Physics 2310 Lab #6: Multiple Thin Lenses Dr. Michael Pierce (Univ. of Wyoming)

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1 Physics 2310 Lab #6: Multiple Thin Lenses Dr. Michael Pierce (Univ. of Wyoming) Purpose: The purpose of this lab is to investigate the properties of multiple thin lenses. The primary goals are to understand the relationship between image distance, object distance, lens separation, and image scale. Theoretical Basis: You may be familiar with simple lenses and how they form images from the previous lab. The properties of multiple thin lenses are somewhat more complex since the lens separation now represents an additional variable. See the lecture notes of your textbook for a more complete description. Let s explore some of the properties of multiple lens systems. Begin by noting the equipment at each lab station. Like the previous lab, you will find a light source, a selection of lenses, a white screen for viewing the images, and that funky splitscreen thing. We ll come back to it later. We will use the scale on the bench to measure the location of each component and compute the object and image distances as well as the lens separation. Note that there is a small offset between the light source support and the actual location of the illuminated screen that serves as the object. Your lab TA will give you the value of this correction. Note that the light source object has a mm scale and two arrows at right angles so that you can determine the size of the image and its orientation on the screen when imaged by the lens. Procedure 1: Simple Telephoto Lens Put the converging lens back on the optical bench. We will be adding the negative, diverging lens in a moment. The order of the mounts will be: OBJECT - CONVERGING LENS - DIVERGING LENS SCREEN, like in the figure below. However, there are many possible positions of the two lenses that will allow you to form an image on the screen and it can be confusing so lets start by placing the convex lens and screen where you got a nice image in part 1. That is, refer to your EXCEL data and pick one combination of object and image distance. Verify that you get an image to form on the screen. Now add the negative, concave lens between the convex lens and the screen, just like in the figure below. Now slide the screen back

2 and forth until you get an image. 6 * Measure the distance from the object to the converging lens, do1. * Measure the distance from the converging lens to the diverging lens, d. * Measure the distance from the diverging lens to the screen, di2. * Measure the height of the image, hi. How does it compare to the image height you found with just the convex lens? * The 1-st thing you need to do is calculate the image distance, di1, for the converging lens. See Image 1 in Figure 4. This is the image that would form if the diverging lens weren't there, that's why it's grayed-out in Figure 4. Using your do1 and the average focal length for the converging lens (from the your earlier lab), calculate di1.in order to do calculations for multiple lens systems you have to treat the image for the 1-st lens as the object for the second lens. That's why do2 is labeled the way it is in Figure 4. (The object distance is from Image 1 to the second lens). * Using the data you took from the previous lab, calculate do2 based on the Figure 4. OK, Image 1 is now considered to be the object for the second lens (call it Object 2). The light that is trying to form Image 1/Object 2 is entering the diverging lens from the left. Since Object 2 is on the opposite side of the lens that the light is coming from, it is considered to be a virtual object. Any virtual object has a negative object distance so make sure you use a negative in the calculation below. * Apply the thin lens equation to the second lens using your data, and calculate the focal length, f2, of the diverging lens. Compare this focal length to the one given on the lens mount. Compute the percent error. Is this value you calculated for f2 consistent with what you know about the focal length for a diverging lens?

3 * The total magnification for a multi-lens system is the product of the separate magnifications. Find the total magnification (include any negative signs). Is the total magnification of the system consistent with the size and orientation of the image (inverted or upright) on the screen? Explain. * We should also be able to calculate the total magnification of the system by using the empirical data, m = hi/ho. Calculate the total magnification with this equation and compare it to your theoretical prediction. Calculate a percent error for the magnification using the two results from above. Procedure 2: Focal Reducing Lens In this lab we want to use to positive lens separated by a distance d. We will need a second positive lens for this part of the lab. Set them up like the figure above but with the negative lens replaced with the second positive lens. Again, you can use your measures of the focal length from your previous lab from your Excel spreadsheet. This time, make sure that d, the separation of the two lenses, is about half of the focal length of the two lenses. Now slide the screen back and forth until you get an image. Repeat the procedure you used above: * Measure the distance from the object to the 1-st converging lens, do1. * Measure the distance from the 1-st converging lens to the 2-nd converging lens, d. * Measure the distance from the 2-nd converging lens to the screen, di2. * Measure the height of the image, hi. How does it compare to the image height you found with just the 1-st converging lens in the previous lab (recomputed it for the new object distance if necessary)? * Again, calculate the image distance, di1, for the converging lens. See Image 1 in Figure 4. This is the image that would form if the diverging lens weren't there, that's why it's grayed-out in Figure 4. Using your do1 and the average focal length for the converging lens (from the your earlier lab), calculate di1. Like the previous example we will do calculations for multiple lens systems by treating the image for the 1-st lens as the object for the second lens. That's why do2 is labeled the way it is in Figure 4. (The object distance is from Image 1 to the second lens). * Using the data you took from the previous lab, calculate do2 based on the Figure 4. OK, Image 1 is now considered to be the object for the second lens (call it Object 2). The light that is trying to form Image 1/Object 2 is entering the 2-nd converging lens from the left. Since Object 2 is on the opposite side of the lens that the light is coming from, it is considered to be a virtual object. Any virtual object has a negative object distance so make sure you use a negative in the calculation below. * Apply the thin lens equation to the second lens using your data, and calculate the focal length, f2, of the converging lens. Compare this focal length to the one given on

4 the lens mount. Compute the percent error. Is this value you calculated for f2 consistent with what you know about the focal length for a converging lens? Describe in your own words how the second conversing lens adds to the power of the first lens. This configuration is used in some astronomical instrumentation to reduce the focal length of a telescope in order to better match that needed for a particular instrument. Explain how this would work and why the second lens in this configuration is called a focal reducer.

5 Apply graphical ray tracing techniques to understand the behavior of these thin lens combinations. Figure 3: Graphical ray tracing with two positive thin lenses.

6 Figure 4: Graphical ray tracing with both positive and negative thin lenses

7 Figure 5: Graphical ray tracing for telescope and microscope configurations.

8 Appendix: The Standard Error When you make several measurements of a quantity you can compute the error in your measurements from the deviations, or spread, from the average. For example, we could average various measurements of the focal length of a lens to find our best estimate of the focal length. We can also use these measurements to estimate how accurate our result is. One way to estimate the precision of such an average is to find the standard error of the estimate. Begin by subtracting the average focal length from each of the estimates to find the residuals and record them. Now square each of the residuals and record them. Add these up to find the sum of the residuals squared S r and divide this by the number of points. This quantity is the average of the residuals squared and is called the variance. If we take the square root of the variance we get the standard deviation, or standard error (σ). We might expect that our measurements would cluster around the correct value and that the average of the measurements would thus have a lower error than any given measurement. Thus, if we took more and more measurements the average would get closer to the true result. Measurement theory, i.e., statistics, predicts that unbiased measurements will follow a Gaussian distribution centered on the correct value. In this case, the standard error of our average will improve with the square root of the number of measurements. standard error of average = (σ) / square root of ( N-1) In our case, N-1 is the number of measurements minus 1. If the standard error is typically about 1 mm, then about 68 percent of the measurements will fall within 1 mm of the average focal length. Rewrite the focal lengths with their standard errors in proper form, for example and adopt this error for all your remaining measurements. Notice that the standard error gives us a measure of the precision of our observations but cannot warn us of systematic errors that affect all of our observations by the same amount. An example of this might be an error in our measurement scale or in our meter sticks. In this case, we have a systematic error and all bets are off! Not really, we just apply this systematic error to all of our measurements to correct for it.