5. Convex, Concave Lenses and the Lensmaker s Law

Transcription

1 5. Convex, Concave Lenses and the Lensmaker s Law 5.. Equipment light ray source, Pasco convex and concave lens slices, ruler,.2m optics track with lens holder and white screen, 0cm lens 5.2. Purpose. To measure the focal length of a convex and a concave lens. 2. To predict the focal length of a convex and a concave lens using the lensmaker s law. 3. To examine the behavior of combining the convex and concave lenses. 4. Test the lens law with error Theory The lensmaker s law used to calculate the focal length f and f is: f = f = (n )( ) R R 2 uses the index of refraction n of the lens material and the radii of curvature of both surfaces to predict the focal length of a convex, concave, plano-convex or plano-concave lens. It assumes that the lens thickness d can be treated as insignificant and the lens is surrounded by air. The Gaussian sign convention indicates that for a double-convex lens, R 2 is negative and for a double-concave lens, R is negative. The Basic Optics kit contains a pair of lenses that allow the measurement of focal length and radii of curvature is a very quick manner Procedure. Place the ray source (three rays) on a large sheet of paper and direct the rays toward the doubleconvex lens as shown in Figure 5.. One edge of the lens is more flat and will be more stable when used as the bottom. Note: Allow room for a total of four trials, two for each lens. 2. Carefully trace around the lens using pencil. This trace will be used to measure the curvature later. 3. Mark the ray edges and then use a ruler to draw the rays on the paper. Extend the center ray well beyond the focal point. Record the f distance in mm under the diagram. 4. Put a reasonable error on the focal length value. 5. Repeat steps one to four. Average the results. 6. Use a compass to determine the radius of curvature of both sides. Record them as R and R 2 with surface being the first surface the light hits. If desired you can retrace the lens on a separate trial somewhere. Explain any changes to procedure you may have made, right on the diagram. (5.) 22

2 5. Convex, Concave Lenses and the Lensmaker s Law 7. Repeat the above steps for the double concave lens with the following changes. Assign reasonable ± errors to measurements of f and R. a) The rays will have to be extended back to a focal point after removing the lens. b) The radius of curvature can be determined using the faint ray that reflect off the first surface. Trace these rays too and use them to measure R. c) The manufacturer states that for the double-concave lens R 2 = R. 8. On the paper, table your values and averages with their errors. 9. Report the thickness at the center of each lens in a table. 0. Nest the two lenses together (touching) and examine the resulting ray pattern that is produced by rays exiting both the lenses. Note this step as Step #0 on the paper and your report. Indicate if the rays are: diverging, converging or parallel after going through both lenses.. Now gradually pull the lenses apart horizontally, and note any changes to the rays exiting the lens pair. 2. Reverse the positions of the two lenses and repeat steps 0 and Calculations. Use the lensmaker s law with the measured radii and focal lengths of both lenses to calculate the index of refraction n for each lens. 2. What does the test is step 0 tell you about the lenses? 3. Do the rays do anything different comparing steps and 2? 5.6. Lens Equation The standard lens equation employed to calculate image and object positions or the focal length of a lens is: s + s = f Often f will be f, which is the focal length on the image side of a converging lens EXERCISE: Verification of Lens Law, Graphing and Measuring Error, plot of 00mm lens data The main goal of this exercise to generate reasonable errors on values of s and s. It is also a refresher of the lens law. The actual values of s and s used are given and are to be confirmed.. Set up the lens holder and light source and the white screen as shown in Figure 5.3. Set the light source to illuminated object. 2. For the rows in Table 5. with a *, set s to be the value shown. 3. Verify that the s value recored in Table 5. is similar to yours. 4. Determine a reasonable error in mm for both s and s. Record these values in the table. It is not necessary to examine all pairs of s and s, but watch how the error changes as you move down the table. 5. In xmgrace, enter the values of s and s with error bars. Use set type XYDXDY with s in the X column. For errors on all points, use the nearby error value from your tests. Include a graph of s as a function of s in your report. (207: this graph is not required.) 23

3 5. Convex, Concave Lenses and the Lensmaker s Law object distance s [mm] Table 5..: 00mm lens data error in s [mm] image distance s [mm] error in s [mm] calculated f * * * * * (?) 75 no focus 6. Use xmgrace to calculate values and plot a second graph of /s as a function of /s with error. This set will also be of type XYDXDY. Consult the appropriate tutorial sections for assistance. 7. List three sources of error for s and s Question. Why is the s=75mm setting not provide a focused image? 5.7. Submission Submit a LYX generated pdf report. The paper diagrams can be attached separately. 24

4 5. Convex, Concave Lenses and the Lensmaker s Law Figure 5..: Thin lens equipment setup Figure 5.2.: Double-concave lens radii Figure 5.3.: Len s equation test setup 25

