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Properties of Lenses and Telescopes Overview: SAFETY WARNING: NEVER LOOK AT THE SUN THROUGH A TELESCOPE. In this laboratory exercise we will investigate the behavior of lenses and mirrors that are used in optical telescopes using a basic telescope kit. When you finish this exercise you will learn how a real image is formed by a lens and measure the focal length of the lens. We will also learn how a virtual image is formed. We will relate the size and distance of the image from the lens to the size and distance of the object from the lens. We will learn how to construct ray diagrams and determine the magnification that results when two lenses of different focal lengths are used to form a telescope. Finally, the increase in light gathered with the kit telescope will be computed. Learner objectives: Make observations with an astronomical telescope. Understand the operation of the parts of the telescope. Make quantitative measurements of image formation by the lenses of the telescope and compute its magnifying power from these measurements. Measure the magnification and compare to the computed power. Materials Needed: You will need: The telescope kit “Star and Planet Locater” Ruler with cm scale. Masking and transparent tape (and also possibly paper glue) Long sturdy rubber bands Three boxes of aluminum foil, small 25 square feet size the same thickness as the larger tube. Image Formation by a Converging Lens In Figure 3.1 a light ray is shown entering a prism-shaped piece of glass. The dotted lines are at a 90 degree angle to the surface at the point where the ray enters and leaves the glass. Note how the ray of light is bent TOWARD this line as it enters the glass. If you stick a drinking straw or pencil into a cup of water, the pencil will appear bent by the same process which is called refraction. This bending happens each time a light ray travels from a medium of one density into another of different density (in this case air and glass). Figure 1. Bending of a light ray by a prism Now note how the ray is bent AWAY from the 90 degree line as the light leaves the glass on the other side of the prism. Finally, you can seehow the beam had a net change in direction downward after it left the prism. Similarly, if the prism were turned upside down, the beam would be bent upward. This net bending depends on the angle between the two faces of the prism. The larger it is the more the beam is bent. If the two faces are parallel, no net change in direction occurs (like light passing through a window pane). Now a convex lens is much like two such prisms placed base to base. We sketch a convex lens in Figure 2 with three parallel rays entering the lens at the top, middle and bottom. Parallel rays like this are typical for extremely distant astronomical sources. Figure 2 Focal point of a convex lens The top and bottom rays are bent down and upward as mentioned earlier. The middle ray hits the portion of the lens with parallel faces that does not bend the ray at all. If the lens is made correctly, all the parallel beams cross one another at a single point, F, called the ``Focus'' (We will use these two facts later). For initially parallel rays, the distance from point F to the center of the lens is called the ``focal length'' (f). A white card (screen) held at point F will show a point of light. This point of light is called a ``real image'' because light rays cross at that location. Lenses that are fatter in the middle which bring light rays to a focus as shown are often called “converging” lenses. As shown in Figure 3 below, there are two lenses with the telescope kit, you purchased for this course. Both are converging but one bulges out more in the middle than the other. Figure 3. Telescope kit with smaller “fat” and larger “flatter” lenses plus tubes. “Fat” is eyepiece lens. “Flatter” is objective. Figure 4 Set-up studying focal length and image. Several thick books, a ruler, and a small Post- it note pad were used. Photographed outside using the Sun. Use an indoor set-up with an illuminated window on the other side of the room to measure image size of and whether the image is up-right etc. You may not need books on the left if the source is horizontal. Take the flatter, wider of the pair of convex lenses and examine it. If a lens is used to form an image of something infinitely far away, the distance from the lens to the image is defined to be the focal length, f. For now, infinitely far will mean the view through a window>>f. Use the lens to form an image of the view through a window. To do this, hold the lens in front of a blank page on a small pad until a sharp image is formed on it as shown in Figure 2 and 4. Read off the focal length using the scale on the ruler as shown with the ruler pointing at the source. Put zero cm at the pad location and make sure the pad is 90o to the ruler to read off the focal length from the position of the lens. Also do this for the smaller “fatter” lens. Focal Length Section. See if you can answer these questions and measure this data. First “fat” lens data: Focal length in cm= fest,1 = ____________ Do the best you can for the “fat” lens. Is the image right side up?