o convex lenses in Table.7. During Reading Comparing Graphics and Text Read the paragraph on orming a real, and then look at the graphics beneath the explanation. Which was easier to understand the word text or the graphics? Would you be able to understand one eature without the support o the other, i.e., words without graphics or graphics without words? How did each graphic help you to understand the idea more ully? screen candle (upside down) Suggested Activity D24 Quick Lab on page 459 igure.57 As you can see in this illustration, there is one drawback to convex lenses. The is upside down! Table.7 Images ormed by Convex Lenses Distance o Object rom Lens Type o Image ormed How Image Is Used Ray Diagram More than two ocal lengths Smaller, inverted, real A camera uses this distance to make smaller s o an object. object Between one and two ocal lengths Larger, inverted, real Photographic enlargers, slide projectors, and movie projectors use this distance. object Less than one ocal length away Larger, upright, virtual Magniying glasses and reading glasses make use o this distance. object Ray diagrams model the behaviour o light in mirrors and lenses. 453

7 Drawing a Convex Lens Ray Diagram You can ollow the steps in igure.58 to draw a ray diagram o a convex lens.. The irst ray o a convex lens ray diagram travels rom the tip o the object parallel to the principal axis (ray ). When it emerges rom the lens, it passes through the principal ocus. 2. The second ray travels rom the tip o the object through the optical centre o the lens ans not reracted (ray 2). 3. Draw the real where the rays appear to intersect. Suggested Activity D25 Inquiry Activity on page 460 object ray ray 2 igure.58 Convex lens ray diagram Learning Checkpoint. Describe the dierence in shape between a convex lens and a concave lens. 2. Which lens, convex or concave, can also be called a diverging lens? 3. Why do light rays bend twice when lenses are used? 4. Draw a ray diagram or a convex lens when the object is situated: (a) more than two ocal lengths away rom the lens (b) exactly two ocal lengths rom the lens Thin Lens Equation The distance o the object rom the lens, d o, the distance o the rom the lens,, and the ocal length o a lens,, can all be related using the thin lens equation. Given any two o these quantities, you can use the thin lens equation to solve or the third: = + d o 454 UNIT D Light and Geometric Optics

8 Keep in mind the ollowing points when working with the thin lens equation (see also Table.8). A concave lens has a negative ocal length and a negative distance to the. A convex lens has a positive ocal length and either a positive or negative distance to the, depending where the object is placed. The distance is positive i the is real and negative i the is virtual. Table.8 Images ormed by Convex Lenses Lens Type ocal Length Distance to Object Distance to Image Convex positive positive positive or negative depending on object location Take It urther Large glass lenses can be extremely heavy. However, because reraction only occurs at the surace o a lens, the middle material can be removed, as long as the lens surace remains correctly curved. These lenses are called resnel lenses, and they are usen lighthouses and overhead projectors. ind out more about how resnel lenses are constructed and used. Begin your research at ScienceSource. Suggested Activity D26 Inquiry Activity on page 46 Example Problem.9 A convex lens o a magniying glass is held 2.00 cm above a page to magniy the print. I the produced by the lens is 3.60 cm away and virtual, what is the ocal length o the magniying glass? Given Distance o the object rom the lens, d o = 2.00 cm Distance o the virtual rom the lens, = 3.60 cm Required ocal length o the lens, Analysis and Solution The correct equation is = + Substitute the values and their units, and solve the problem. = d + o = = cm Take the reciprocal o both sides. = 4.50 cm 3.60 cm d o Paraphrase The ocal length is about 4.50 cm. Practice Problems. A powerul magniying glass produces a real 4 mm rom the convex lens. I the object was placed 28 mm away, what is the ocal length o the lens? 2. Determine the ocal length o a convex lens that produces a virtual at a distance o 30 mm when the object is placed 5 mm away. 3. The objective lens o a microscope is convex. The light rom a specimen 4.0 mm rom the lens orms a real 0.0 mm rom the lens. What is the ocal length o this lens? Ray diagrams model the behaviour o light in mirrors and lenses. 455