5 6.. Equipment water, beaker, small weight (dollar coin, etc), eye-dropper (optional), from Basic Optics Kit: Light Source (rays), hollow lens, clear plastic box (items removed), white plastic sheet 6.2. Purpose To explore the properties of a lens by changing the index of refraction of the lens and the surrounding media Theory A conventional lens is made of material whose index of refraction is higher that the air surrounding it. Examples of this include magnifiers, corrective glasses which may be made of plastic or glass. Most glass materials have an index of refraction between roughly.4 and.9, while air is about However a lens can also made of material whose index is less than the surrounding medium. In this lab a hollow lens of air will be partially or fully surrounded by water whose index of refraction is.333 at 589nm. This will be compared to a lens of water situated in similar surroundings. Figure 6. shows a lens of material n situated between two media of different refractive indices n and n. Figure 6..: general lens parameters. C and C 2 are the centers of the spherical surfaces of the lens; The respective radii of curvature begin at these centers and reach across to the far surface which is their matching lens surface. F is the focal point of the lens (both refractive surfaces are involved) F is the focal point for rays striking the first surface (left) and continuing forever in glass thereafter. Thus these ray never see the second surface. F 2 is the focal point for parallel rays striking the second boundary (right). Thus these rays don t involve refraction at the left (first) surface. The Gaussian formula 2 for a thick lens is used as a starting point and relates the focal lengths to the indexes of refraction and the thickness of the lens d. For the first surface: p85, Jenkins & White, Fundamentals of Optics, Fourth Ed. 2 p84, Jenkins & White, Fundamentals of Optics, 4th Ed. 26

6 and similarly for the second surface, n f = n f n f = n f + n f 2 + n f 2 dn f f 2 dn f f 2 Treating d as small compared to the product of f f allows us to drop the last term in equation 6.. Using the fact that for each single surface of the lens we know that n f = n n R and n f = 2 n n R 2 we have for the first focal point 3 : (6.) (6.2) n f = n n + n n (6.3) R R 2 When the media on either side of the lens are different, the two focal lengths are different and have a ratio corresponding to their refractive indices 4. This means n f = n f (6.4) Note that for n n then it must be that f f. However the ratios are equal. Thus the f-points of a thick lens are not symmetric. Using the right-hand sides of equations 6.3 and 6.4: n f = n n R + n n R 2 (6.5) With air on both sides of the lens (n = n = and now f becomes simply f ), we have the familiar lens-maker s formula: f = f = ( n )( ) R R 2 Returning to the case of a thick lens, Equation 6.6 becomes f = f = ( n )( + R R 2 ( (n )) )d n R R 2 The equation relating the object distance s and the 6.7 image distance s is 5 : 6.4. Questions on theory (6.6) (6.7) f s + f s = (6.8). Use equations 6.3, 6.4, 6.5 and 6.5 to find the second focal length for a double-convex lens with the following specifications and media. R = 6.58cm, R 2 = 70.3cm. Use the Gaussian sign convention. Assume the index of refraction for the lens is n =.5. a) Air on both sides of the lens. b) Water on both sides of the lens 3 p75, Jenkins and White, Fundamentals of Optics, 4th Edition 4 p8 of Jenkins & White, Fundamentals of Optics, 4th Ed. 5 Wes Wong Optics Lab manual, Experiment 2, p0 27

7 c) air on the first (left) side and water on the second (right) side. d) Water on the left side and air on the right side of the lens Procedure Hollow Lens Behavior A hollow lens has three sections as shown in figure 6.2:. a plano-concave section 2. a plano-convex section with a small R value near R 30mm 3. a plano-convex section of larger R value near R 48mm Figure 6.2.: The hollow lens apparatus. Place the hollow lens in front of the light ray source with the flat side closest to the source. The sections will be alternately contain air or water. Select five or three rays, whichever performs best. 2. For testing with water surrounding the lens, place the lens inside the plastic rectangular box with the white plastic sheet on the bottom as shown in Figure 6.3. Place coins on top of the lens to keep it from floating as necessary. 3. Fill in Table 6. with your observations as to whether a diverging or converging lens is created. Figure 6.3.: Hollow lens in box for surrounding it with water 28

8 Thick Lens. Remove the lens from the plastic box and rotate it so that section faces the light 6 and place the ray source and lens apparatus on a large sheet of paper. 2. Trace around the lens footprint so that its location is shown on the paper. 3. Fill section 2 and 3 with water to make a double-convex lens. Section should have air in it. Trace three rays to their focal point. 4. Measure the focal length of this water dbl-cvx lens in mm using your best guess as to the center of the lens: f dbl cx = mm 6.6. Results Lens surrounded by air water fill section with Table 6..: Observations fill section fill section 2 with 3 with water air air air water air air air water water air water air water water water air water water water air Lens type C = converging or D = diverging 6.7. Questions and Calculations. Under what conditions is the is the plano-convex lens converging? 2. Under what conditions is the plano-convex lens diverging? 3. If a plano-concave lens of unknown material is a diverging lens when surrounded by air, is it possible to know whether the lens will be converging or diverging when placed in water? Explain why or why not. 4. The double-convex lens in this experiment has R = 30mm±3mm and R 2 = 48mm±3mm. a) Use Equation 6.6 to calculate the focal length for the double-convex lens measured in subsection We want a flat surface facing the light source. 29

9 b) Is it appropriate to use 6.6 this formula for this situation? Explain why or why not. c) Use equation 6.7 to calculate the focal length of the thick lens. The thickness for the double convex lens in this case is d = 25mm. Was there better agreement with the measured focal length? 30