Experiment 2 Simple Lenses Introduction In this experiment you will measure the focal lengths of (1) a simple positive lens and (2) a simple negative lens. In each case, you will be given a specific method of measurement, and be told how to calculate the focal length from this measurement. Material on thin lenses can be found in J. &W. 4.1-4.9, 4.14, 4.15 and P. &P. 3-9, 3-10, and 3-11. Focal Lengths of Simple Lenses There are six sections in this lab. In each section you will measure the focal length of a simple lens. 1. Positive Lens Procedure For the first section, you will make several measurements of image and object distances for a simple positive focal length lens (f = +20 cm). Place an illuminated object at one end of the optical bench. Then place the positive lens 60 cm from the object. Place a large white screen on the other end of the bench and move it toward the lens until you find the image. Record both the object distance and the image distance. Move the screen a few centimeters away and then find the image position again. Take total of five measurements of the image distance. Repeat the above steps for an object distance of 30 centimeters. Make only one careful measurement of the image distance for object distances of 55, 50, 40, 45, and 35 cm. You will continue to move the lens toward the object but now the images will become virtual, since you will be moving inside the focal length of the lens. Since you can no longer use the screen to find the image, you must locate the image by an alternative method. Here, you will employ a device called a range finder. The operation of the range finder will be explained in lab. The range finder should be located 20 cm from the simple lens (see Figure 2 and 3). Replace the illuminated object with the white screen. Note that the screen has a black line on the back. This line will be the new object. Place the lens 16cm from the object.
Figure 1: Positive Lens, Real Image Figure 2: Positive Lens, Virtual Image
Measure the image distance. Remember the range finder will tell you the distance of the image from the range finder, you must subtract the distance of the range finder from the lens in order to find the image distance. Take a total of five measurements of the image distance. Repeat the above steps for an object distance of 8 cm. Make only one careful measurement of the image distance for object distances of 14, 12 and 10 cm. Analysis In order to find the focal length of the lens from these data, the simple-minded approach would be to use the lens equation, compute several values for the focal length and average. This is not what you are going to do. Instead, you will plot the data and find the focal length from the intercepts. Plot the values of (1/u) vs. (1/v). The intercepts of this graph will be (1/f). Of course there are two intercepts that can be used. Compute f from both these intercepts. Be sure to estimate the error in the intercepts using a least squares fit. Then from the error in the intercepts calculate the errors in the focal length. If you need help with this, see your instructor. See the Report Outline for additional instructions. 2. Negative Lens Procedure In the second section you will repeat Section 1, but you will use a negative lens (f = -20cm) instead of a positive lens. Make five measurements of the image distance for object distances of 55, 35, and 15 cm. Then make only one measurement with object distances of 50, 45, 40, 30, 25, 20 cm. You will note that a negative lens always produces a virtual image, so you will need to use the range finder to measure all of the image distances in this section. Figure 3: Negative Lens
Analysis The method for computing the focal length is the same as Section 1. However, you will note that in this case you are computing the focal length by extrapolation, while in Section 1 you're using interpolation. Do you see any difference in accuracy? 3. Spherometer Procedure In this section you will use the spherometer to measure the focal length of both lenses. Actually you will measure the radius of curvature, and using an assumed value for the refractive index of glass, calculate the focal length. Figure 4: Spherometer Measure the distance from the center of the spherometer to one of the legs. You can measure the distance r with a simple scale. Place the spherometer on a plane mirror and record the reading. This reading on the micrometer corresponds to zero curvature. Place the spherometer on each lens and record the reading. (This is a special case. You only need one careful measurement here. We have found that repeating the measurement many times would only give the same result many times. There are no averages to be taken in this section.) Also note that you only need to measure one side of each lens. You will assume that they are symmetric. (they are indeed close). Analysis Now from your readings you can calculate the focal length using the lens makers formula. The radius of curvature can be calculated using,
= h + 2h For a symmetric lens, R 1 = -R 2, so the lens makers formula is just, 1 =2 1 where h is the difference between the spherometer reading for flat glass and the spherometer reading for the lens. r is the distance from the center of the spherometer to each leg. The refractive index, n, is assumed to be 1.517. Compute the error in your result by using the standard propagation of error method. In other words, given the precision of measurement of h and r, compute the error in R and the compute the error in f. A side note here, the percent error in R is the same as the percent error in f. That might help a little. 4. Object at Procedure This is the quick and dirty method of getting the focal length of a positive lens. Figure 5: Object at Place the image on the optical bench and place a large white screen behind the lens. Form an image of the far wall on the screen. Record the image distance.
Repeat this measurement five times. It is best here to let each lab partner take a couple of measurements independently to reduce systematic error. Record the distance from the lens to the wall which you are using for an object. This is not used to get the focal length (you are assuming the object is at ). This measurement is for the error analysis only. Analysis Since you assume that the distance to the wall is infinite, the image distance equals the focal length =. Compute the average image distance, the standard deviation, and the standard deviation of the mean. Compare the average focal length to the more exact value obtained by using the finite distance to the wall. How big is the error caused by assuming that the object distance is infinite? 5. Autocollimation Procedure The autocollimation method allows us to place an object exactly in the focal plane of a lens. This means that light from the object will be parallel (collimated) when it emerges from the lens. Figure 6: Autocollimation Place the illuminated object at one end of the bench. Place the positive lens about 20 cm from the object. Now place the large plane mirror behind the lens. Rotate the mirror until you can see that the light from the object is reflected back on itself.
Move the lens back and forth until you get a sharp focus. There is a trick to this: place a piece of white paper next to the object so that you can see an image on the paper. The distance from the object to the lens is the focal length. Repeat this measurement five times. After you have taken your fifth data point, remove the large mirror. You should now have an object which is in effect infinitely far away. Place a 10cm lens on the bench and a screen behind it. Focus the image of the object on the screen. Check to see if indeed the image distance is 10cm. Analysis Compute the average as in Section 4. 6. Two Position Method Procedure This method will be used again in later experiments. Be sure that you understand the principles involved and keep this procedure available for future reference. Figure 7: Two Position Method Place the illuminated object at one end of the optical bench. Place the screen 100cm from the object on the other end of the bench. This is the distance L. Place the positive lens about 20cm from the object. Move the lens away from the object until you get a sharp image on the screen. Record the position of the lens on the bench.
Move the lens away from the object until you get the other image. Again record the position of the lens. Now the difference in bench readings is the distance d. Repeat this five times. Do not move the screen or the object. Analysis Using the formula = /4, compute five values of the focal length and average. Addendum The range finder can be used to measure distances with the following formula: Where Range R R f = + = distance from the range finder objective lens to the object = reading for an object at infinity = reading for the distance you are measuring = focal length of the range finder objective lens i = D ir - Range where D ir = the distance between the simple lens and the Range finder objective