THIN LENSES: BASICS. There are at least three commonly used symbols for object and image distances:

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1 THN LENSES: BASCS BJECTVE: To study and veriy some o the laws o optics applicable to thin lenses by determining the ocal lengths o three such lenses ( two convex, one concave) by several methods. THERY: Each point o a sel-luminous object or relection object () is a source o light with a large number o rays emanating rom it in all directions. A lens will alter the direction o those rays which strike it, and an image () o the object may be ormed. rays rom a single point on the object strike dierent parts o the lens and eventually all intersect at some other point in space, a real image is ormed. Real image ormation is illustrated in ig. 1. A virtual image is ormed i the projections o the rays rom a single point on the object intersect at a single point in space. A screen placed at this point will not reveal an image, but the image is visible to the brain i observed rom the proper position such that the diverging rays enter the eye. See ig 2. A thin lens is one whose thickness is negligible in comparison to the image and object distances [object distance is measured rom object to lens; image distance is measured rom lens to image position]. A convex lens is thicker in the center than at the edges. Such a lens is also called a positive lens or a converging lens. A concave lens is thinner at the center than at the edges. t is also called a negative lens or a diverging lens. The axis o a lens is the line drawn through the centers o curvature o its reracting suraces. a beam o rays parallel to the axis is incident on a converging lens, the beam is brought to a ocus at a point on the axis called the ocal point (). a beam o rays parallel to the axis is incident on a diverging lens, the rays diverge as though radiated rom a point on the axis. This point is also called the ocal point (). The distance along the axis rom the ocal point to the center o the lens is known as the ocal length (). A converging lens has a positive ocal length; a diverging lens has a negative ocal length. There are at least three commonly used symbols or object and image distances: object distance: s (used here), o (sometimes used), p (sometimes used) image distance: s' (used here), i (sometimes used), q (sometimes used) These distances and the ocal length are related by the thin lens equation, 1/s + 1/s' = 1/. (1) bject distances, image distances, and ocal lengths can be positive or negative. We will see examples o both positive and negative values in this experiment. The ratio o the image size (h ) to the object size (h) can be predicted by use o the lateral magniication actor M exp = h / h. (2) [NTE: the image is inverted relative to the object, then h is negative, making M negative.]

2 Thin Lenses: Basics 2 As seen in ig. 1, geometry gives a theoretical lateral magniication in terms o the object and image distances: M the = -s' / s. (3) [NTE: The minus sign in Eq. (3) is necessary to keep the signs o the experimental and theoretical magniications consistent. That is, when s is positive, h is negative.] h h ig. 1: Real mage ormation ig. 2: Virtual mage ormation ig. 3: Conjugate oci s D s h h (a) with converging lens d s s h h (b) with diverging lens

3 Part 1: Convex Lens Thin Lenses: Basics 3 Three methods will be used to measure the ocal length o a convex lens: 1. Parallel rays: the object is suiciently ar away so that s>>, the incoming rays will be almost parallel, and the object will be ocused at approximately the ocal point o the lens. This method is used in Step The thin lens equation: The image and object distances will be measured and calculated with Eq. (1). This method is used in Step 2. (Since object and image distances will be measured, we will also investigate lateral magniication in this step as an added bonus!) 3. Conjugate oci: The locations o the object and image on the axis o a lens are called the conjugate oci. or a given object distance, an image is ormed at a distance determined by the thin lens equation. the object is moved to the image location, a new image will be ormed at the original object position. Thus, a lens has a symmetry with respect to its conjugate oci (s and s' can be interchanged in Eq. (1)). n the lab the position o the lens will be changed in order to eect the interchange o object and image. Reerring to ig. 3, i D = s + s' and d equals the shit in position o the lens, then one can show that This method is used in Steps 3 and 4. PRCEDURE: = (D 2 - d 2 ) / 4D. (4) 1. See how each o the three lenses (+5 cm convex, +10 cm convex, and 15 cm concave) acts when used as a magniying glass. Do all the lenses magniy? Which is the "stronger" convex lens, i.e. which magniies more when used as a magniying glass? Which is the weaker convex lens? 2. Place only the +10 cm convex lens and the white screen on the optical bench and aim the lens at some object outside the room. ( possible, perorm this part o the experiment in the hallway with the hallway lights o and with the lights in a ar room on.) Move the lens or screen along the bench until the image is ocused as sharply as possible. an image can be ocused, measure and record the image distance, which in this case is approximately the same as the ocal length. Can you justiy this assumption? [HNT: Consider the thin lens equation.] Replace the +10 cm convex lens with the +5 cm convex lens and repeat the above procedure. Replace the convex lens with the concave lens and repeat the above procedure. 3. R PARTS 3, 4 & 5, USE NLY THE WEAKER CNVEX LENS. a) CAL LENGTH. Place the slide on the optical bench. By shining a light through the slide, we will use the +10 cm convex lens to orm an image o the slide on the screen. Hence the slide will become our object in this part. Next place the lens at some position beyond the ocal length rom the object and ocus an image o the object on the screen by moving the screen. Measure object and image distances, s & s', and calculate the ocal length,.