__________ Measure the image size.__________ Is the image real ? _______ Is the image right-side up or up-side down?____________________ Second larger “flat” lens data: Focal length= fest,2 = ____________ Is the image right side up?_________ _ Measure the image size.__________ Is the image real ? _______ Is the image right-side up or up-side down?____________________ 1. Which lens in the Telescope kit had the shorter focal length, the “fat” lens or the “flatter” less bulgy lens? a) fat or more bulgy b) flatter or less bulgy c) Both the same focal length. 2. Which lens in the Telescope kit had the larger image of the distant object? a) the long focal length lens. b) the shorter focal length lens. c) There was no difference because both had the same focal length. 3. Relative to the original distant object or light source, what was the image orientation when you formed them with the Telescope kit lenses? a) right-side up (same orientation as original) b) up-side-down (inverted relative to original) c) sideways (turned through 90 degrees relative to original). Non-celestial, everyday point light sources such as a candle flame or light bulb are not at an infinite distance from the lens. In Figure 5, we sketch three rays from a point source that is not at a large distance from the lens but is farther than the focal length, f, from the lens. Figure 5. Image and Object Distances The middle beam traveling along the symmetry axis of the lens is unbent and the other two are bent to bring the point light source to a focus as before. However, with the parallel rays in Figure 4, the image was at the focal length, f, from the lens. In this case the incoming rays are diverging and not parallel so the lens forms an image a distance larger than f. The increase in the distance of the focal point if the object is closer has an effect on our vision. The human eye can adjust its lens shape to accommodate an object being closer but there is a limit. For most people, the closest distance to comfortably view an object is about 25 cm (unless one is very near sighted). Image formation processes Now there can be, of course, more than one source of diverging light rays and they do not have to be on the central line of symmetry of the lens. In Figure 6, a lens is shown imaging two sources of light which also could correspond to points on the ends of an arrow with the head on top at O and the tail below at O’. Following the dashed beams from O, the middle beam passing through the center parallel faces of the lens is not bent. However, the other two are bent so that all three come to a focus I below the axis of the lens. By tracing the three solid rays from the tail, you can see that those from the tail O’ below are brought to a focus I’, above the axis i.e., the point images are upside down or inverted. By imagining light coming from points all along the arrow, Figure 6 shows how an inverted image of an extended object like an arrow is formed. Figure 6. More than one point source or sources at different locations on an extended object.. Solid rays come from lower end of objet; dashed from upper end. They converge on opposite (reversed) ends of the image as shown. An observer near the axis of the lens and on the same side of the lens but beyond the focus of the rays will see what appears to be an arrow extending from above the lens axis to a point below. A white card at the focus will display an arrow to an observer off to one side. This type image where rays cross is called a “real” image. Drawing and Using Ray Diagrams The method used in Figure 6, a ray diagram, is a useful way of checking some of the results you observe with lenses in this exercise. The question is, when light from an object passes through a converging lens, how and where does the image form? Figure 7 shows how you use knowledge of the focal length, f, of the lens to make this judgment. Points F and F' are at a distance equal to f from the lens. We have also drawn an arrow with the tip of the arrow marked O. The position of the image can be located using two of three rays from each point on the object as illustrated in Figure 7 for point O. Figure 7. Ray diagram. Carefully referring to Figure 7, to find the image position of any other point on the arrow you need to draw two rays with the third serving as a check, where the three rays are defined as follows: (i) Draw a ray from the point on the object to the lens parallel to the axis of symmetry of the lens. After passing through the lens this ray must pass through F. In Figure 7 this is line OL. (ii) Draw a ray from the point “O” on the object through the point F' toward the lens. After passing through the lens this ray must emerge on a path parallel to the symmetry axis of the lens. The intersection of the two rays locates the image position of a point. In Figure 7 point O on the arrow is images at point I behind the lens. (iii) A third ray which will also locate the image position of a point can be drawn from the point on the object to the center of the lens. It will