Part I: Thin Converging Lens

Transcription

1 Laboratory 1 PHY431 Fall 011 Part I: Thin Converging Lens This eperiment is a classic eercise in geometric optics. The goal is to measure the radius o curvature and ocal length o a single converging lens rom which you can calculate the inde o reraction n. We shall eplicitly consider the errors that accompany any measurement and how errors are analyzed to yield a quantitative estimate o uncertainty. This includes quantities derived rom measurements, in this case the inde o reraction. In the procedures or this lab, you are asked to estimate the uncertainties. Please see Appendi I and II or reerence material and some relevant equations. The questions, labeled Q1, Q, should be eplicitly addressed in your write-up in the Analysis & Discussion section. Procedures A. By deinition, the ocal length o a lens is the image distance rom the lens center or an ininitely distance object. To obtain a rough estimate o, project an image o the trees outside the lab onto the white paper near the door. Use a 3 diameter lens labeled A,B,C, or D. Q1. Why do the trees appear upside down? Describe a method to measure the height o a tree based on this imaging method. Apply this method to estimate the height o a tree around the BPS building. Estimate the image size o the moon and Sun using this given lens? B. Use a spherometer to measure the radius o curvature o both suraces o your lens. See Appendi I. You begin by inding the zero position, 0, using a scratch-ree spot on your bench (a good approimation to a lat surace). Then perorm the measurement with your lens in place, 1 ; the distance h is then 0-1. Repeat the measurements a ew times to obtain an estimate o the spherometer s precision. Rulers, calipers, graph paper, and markers are available. Q. Having estimated measurement uncertainties δ 0 and δ 1, write an epression or h and evaluate it using your data. C. Arrange an object (the T on the lamp window) and screen with a separation greater than 4 on the optical rail. Locate the lens position which gives a sharp image on the screen. Record the object and image distances measured rom the center o the lens including uncertainties. Use the thin lens equation to calculate and. Repeat this or 4 positions o the screen increasing the object-screen separation in units o about cm. Find the best value or the ocal length. Q3. How does the best value or compare to your original rough estimate? D. Insert a variable iris beore/ater the lens. Observe the image as the aperture size is changed. Speciically note whether it aects your ability to ocus the image. Q4. What is the meaning o the term depth o ield (DOF)? Estimate the depth o ield or your setup or a camera. Res: Depth o Field Calculator - Depth o ield - Wikipedia - The -number (-stop, /#) is the ocal length divided by the "eective" aperture diameter. 1

2 Laboratory 1 PHY431 Fall 011 E. Place the light source a distance less than rom the lens. Try to position the screen to bring the object into ocus. Q5. How do you eplain your observations? Calculate the inde o reraction (including uncertainty) or the glass o your lens using the lensmaker s Equation. Compare with known values or n.

3 Laboratory 1 PHY431 Fall 011 Part II: Thin Divergent Lens For a divergent lens, all principles and conventions used or a convergent lens will apply. The key dierence is that a divergent lens cannot by itsel orm a real image o a real object. Hence, in this eperiment we will measure using a virtual object. The virtual object and real image are on the same side o the lens. You will measure the radius o curvature and ocal length, then calculate the inde o reraction o the glass. Be sure to consider error propagation in your inal results. Procedure: A. Use a spherometer to measure the radius o curvature o a divergent lens. Determine the ocal length using the lensmaker s equation. B. Use a convergent lens L 1 to orm a sharp image i 1 o your object on a screen using the lamp as a source. Net, place a divergent lens L between L 1 and i 1 as shown below. Measure the distances to i 1 (=S i1 ) and i (=S i ), the distance between two lenses (d), and the object distance (S 0 ) to calculate or the divergent lens. [You may use the ormula given in Appendi II-1.] Repeat this or 3 positions o i by changing the lens-screen separation in units o about 1 cm. Find the best value or the ocal length using the thin-lens equation. Q6. Make a quantitative comparison o the L ocal length obtained by the two dierent methods. Do they agree within eperimental error? How could you improve either measurement? C. Calculate the inde o reraction (including uncertainty) or the glass o your lens using the lensmaker s equation. Compare this with the value ound in Part I. Beore leaving the laboratory, make rough estimates o all quantities that need to be calculated or included in your report. 3

4 Laboratory 1 PHY431 Fall 011 Thin Lens Equation Appendi I-1: Lens Equations s s ' = 1 Lensmaker s Equation = ( n 1), assuming that n air = R R 1 Spherometer Equation R b h = + h From Pythagoras: R = ( R h) + b R = R Rh + h + b R b h = + h 4

5 Laboratory 1 PHY431 Fall 011 Appendi I-: Error Analysis Random luctuations in the measurement process lead to a Gaussian distribution about the true value. This distribution gives us a parameter,, called the standard deviation. (Systematic errors lead to a non-gaussian distribution.) Essentially, i many measurements i are taken, 68% o the data points lie within ±, where is the mean value o. Now, suppose an arbitrary unction (,y) depends on the variables and y, assumed to be independent o each other. How do we compute the uncertainty in,, given y and? Under the assumption that the uncertainties are small compared to the absolute value o the quantities in question = + y y For errors that are much smaller than the measured values, speciic unctions yield: = a + by = a + b y = cy y = + y = a b c y a b y = + y = ce b = b = ca b = ( bln a) When we make N measurements o the same quantity, each with an uncertainty δ, we epect that ater averaging the measurement will have uncertainty smaller than δ. In act, the value o varies as 1/ N when N is large. 5

6 Laboratory 1 PHY431 Fall 011 Appendi II-1: Thin Lens Combination The image position or two thin lenses is s i = d s o1 1 (s o1 1 ) s d o1 1 (s o1 1 ) where s o is the position o the object (beore either lens), and d is the distance between the lenses. I s o =, and d= 1 +, then s i =. This is a Galilean telescope. 6

7 Laboratory 1 PHY431 Fall 011 (Source: Ch. 5. Optics, by Hecht) 7