9 Example Problem.0 Practice Problems. A convex lens has a ocal length o 5 cm. An object is placed 20 cm rom the lens. What type o is ormed? How ar rom the lens is the? 2. A convex lens ocusses the light rom the o a bacterium that is 0.02 cm rom the lens. I the ocal length o the lens is 0.03 cm, how ar rom the lens is the? 3. A convex lens has a ocal length o 5.0 cm. I a penny is placed at the ocus, where is the o the penny ormed? A convex lens has a ocal length o 60.0 cm. A candle is placed 50 cm rom the lens. What type o is ormed, and how ar is the rom the lens? Given ocal length o the convex lens, = 60 cm Distance o the object rom the lens, d o = 50 cm Required Distance o the rom the lens, Analysis and Solution A convex lens has a positive ocal length, so =60.0 cm Use the thin lens ormula: = d + o Rearrange the ormula or d : = i d o Substitute the values and their units, and solve the problem. d = i d o = = cm cm Take the reciprocal o both sides o the equation to eliminate the ractions. = 300 cm Paraphrase Since is negative, the is virtual ans located 300 cm rom the lens. 456 UNIT D Light and Geometric Optics

10 Example Problem. A camera with a 200-mm lens makes a real o a bird on ilm. The ilm is located 20 mm behind the lens. Determine the distance rom the lens to the bird. Given ocal length o lens, = 200 mm Image distance, = 20 mm Required Object distance o the bird rom the lens, d o Analysis and Solution Use the thin lens ormula: + = Rearrange the ormula or d : = o d o Substitute the values and their units, and solve the problem. d = o d = o mm d o = mm Take the reciprocal o both sides o the equation to eliminate the ractions. d o = mm Paraphrase The birs about 40.2 m away rom the camera lens. d o Practice Problems. A convex lens with a ocal length o 8 mm produces an 25 mm rom the lens. How ar rom the lens is the object? 2. Where is the object placed i a convex lens with a ocal length o 7.00 cm produces a virtual 3.00 cm rom the lens? 3. An o a candle is produced by a convex lens 4 cm away. I the ocal length o the lens is 7.0 cm, how ar rom the lens is the candle placed? D22 STSE Science, Technology, Society, and the Environment Two-Way Mirrors A two-way mirror is partly relective and partly transparent. The relective coating o a two-way mirror is not nearly as thick on a regular mirror so that some light gets relected and some passes through. or the mirror to work properly, there must be a signiicant dierence in the brightness o light on the two sides. The side where the observers are positioned must be dark so that no light goes through the mirror to the other room. The other side, where the person being observes located, must be bright so that enough o the light is relected and the person does not see through to the other side.. How might you be able to tell i you were looking at a two-way mirror, without looking at it rom the other side? 2. Two-way mirrors are sometimes usen training hospitals so interns can watch operations being perormed. Brainstorm other situations in which two-way mirrors would be useul. 3. What issues do you think arise rom the use o two-way mirrors? Ray diagrams model the behaviour o light in mirrors and lenses. 457

11 D23 Skill Builder Activity Drawing Ray Diagrams or Convex Lenses Convex (Converging) Lenses. Copy igure.59(a) into your notebook To determine where the o the top o the arrow will be, draw the irst ray parallel to the principal axis until it strikes the lens and reracts through the ocal point. object igure.59(a) Draw the irst ray. lens 2. Draw the second ray rom the tip o the arrow through the optical centre o the lens (igure.59(b)). object igure.59(b) Draw the second ray. 3. Draw the real where the rays appear to intersect. A real is shown as a solid arrow (igure.59(c)). object igure.59(c) Draw the real. 4. Optional: You can add a third ray to check your work. Draw the third ray travelling rom the top o the arrow toward the secondary ocus on the ar side o the lens. When this ray emerges rom the lens, it travels parallel to the principal axis (igure.59(d)). object igure.59(d) Optional: Draw a third ray to check your work. 458 UNIT D Light and Geometric Optics

12 D24 Quick Lab ocal Length Purpose To ind a good approximation o the ocal length o any convex lens and see the relationship between the curvature o the lens and the ocal length Materials & Equipment ruler mm blank paper several lat convex lenses o dierent ocal lengths CAUTION: Do not shine bright light into anyone s eyes. Incandescent light sources can become very hot. Do not touch the bulbs or block air low around the light bulbs. Procedure. Copy the data table below into your notes. Give your table a title. Lens ocal Length (cm) ray box with several parallel rays o light mm blank paper Radius (cm) 2. Use your ruler to draw a straight line lengthwise across the middle o the mm paper. This line will act as the principal axis or each lens. 3. Near one end o the paper, draw a vertical line that intersects the irst line at 90. This line will act as the axis o symmetry or each lens. 4. Place the irst convex lens on the principal axis aligned with the axis o symmetry as shown in igure.60. igure.60 Step 4 5. Place the ray box o the paper so that the rays will shine parallel to the principal axis through the lens. An equal number o rays should be above and below the principal axis. 6. Look to see where the rays converge on the principal axis. I they converge above or below the axis, adjust the ray box a little to make them converge on the principal axis. 7. Place a dot at the location where the rays converge. This is the ocus. Remove the lens, then measure the distance rom the vertex, which is the intersection o the principal axis and the axis o symmetry, to the dot. Record this as the ocal length. 8. Determine the radius o the lens by placing the lens on a sheet o mm paper and tracing one side o the lens to create an arc. Reposition the lens so that the arc is extended until you have traced a ull circle. Measure the radius o this circle, and recort in your table. 9. Repeat steps 4 to 8 or the remaining lenses. Questions 0. Explain how you determined the radius o the lens once you drew the circle.. What is the relationship between curvature (radius) o the lens and ocal length o the lens? 2. Explain how you would modiy this lab to make it work or: (a) concave lenses (b) concave mirrors Ray diagrams model the behaviour o light in mirrors and lenses. 459