emerge from the lens with no change in direction. Note this in Figure 7. (d) The intersection of this central ray with either of the two previous rays will locate the image of the point on the object. As you can see from Figure 7 all three rays from point O intersect at point I, but only two of the three need to be drawn to locate point I. When the rays cross, a real image is formed. A screen at crossing would show the image. Figure 8 Ray diagram for the upper point of the object O and lower part of image I. Space is left to complete the diagram for lower part of object O” and upper part of image I”. See questions below. Try answering the following questions. Refering to the figure above, pick a, b, c, d and e answers which match in Figure 8 and questions 4 and 5. 4. Partially construct the upper end of the image to the right of a lens due to an upwadly pointing vertical arrow outside the focal length to the left of a convex converging lens. Draw a ray from the low tail of the arrow to the lens parallel to the axis of symmetry of the lens. This ray must pass through a point _______ a) one focal length on the side of the lens toward the arrow. b) in the middle of the lens. c) one focal length from the lens on the side away from the arrow. d) in the upper half of the lens but not as high as the arrow point. e) in the lower half of the lens below below the middle a distance equal to the arrow tail. 5. Assume a vertical arrow (pointing upward) to the left of a convex converging lens outside the focal length. One step in using a ray diagram to construct the upper end of the image to the right of the lens is to draw a ray from the lower tail toward the lens parallel to the axis of symmetry of the lens. After passing through the lens, this ray must pass through a point a) one focal length on the side of the lens toward the arrow b) in the middle of the lens c) one focal length from the lens on the side away from the arrow 6. Assume a vertical arrow (pointing upward) to the left of a convex converging lens outside the focal length. If you draw a ray from the end of the arrow point through the focal point one focal length from the lens toward the arrow, the ray will encounter the lower half of the lens. After passing through the lens, this ray must emerge on a path _______ a) parallel to the symmetry axis of the lens b) 90 degrees to the axis of symmetry of the lens c) downward through the focal point on the opposite side of the lens. 7. A ray which will help locate the image position of a point can be drawn from the lowest point on the object to the center of the lens. It will emerge from the lens _________________ a) parallel to the symmetry axis of the lens b) 90 degrees to the axis of symmetry of the lens c) proceeding in the same direction that it entered the lens. 8.For an object to the left of a convex lens farther than the focal length, the image is found ___________ where the light rays cross. a) to the right of the lens b) to the left of the lens c) at the center of the lens The Telescope Eyepiece (Magnifier) and a Virtual Image In astronomical telescopes the front lens or mirror forms an image of a celestial object as we have observed. This front objective lens is called the “objective” lens or mirror. In many cases a CCD (like in a camera) is placed where the real image is formed. In most every-day telescopes/ the real image is viewed by a person using an eyepiece lens which magnifies the image revealing small details. This is simply the familiar magnifying glass. Larger aperture objective shows finer detail than the smaller aperture eye alone. Typically, one moves the eyepiece moved inward and outward until the eye comfortably views the image through the magnifier i.e. the object is seen sharply. This arrangement is shown in Figure 9 of an objective lens and magnifying eyepiece in a telescope. Typically the observer’s view is sharpest and most comfortable when the eye piece is somewhat closer to the intermediate real image than the eye piece’s focal length. In Figures 6, 7 and 8, the object is farther than the focal length from the lens. If the object is closer than the focal length it is not possible to form a real image that can be seen on a screen. Figure 10 shows the situation in a telescope where the real image “source” is closer than the focal length from the eyepiece lens. Figure 9. Arrangement for a visual telescope. From left: object, objective lens which forms a real image.. , real image formed by it and magnifying eyepiece, then finally the observer. The virtual image created by the eyepiece is seen. The f’s of objective and eyepiece are shown. Image or object at comfortable 25 cm on objects very far away Figure 10a. Typically, the eye can comfortably focusviewing distance. It down to about 25 looks closer. The small angular size is indicated by lines to the point and tail. cm but not tiny. Convex lens permits bringing the image or object closer. Ray diagram shows how giant virtual image at 25 cm results. Figure 10b Magnified virtual Image ray diagram. Convex lens permits bringing the image or object closer than 25 cm. Ray diagram