4 Thin Lenses: Basics 4 b) MAGNCATN. The length o one o the object arrows is h = 30 mm. Measure the length o the image arrow, h. ( you cannot image the entire arrow clearly, then use the large or small circle as your image and measure the diameter o this image circle. This diameter is the value o h. The object circles have diameters o 20 mm and 10 mm. Use the appropriate value or h.) Note whether the image is inverted and use the proper sign or h. Calculate the experimental and theoretical magniications using Eqs. (2)&(3) and compare the values. Repeat these measurements and calculations or at least two more object distances. Be sure to have at least one case where the image is magniied (bigger than object) and at least one case where the image is demagniied (smaller than object). Average your ocal length values. 4. Reerring to ig. 3, i the image and object distances are equal (s = s ), then D = 2s, d = 0 and Eq. (4) gives = s/2. (The same result is ound rom the thin lens equation.) This would put the image a distance o 4 rom the object. See i this is true experimentally. Using the average value o rom Step 2, place the lens a distance o 2 rom the object and move the screen until the image is ormed. Measure the distance between object and image. s it 4? Record the dierence between 4 and the measured object to image distance. 5. Place the screen at a distance rom the object that is greater than 4. Call this distance D. By moving the lens, ind the two lens positions that give sharp images. Measure the distance between these two lens positions. This is distance d. Using Eq. (4), compute the ocal length and compare with the values obtained above previously. REPRT: 1. or each o the procedures above: draw a diagram, record your data, show your calculations, state your results, and answer any questions posed. n each case, be sure to include a discussion o possible errors and the estimated accuracy o your data and how this uncertainty in data aects your results. 2. Compare the values o the ocal length rom the our procedures. Comment on whether they agree with one another within experimental uncertainty. 3. Comment on your values or magniication. s the theoretical value the same as the experimental value within experimental error? s there just one value or magniication or this lens? not, what does the magniication depend upon?

5 Thin Lenses: Basics 5 Part 2: Concave Lens The method o determining the ocal length o a concave (negative, diverging) lens will involve using it in conjunction with an auxiliary convex (positive, converging) lens. n ig. 4, 1 is the image o the object 1 which would be ormed by lens #1 i the diverging lens #2 where not present. But with lens #2 in place, the rays do not converge as quickly, and the image 2 is ormed arther away. When analyzing multiple lens systems, the image ormed by the irst lens is considered to be the object o the second lens ( 1 2 ), and the thin lens equation is used. PRCEDURE: 1. MAGE AS BJECT. (a) Beore we consider a concave lens with a convex lens, let's work with two convex lenses to see more clearly how to work with two lenses. Start with the weaker o the two convex lenses and ocus an image on the screen. Try to get an image that is somewhat close in size to the object. Record the object distance, image distance, object height, and image height. Calculate the ocal length o this weaker lens. Compare to the values you have already obtained or it in Part 1. (b) Next place the stronger convex lens about 10 cm beyond the image o the irst lens and record this position. The distance rom the screen to the second lens will be the object distance or the second lens. Now relocate the screen so that it is beyond the second lens and see i you can ocus an image. you can, record the position o the screen and the inal image height, and calculate the image distance or this second lens. Now calculate the ocal length o this second (stronger) lens. Should the stronger lens have a ocal length bigger or smaller than the thinner lens? Does it? s this ocal length close to what you determined in Part 1, procedure 1 (hallway method) or this stronger lens? 2. CAL LENGTH CNCAVE LENS. (a) Remove the stronger convex lens and orm an image with the weaker convex lens, just as you did above. Record the position o the image since this will become the object or the concave lens. Without moving either the object or convex lens, place the concave lens on the bench BETWEEN the convex lens and the screen. (NTE: We cannot position the lens behind the image since a concave lens does not converge the light diverging rom the image the way a convex lens does. nstead, we will use the concave lens to diverge the already converging light coming rom the convex lens beore it orms an image. the concave lens is weaker than the convex lens, the rays will still orm an image but urther away as shown in ig. 4.) (b) Since the object or the concave lens (image rom the convex lens) is on the "wrong" side o the lens, we must make the object distance or the concave lens negative. Now reposition the screen until a new image is ormed. Record the position o the concave lens and o the new image. By remembering that the initial position o the screen is the location o the virtual object o the concave lens, compute the ocal length o the concave lens. A concave lens should have a negative ocal length. Does yours? (c) Repeat the above using a dierent distance between the object and the convex lens. Compare your two values or the ocal length o the concave lens. Are they the same? Should they be?

6 Thin Lenses: Basics 6 3. MAGNCATN. (a) Using h' 1 (image size o convex lens) as the object size or the concave lens (h 2 ), determine the magniication (both experimentally and theoretically) or the concave lens. Are the object and image both upside down? Should M concave be positive or negative? (b) Now determine the experimental magniication o the system o lenses. This is the inal image size (h' 2 ) divided by original object size (h 1 ). Does M inal = M convex + M concave or does M inal = M convex * M concave? h h' 1 h' 2 igure 4 REPRT: 1. or each o the steps above: record your data, show your calculations, state your results, and answer any questions posed. n each case, be sure to include a discussion o possible errors and the estimated accuracy o your data and how this uncertainty in data aects your results. 2. Show rom theory that M inal = M convex * M concave. (n act, or a system with N lenses, the inal system magniication is simply the product o all o the individual magniications, M M M... M M. inal 1 2 N N j 1 j

Reflection and Refraction

$Reflection and Refraction$ Relection and Reraction Object To determine ocal lengths o lenses and mirrors and to determine the index o reraction o glass. Apparatus Lenses, optical bench, mirrors, light source, screen, plastic or