13 D25 Inquiry Activity Convex Lens Images Skills Reerence 2 SKILLS YOU WILL USE Conducting inquiries saely Using appropriate ormats to communicate results Question How does the distance between an object and a convex lens aect ormation? Materials & Equipment cardboard stand sheet o unlined white paper adhesive tape light bulb and socket Procedure battery and wires 2 convex lenses with dierent ocal lengths modelling clay metre stick CAUTION: Do not shine bright light into anyone s eyes. Incandescent light sources can become very hot. Do not touch the bulbs or block air low around the light bulbs.. Prepare a data table that will allow you to record the ollowing values. Give your table a title. Distance (cm) rom bulb to lens (d o ) Distance (cm) rom the lens to the screen ( ) Size o the glass part o the light bulb (h o ) Size o the on the screen (h i ) Orientation o the (inverted or upright) 2. Tape the paper onto the cardboard stand. This is your screen. 3. Measure the height o the glass part o the bulb. Record this in your notebook as the object height. 4. Determine the ocal length () o your lens as shown in igure.6: Using the modelling clay or support, place the lens in between the stand and the bulb. Move the screen and bulb slowly inward, then outward, keeping the lens in the middle. At a certain distance, an invertemage o the same size as the actual bulb will come into ocus on the screen. Measure the distance between the bulb and lens. This measurement is equal to 2. Divide the value by 2 to determine the ocal length o your lens. Record this value or in your notebook. 5. Record values in your data table or the ollowing placements o the light bulb: 2.5, 2,.5, and 0.5. In each case, you will need to move the screen until the comes into ocus beore recording your data. I at any time you cannot get an on the screen, look at the bulb through the lens. I you see an through the lens, estimate h i, but do not record a value or. 6. Repeat steps 3 to 5 with another convex lens o dierent ocal length. Analyzing and Interpreting 7. (a) Is the ormed by a convex lens always inverted? (b) I not, under what conditions is it upright? 8. (a) What happens to h i as the bulb is moved toward the lens? (b) What happens to? 9. What type o is ormed when the bulb is placed closer than one ocal length? 0. How does ocal length aect convex lens ormation? Skill Practice. Draw ray diagrams to represent the s ormed at the dierent bulb placements or both lenses. orming Conclusions igure.6 inding the ocal length o the lens 2. (a) Convex lenses are oten usen computer projectors. Explain why an changes size and must be reocussed when a projector is moved closer to or arther rom the screen. Use the data you have collecten this activity. (b) Explain why dierent projectors might have convex lenses o dierent ocal lengths. 460 UNIT D Light and Geometric Optics