shows how a giant virtual image at 25 cm results. The object is closer to the lens than the focal length. F. In this case, the lens is unable to make all the rays passing through the lens converge after passage through it, and no image is formed that can be projected on a screen on the right side of the lens. Figure 10 shows a ray diagram of the formation of a virtual image of an object closer than the focal length F. In particular two red rays are shown from the tail of the arrow, one from the bottom through the center and another from the tail parallel to the axis which is bent slightly upward to pass through F on the observer side. The back extrapolation of these two red lines diverge from a point about 25 cm from the observer. So as the red and black lines show, an observer can see an enlarged or MAGNIFIED image with the ends of the image arrow being farther from one another than the ends of the REAL arrow. In other words, the eye sees an apparently larger image because it perceives the final direction that the light rays come from, and has no awareness of any bending that might have happened to the rays on their way from the object to the eye. Because the light rays do not actually cross in this type image and an image cannot be formed on a screen this type of image is called a VIRTUAL image. Try this yourself. Place the Telescope kit short focal length convex lens on, say, a printed page or other object and move it outward with your eye at a constant distance from (fairly close to) the lens Note when the magnified image appears and its qualitative properties when the lens is less than one focal length away from the page. Continue to move the lens away and note its distance from the paper when the image ceases to be magnified.. Eyepiece View Section. Close-up view: “Fat” eyepiece lens data: Focal length= fest,fat = ____________ Is the image real or virtual? _______ Is the image right-side up or up-side down?____________________ Use these results above to try answering the following question. Refer to the figure above and pick a, b, c, and d answers. 9. Assume you have a magnifier, such as the telescope short focal length convex eye piece lens which you want to use to get a magnified view of a printed page. The eypepiece lens placed, close to, a printed page, should show a (an) _______ magnified image of the page if the lens is within ________ of the printed page. a)upright; 3 times the focal length b) upright, 2 times the focal length c) upright: somewhat less than the focal length. e)inverted; less than the focal length . Telescope Optical Assembly and Arrangement of Telescope: Now construct your telescope following the directions in the kit. Refer to Figure 3 earlier and Figure 11 below. You can follow the printed assembly instructions exactly if you wish using glue but note that “thin” and “thick” refer to the width of the lenses. Here is a a simplified assembly procedure not requiring glue that you may wish to use. Slide the smaller eye piece tube into the larger objective tube. Slide the yellow mount tubes onto the black tube ends almost all the way so they stick out about ¼ inch. Holding the larger black tube vertically with the yellow tube at the top, drop the wider, flatter objective lens into the opening. Push in the two black rings to keep the objective lens from falling out. Slide the yellow tube in until the rings are flush with the outer end of the yellow tube. As shown in Figure 11, secure the rings with small tabs of tape that don’t touch the lens along with a tabs to keep the yellow tube from sliding. Do the same for the eyepiece end using the fatter eyepiece lens and blue rings. Again, secure the rings and yellow tube by small tape tabs. When you look into the eye piece, your eye should be back from the eye piece a small amount to maximize clarity. Now that you have constructed your telescope, point the long focal length objective lens end of the tube away from you toward the object to be observed. Turn short focal length eyepiece lens end of tube toward you. Set up as shown <) ()______________________()______________ Object Eye Eyepiece Magnifier Objective Lens The eyepiece magnifies the real image formed by the objective. Starting with a distance between the eyepiece and objective a distance apart slightly less than fe + fo , slide the objective toward the eyepiece while looking through the eyepiece at the end. Stop when the distant object comes comfortably into focus. The eyepiece acts as a magnifier of the objective image. Examine the image carefully. Figure 11. Assembled telescope plus simple mount using three small aluminum foil boxes and two large rubber bands. Objective is in red fitting slipped over cardboard tube. Eyepiece is at other end of smaller tube inserted into objective tube. See if you can answer the following questions. Refer to Figure 9 and your observations through the telescope. 10. The telescope objective lens in the “Telescope” module produces a (an)________ image. a)upright; real b) upright, virtual c) inverted; real e)inverted; virtual . 