Optics II: Lenses, Mirrors, and Telescopes Purpose: The purpose of this lab was to investigate the properties of lenses and mirrors to find the various relationships between focal length, concavity, radius of curvature, image size, object size, distance to the object, and distance to the image. Procedure: -
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In Exercise 1, we reflected a beam of light off a plane mirror to compare the angle of incidence to the angle of reflection. Then we reflected a beam of light on concave and convex mirrors to find their focal length, and compared that to their radius of curvature. In Exercise 2, we shined beams of light through concave and convex lenses to find their focal lengths. We also saw what would happen if we nested a concave and a convex lens with similar curvatures together and shined a light through it. In Exercise 3, we used a lens to focus an image at various distances to confirm the relationship between focal length, distance from the object, and distance from the image. In Exercise 4, we built a simple telescope and determined its magnification by estimating the ratio of the size of a far-away image to its size viewed through the telescope.
Data: -
Exercise 1: o Plane Mirror:
Trial 1: Angle of Incidence (degrees): 14 Angle of Reflection (degrees): 15 Uncertainty: +1 degree o Cylindrical Mirror: Focal Length: Concave Mirror: 6.1 cm Convex Mirror: 6 cm Uncertainty: +.1 cm -
Trial 2: 45 43
Trial 3: 70 70
Radius of Curvature: 12 cm 12.1 cm
Exercise 2: o Double convex lens: focal length = 14.6 cm o Double concave lens: focal length = 13.6 cm o With both lenses nested: Rays are nearly parallel, bend outward a tiny bit Uncertainty due to imperfections in the lenses o With both lenses at various separations: Rays always converge Uncertainty due to imperfections in the lenses
o Blocking various rays with the convex lens: rays all converge to about the same point - Exercise 3: o Focal length of lens from an object at infinity: di=8.3 cm (listed focal length 100 mm) Uncertainty: +1 cm o Focal length of mirror from an object at infinity: di=35 cm Uncertainty: +3 cm o Focal length by plotting 1/do vs. 1/di: Screen do di Object Image size Magnification from image and Magnification Position (cm): (cm) (cm) size (mm) (mm) object size: from di and do: 0 12.1 96.9 5 39 7.8 8 0 97 12 5 .5 .1 .1 66 21.5 21.5 5 5 1 1 56 15 38 5 13 2.6 2.5 56 38.4 14.6 5 2 .4 .4 46 13.9 49.1 5 18 3.6 3.5 46 49.5 13.5 5 1.5 .3 .3 36 13.1 59.9 5 23 4.6 4.6 36 60 13 5 1 .2 .2 26 12.8 70.2 5 28 5.6 5.5 26 70.5 12.5 5 .6 .12 .18 16 12.5 80.5 5 33 6.6 6.4 16 80.8 12.2 5 .5 .1 .15 Uncertainty: +.1 cm in object and image distances, +.1 mm in image size 1/do 0.082645 0.010309 0.046512 0.066667 0.026042 0.071942 0.020202 0.076336 0.016667 0.078125 0.014184 0.08 0.012376 -
1/di 0.01032 0.083333 0.046512 0.026316 0.068493 0.020367 0.074074 0.016694 0.076923 0.014245 0.08 0.012422 0.081967
(from Excell) Exercise 4: o Telescope magnification measured at about 25X
Analysis:
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Exercise 1: o We found that when a beam of light is reflected off a flat surface (like a plane mirror), the angle of incidence of the light is equal to the angle of reflection. Any error we experienced was likely the result of inaccurate tracing of the incident and reflected rays. o We also found that the focal length is about half of the radius of curvature for a cylindrical mirror (concave or convex). Since the radius of curvature of a plane mirror is technically infinite, its focal length would also be infinite, meaning any parallel rays reflected off it would never converge. We tested this and found it to be the case (as far as we could observe). Exercise 2: o We found that the focal lengths of the double concave and the double convex lenses were roughly the same (give or take some uncertainty due to tracing inaccuracies). We confirmed this when we nested the lenses together. When we did this, the refracted rays were nearly as parallel as the incident waves. Because the focal lengths were so similar, the refractions that took place within each lens cancelled each other out, making the outgoing rays nearly parallel again. The rays were actually bent outward a little bit, but we figured this was due to imperfections in cheap plastic lenses. When we moved the lenses to various separations, the difference in their curvature became more apparent, and the rays diverged when they should have remained parallel. o When we traced the focal point of various sets of rays that went through the convex lens, we found that they converged to about the same point (again allowing for a small amount of error), which shows that the lens was fairly uniformly curved. We found the same result using the colored light. Exercise 3: o When we found the focal length of the lens using an object at ‘infinity’, we got fairly close to the actual figure considering how difficult it was to focus the image and measure the distance. It was also very interesting to see the projected image from the curved mirror. o The intercepts from the graph of 1/do vs. 1/di were about 9.5 cm and 9.3 cm, averaged at about 9.4 cm, only .6 cm away from the listed focal length. This focal length was also about 1 cm different from our rough estimate from the object at infinity, again not too bad for a method that was so difficult to measure. We also found that our measurements for magnification were generally pretty close together whether we got the value from the ratio of image size to object size, or from the ratio of image distance to object distance. We experienced more and more error as we had to measure smaller and smaller images as the lens approached the screen. This added to some inaccuracies, but overall our results were sound. o From a purely mathematical perspective, there must be two points at which the image is in focus for a given screen-object distance because there are two combinations of (one over) image distance and object distance whose sum is one over the focal length. The image will be in focus at a certain object and image distance, and it will be in focus again when those distances are switched. If the object is less than the focal length away, then there is no
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combination of image and object distance that will do the job. There is simply not enough length to work with. Exercise 4: o This exercise was mostly just fun and interesting, to see an application of lenses and optics. The simple telescope we built worked quite well, even if it was a little flimsy. We were able to roughly estimate about a 25 times magnification from the telescope by looking at the lined paper down the hall. This seems like a pretty reasonable figure, though we didn’t have time to measure the magnification using the objective and eyepiece lenses.
Conclusion: I found this lab to be quite interesting. I remember learning many of these concepts in high school, and it was nice to have a refresher. I do wish we’d gone over some of the material in class beforehand though. That would have been quite helpful. I found this lab to be a little more tedious than the others, given the immense amount of length-measuring we had to do, but overall it was still a good learning experience. I feel this lab has reinforced many of the key concepts in optics.
