THE TELESCOPE OBJECTIVE: As seen with the naked eye the heavens are a wonderfully fascinating place. With a little careful watching the brighter stars can be grouped into constellations and an order seen in what at first appears to be a chaotic, myriad jumble of lights in the sky. With this order in the sky there often comes a feeling that the heavens are not so terribly strange after all. But, when the heavens are seen through a telescope, the vastness of space and the multiplicity of new-found stars again stagger the mind. To begin to understand these new observations, we must first understand how the telescope works and what the telescope really does. In order to accomplish this objective, the lab is broken up into three parts: (1) the eye and visual acuity; (2) a lens: its focal length and image formation; (3) two lenses: the refracting telescope. PART 1: The Eye and Visual Acuity We can think of light as consisting of rays. A point source of light will emit these rays in all directions. What the eye does is to receive these rays, and then bend them so that they again come together at the back of the eye where there are light-sensitive cells (called cone cells) which send messages to the brain. See Figure 1 below. Point Source Pupil (opening in iris) back Retina Fig. 1 Cornea Iris If there are two point sources of light, the eye will "focus" the light onto two places on the retina, and the brain will recognize that there are indeed two point sources. However, there are two LIMITATIONS on the eye. One comes from the fact that the eye cannot perfectly focus the light to one point on the retina. The second comes from the fact that the receptor cells have a size. Both limitations have the same effect: if two point sources of light are too close together, the eye cannot tell whether there is one or two sources of light. But how close is too close? Is a millimeter apart too close? A sixteenth of an inch? In this experiment we will look at the smallest markings on a ruler (which are separated by either 1/16 of an inch or 1 millimeter) and see if we can resolve them (see two consecutive marks clearly). Obviously, you can see the individual markings on a ruler -- if the ruler is close enough! Thus the distance between the two point sources (which we will label as h) is not the whole story. We must also include the distance of the point sources from the eye (which we will label as d). The separation of the markings gets easier to see as h gets bigger (meaning proportional to h). Also, the separation of the markings gets easier to see as d gets smaller (meaning inversely proportional to d). Thus the ease of distinguishing the markings depends on h/d, which is just equal to the tangent of the angle as shown in Figure 2.
The Telescope 2 h 1 d 1 h 2 Fig. 2 d 2 In this first part of the lab, we will try to find the minimum angle that the eye (your eye, that is) can resolve. (If you have glasses or can easily remove your contacts, you may wish to perform this experiment with and without your glasses or contacts.) Decide whether you will work with the sixteenth-of-an-inch markings or the millimeter markings of the ruler and record the appropriate value for h (1/16 inch or 1 mm). Have your partner hold up the ruler. Slowly walk away from the ruler until you can no longer distinguish one marking from the next marking. Measure the distance (d ) from your eyes to the ruler. CALCULATIONS: If your calculator has the INV TAN function, then find the minimum angle (we often use the Greek letter theta,, to indicate an angle) you can resolve by taking the inverse tangent of (h/d). Be sure both h and d are in the same units (inches, mm, cm, or meters). [ Recall from your math background: since TAN( ) = h/d, then = INV TAN (h/d).] smallest angle = (in degrees) = INV TAN (h/d). Your angle should come out in degrees and should be small (much less than 1 ). If your calculator does NOT have the INV TAN function, you can get a very good approximation by using the fact that a small angle measured in radians is almost equal to the tangent of the small angle. Thus to find the minimum angle you can resolve (in radians rather than degrees), simply divide h by d: smallest angle = (in radians) = h/d To convert from radians to degrees you must multiply by 180 and divide by ( 3.14) : (in degrees) = (in radians) (180 degrees/ radians). A better measure of small angles is in arc-minutes or arc-seconds. There are 60 arc-minutes in one degree, and 60 arc-seconds in one arc-minute. Thus, to get your minimum angle in arc-minutes, simply multiply your angle in degrees by 60 arc-min/degree: (in arc-minutes) = (in degrees) (60 arc-min/degree). Show your data (h, d), all your calculations, and state as your result your calculated angle ( ) in arc-minutes.