11. Assume you have a magnifier, such as your kit telescope eye piece lens which you use with an objective lens like the one in your kit to observe an object. Using this telescope, through the eye piece, you see a (an)_______image of the object. a)upright; real b) upright, virtual c) inverted; real e)inverted; virtual Magnification of Telescope: Use a page with equally spaced marks fastened to a wall with masking tape. A brick wall or fence with boards or bricks all the same width or height would also work. A pattern to use for the magnification exercise is on the last page of this module. Pointing your telescope at the wall, look through it with one eye and keep the other eye open. Alternative blinking is another possibility. Count the number of divisions seen without the telescope inside one or two or three etc. as seen through the telescope Example: The horizontal bars are divisions of the scale. Telescope Eye alone (not looking through the telescope) ----------- ------------ ------------ ----------- ------------ ------------ ------------ ------------ In our graphic example above we see that two magnified divisions corresponds to four unmagnified divisions. Therefore, 4/2 = 2x angular size magnification. Record your results below. Magnification = (# without telescope)/(# with telescope) = (_____________) / (____________) = _________ The theoretical magnification of a telescope is approximately equal to the focal length of the objective divided by the focal length of the eyepiece. Record the objective lens focal length fo = ___________________(units?) from your earlier measurement. Record the eyepiece focal length fe = ___________________(units?) from your earlier measurement. Compare your result above to the theoretical telescope angular magnification mag = fo/fe = _____________ / _____________ = _________________ See if you can answer the following question now. 12. If the objective of a lens telescope has a focal length of 60 cm and the eyepiece a focal length of 10 cm, what is the magnification of the telescope a) 10 cm/ 60cm = 1/6 b) 10cm x 60 cm = 600 c) 60 cm/ 10 cm = 6 d) (60 cm/10 cm)2= 36 Change in Objective Size: You know the telescope objective forms a real image. Now mentally see if you can predict what will happen if you cover up half the objective. See if you can fill in the answers below about “Light Gathering Power.” Now, looking through the telescope eyepiece with it focused on a distant object, cover up first ¼ then ½ the objective. Describe how the image changed as you covered up more of the objective. __________________________ Considering how the image changed, explain why this happens. ______________________ The amount of light gathered by a telescope objective depends on its area. Measure the diameter of your telescope’s objective lens in mm or cm, diameter =_________ Compute the radius, r = (diameter/2) =____________________ Compute area of objective = πro² = _______________ The radius of the opened iris of the dark adapted eye is about 3 mm or 0.3 cm. Compute the light gathering power of your telescope compared to the eye = πro²/(πre²) = (ro/re) ² = _____________________ From these results explain why astronomers build telescopes with 10 m diameter objectives to explore deep into space. Now see if you can answer the following question. 13. If the objective of a lens telescope has a diameter of 6 cm and the dark adapted eye 3 mm What is the light gathering power of the telescope compared to the eye ? The telescope’s light gathering power is _______ the eye. Note that 3 mm=0.3 cm. a)1/20 b)20 x c)400 Summary and Additional Remarks. We have defined focal length of a convex converging lens and measured it for the lenses of our telescope kit.. Ray diagrams show how a convex lens forms a real inverted image if the object distance is longer than the focal length. The image is larger the longer the focal length. This is the situation for a telescope objective lens examining an object at great distance. For astronomical observations, usually a CCD or other device is placed at the focus to record the real image. The real image formed by the larger objective contains more detail than the smaller eye alone can reveal. For more every-day observations, a magnifying eyepiece is placed just beyond the real image a distance less than its shorter focal length. We show with ray diagrams how the eyepiece forms an enlarged virtual image. The magnification of the telescope can be computed from the focal lengths of the objective and eyepiece. The light gathering power of our telescope depends on the area of the objective thus revealing fainter stars than the eye alone can see.. Again, a pattern to use for the magnification exercise is on the last page of this module. Later we will learn how to make observations with our telescope or your own binoculars or telescope. You can go ahead and try it on your own. The skymaps.com web site has lists of binocular objects that your kit telescope will also reveal under dark sky conditions. Many of these e.g. M42 Orion Nebula, M31 globular star cluster, M31 Andromeda Galaxy are marked on your Star and Planet locator.
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