14 D26 Inquiry Activity Identiying the Properties o Images Skills Reerences 2, 8 SKILLS YOU WILL USE Interpreting data/inormation to identiy patterns or relationships Identiying sources o error Question How can you use ray diagrams to determine the properties o s ormed by convex lenses? Materials & Equipment 3 sheets o legal paper sharp pencil Procedure ruler calculator. Make an observation table with the rows and columns shown at the bottom o this page. Name the table Convex Lens ocal Length 4.0 cm. 2. Turn a piece o legal paper sideways. Use the ruler to draw a horizontal line across the middle o the paper. This is the principal axis o the lens. 3. At the midpoint o the principal axis, draw a perpendicular vertical line that is 5.0 cm above and below. This is the axis o symmetry o the lens. Label the axis Convex Lens. 4. Measure 4.0 cm to the let o the vertex along the principal axis, and place a dot. This is the secondary ocus,, o the lens. Do the same thing on the other side o the lens or the primary ocus,. Label both ocuses. 5. Place two more dots at twice the ocal length, and label them 2 and 2 accordingly. 6. Repeat steps 2 to 5 twice to create a total o three convex lens diagrams. 7. Using one o the convex lens diagrams, draw a vertical arrow at d o as indicaten the Convex Lens table to the height h o. 8. Complete the ray diagram to produce the o the arrow at the appropriate place on the diagram. Label the object and the. 9. Measure and record and h i. Then, inish the next two ray diagrams or convex lenses using the values in the next two rows o the table. Analyzing and Interpreting 0. Write a general statement that relates the size, orientation, and type o to the object or a convex lens when: (a) d o is greater than 2 (b) d o is between 2 and (c) d o is less than Skill Practice. Use the data you collected to interpolate where the object would have to be placed so that the magniication o a convex lens would be exactly. 2. Calculate and h i using the thin lens and magniication equations or each set o data in the table. Compare these values to the ones you obtained using ray diagrams. Do they agree? I not, suggest reasons why the values are dierent. orming Conclusions 3. Describe how you can use ray diagrams to determine the properties o s ormed by convex lenses. Object Data Image Characteristics Image Data Calculate d o (cm) h o (cm) Side o Lens Real or Virtual Size Upright or Inverted (cm) h i (cm) h i Ray diagrams model the behaviour o light in mirrors and lenses. 46

15 .3 CHECK and RELECT Key Concept Review. (a) What type o lens produces a real? (b) What type o lens produces a virtual? 2. A converging lens produces a real 0 cm rom the lens when the object is placed 30 cm rom the lens. (a) What is the ocal length o the lens? (b) What is the magniication o the lens? 3. An object is placed at each o the ollowing distances rom a converging lens. or each location, draw a ray diagram and state the properties and location o the. (Hint: Choose a convenient value or.) (a) 2.5 (b).5 (c) Suggest one use or each lens set-up in question What is one use or a diverging lens? 6. An object.2 cm high is placed 4.0 cm rom a converging lens that has a ocal length o 3.0 cm. (a) What is the location o the? (b) What is the size o the? 7. A converging lens is placed 2 cm rom a wall chart. The ocal length o the lens is 5 cm. (a) What is the location o the? (b) What is the magniication? 8. A photographer uses his camera to view some deer in a ield. I the o the deer is produced 20.4 mm rom a convex lens that has a ocal length o 20 mm, how ar away are the deer? Question 8 Connect Your Understanding 9. At what distance rom a convex lens must an object be placed so that the is the same distance rom the lens? 0. A student examines a ladybug using a magniying glass with ocal length o 5.0 cm. He holds the magniying glass 3.5 cm above the ladybug. What is the magniication?. A photographer uses a telephoto lens to take a photograph o a 50-m high building that is 000 m away. The on the negative is 2.0 cm high. What is the ocal length o the lens? 2. How can you ind a good approximation o the ocal length o an unknown concave lens by using an unknown convex lens as a starting point? Relection 3. Why do you think it is important to understand how light travels through lenses? 4. What analogy or model could you use to remember the dierences between virtual and real s? or more questions, go to ScienceSource. 462 UNIT D Light and Geometric Optics

16 SCIENCE everywhere Sel-Adjusting Glasses The man in this photograph is wearing glasses that correct his blurry vision. The unusual thing is that he adjusted his own lenses without the help o an optometrist in determining his lens prescription. In act, over one billion people in the world do not have access to an optometrist. Low-cost glasses that can be adjusted by the wearer may be part o the solution or making good quality, low-cost vision correction a reality or the whole world. This pair o sel-adjusting glasses is constructed by sandwiching a luid-illed sac inside a lexible plastic lens. More luid means a thicker lens and a stronger correction in the glasses. To make an adjustment, the wearer uses a small syringe to add or remove the luid, and a small screw to lock the amount o luin the lens. One design hurdle is inding a way to produce the glasses in bulk cheaply. Another diiculty is the bulky rame that is currently needed. Dr. Silver and others are working to overcome these diiculties. The inventor o sel-adjusting glasses is Joshua Silver, a proessor at Oxord University in England. Dr. Silver wondered whether people could correct their own vision i they could manually adjust the ocussing power o the lenses. This would be just like anyone ocussing a camera or a pair o binoculars. By 2008, Dr. Silver had already seen pairs o his sel-adjusting glasses delivered to people in 5 developing countries. By 2020, he would like to see pairs distributed annually at a cost o less than $2 per wearer. Ray diagrams model the behaviour o light in mirrors and lenses. 463