The Telescope 3 PART 2: A Lens: focal length and image formation A single converging lens will form an image behind the lens of any object sufficiently far in of the lens. The image will be focused so that it appears to be at a specific distance (called the image distance, s ) behind the lens. This image distance (s ) depends on the distance the lens is from the object (called the object distance, s) and the strength of the lens. See Figure 3 below. The image may be observed directly by looking through the lens toward the object as in Figure 3. The image may also be observed by placing a screen at the image position and looking at the screen. s lens s h object f Fig. 3 F image h eye We will place on one end of an optical track a light source with a slide of arrows and circles on it so that the light will illuminate the slide. The slide with arrows and circles will be our object. We also place a lens on the track, and then place a screen on the other side of the lens on which the image will appear if we position the screen correctly. 1. Adjust the position of the lens and the screen until a focused image appears on the screen. Record the object position, the lens position, and the screen position. Also measure the height of the image (h', the length of arrow on the screen) as well as the height of the original object (h, the length of the arrow on the slide). (If you cannot image the entire arrow clearly, then use the large or small circle as your image and measure the diameter of this image circle. This diameter is the value of h. The object circles have diameters of 20 mm and 10 mm. Use the appropriate value for h.) Calculate the object distance (s) and the image distance (s ). Is the image upright or inverted? Is the image bigger or smaller than the object? 2. Change the lens position by about 10 cm (either nearer or farther from the lens, whichever works best). Move the screen so that a focused image appears. Again record the object position, the lens position, the screen position and measure the image height. Calculate the object distance (s) and the image distance (s ). Did the image size change? If so, did it get bigger or smaller? 3. Repeat step 2 above for a third position of the lens and screen. Be sure to answer the same questions. 4. Now take the light source with slide off the track, but keep the lens and screen on it. Take the track with lens and screen out into the hall, turn the hall lights out but keep a light on in the room at the opposite end of the hall. This light at the opposite end of the hall will be our object. With this light going through the lens, move the screen to get an image focused on the screen. In this case we have a very large object distance (s ). Record the lens and screen positions, and calculate the image distance (s ). When the object is far away, the
The Telescope 4 image distance is equal to what we call the focal length of the lens. This length is labeled f in Figure 3. Thus the image distance you determined should be the focal length of the lens. Compare your value of f to the value listed on the lens. Are they close to the same? Is the image upright or inverted? Is the image much bigger or much smaller than the object size? Do this for all three of your lenses. List all object, lens, and image positions and the image and object heights. Calculate all object and image distances. Make a table of your object and image distances and image heights, along with the answers to the questions about whether the image is inverted or upright, and about image size. See if you can make a generalization about what you have observed. PART 3: Two Lenses: the refracting telescope A refracting telescope is basically a combination of two lenses. The first lens is used to form an image at a distance of one focal length of the lens away from the lens. The second lens is used as a magnifying glass to examine this image. 1. Use the "hall method" to determine the image distance for each of your two lenses when the object distance is far away. Recall from Part 2 that this particular image distance is called the focal length of the lens. (The focal lengths should be different for the two different lenses.) Also note which of the two lenses gives the bigger image, and which lens is the "stronger" lens (i.e., the better magnifying glass). 2. To make a telescope, we will use one lens to provide an image and use the other lens as a magnifying glass. Therefore we will use the lens that gave the bigger image as the objective lens and the lens that is a better magnifying glass as the eyepiece. The distance between the two lenses should be approximately equal to the sum of the two focal lengths. I suggest you put the eyepiece lens at the 60 cm mark on the optical track, and the objective lens at the mark that is about ( f e + f o ) further out from the eyepiece. That way, you can balance the light track on your shoulder while you look through the eyepiece. Look at something far away using this telescope by placing your eye near the eyepiece, looking through it so that you also then look through the objective lens. You may have to change the distance between the lenses slightly to obtain the best focus. (This part is hard to do. With a little patience you should see that this combination of lenses does indeed act like a telescope.) 3. After you have successfully completed step 2 above, look through your telescope at the ruler with the markings as in the first part of the experiment. You may have to slightly change the distance between lenses to focus the telescope as the ruler is placed at different positions. How far away can you now place the ruler and still make out the individual markings? Using this distance d, calculate your smallest angle that you can resolve with the telescope. [Calculate it just as you did in the first part on visual acuity.] Let us call this smallest angle formed by the light coming into the telescope tele.
The Telescope 5 1. Record the lens and screen positions used to calculate the focal lengths of both lenses, and record those calculated focal lengths. Indicate which of the two lenses gave the biggest image in the hall, and which acted as the best magnifying glass. 2. Record the distance at which you could just distinguish the millimeter markings using the telescope, and calculate (using the procedure of Part 1) tele. 3. Experimentally, the useful magnification of the telescope can be calculated by dividing the minimum angle using just the eye (which you determined in Part 1) by the minimum angle using the telescope: M useful = eye / tele. (1) [NOTE: the distance to resolve the markings with the telescope should be greater than the distance with just the eye, and so tele should be smaller than eye. ] Calculate and record this useful magnification. 4. Theoretically, the magnification of a telescope should be the focal length of the objective lens (the longer focal length) divided by the focal length of the eyepiece (the shorter focal length): M theory = f obj / f eye. (2) Calculate and record this theoretical magnification. 5. Compare the two magnifications, and discuss why they are different.