Hinds_M7A1

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Course: PHYS204 Section: Module 7 Name: Kaitlin Hinds Instructor Name: Dr. Felix Rizvanov __________________________________________________________________________ Title : Radius of Curvature vs. Index of Refraction __________________________________________________________________________ Abstract : The experiment was set up to prove the relationship between index of refraction, focal length, and radius of curvature shown in the Lens Maker’s Equation. Using a computer program, I was able to examine how the index of refraction will change in response to a change in the radius of curvature. The results showed an increase in radius of curvature would cause an increase in the required index of refraction. This outcome proved the validity of the Lens- Maker’s Equation. __________________________________________________________________________ Introduction: The purpose of this lab was to explore the relationship between the index of refraction of the lens and the radius of curvature with the value for the focal length being held constant. This experiment seeks to prove the theory behind the Lens Maker’s Equation. The radius of curvature of a lens and the material’s index of refraction determines the focal length of the lens itself. The report compares the required index of refraction for a given radius of curvature. The requirement for the index of refraction was that after the radius of curvature was changed, it would move the focal point back to the original position. The equation necessary for this experiment is the Lens Maker’s Equation below: 1 f = n lens − n outside n outside ( 1 r L − 1 r r ) Where n represents the index of refraction, f represents the focal point, and r represents the radius of curvature. It was observed that an increased radius of curvature for the lens will increased the required index of refraction to maintain a constant focal length. __________________________________________________________________________ Equation 1

Methods: For this experiment, I used the Lens Maker’s Equation lab from the GeoGebra computer program to test the required index of refraction to maintain a focal point when the radius of curvature were changed. First, I chose a focal length of 6 to set as the original focal point position. The radius of each lens was set to 2 (the radius of the right lens had a negative value to represent that it was curved the opposite direction) and adjusted the index of refraction until the focal point reached 6. I then varied the radii by 1 and recorded the required index of refraction to return the focal point back to its original position. This was repeated until I had 10 different data points. __________________________________________________________________________ Results: For this experiment, I first observed the required index of refraction of a lens with a radius of curvature of 2 to obtain a focal length of 6. Figure 1 shows an index of refraction of 1.17 was required to bring the focal point to six with the radii are set to 2. Again, the right lens has a negative value to represent that it is curved in the opposite direction. The radii of the two lenses are equal. Then, I changed the radius of the right and left lenses in increments of 1. With each adjustment, I used the slide bar to change the index of refraction of the lens until the focal point returned to its original position at 6. It was noticed that throughout the experiment, the required index of refraction increased as the radius of curvature for both lenses was increased. The final radii value tested is shown in Figure 2. As the radii were changed to 11, the index of refraction required changed to 1.92. After my values for the index of refraction were recorded, I used Equation 1 to calculate the focal point. I found that my calculated focal points were within 0.15 of the constant focal length used. This was to be expected as I was not able to get the focal point exactly on the 6- mark every time. Figure 1 Screen shot of the required radius of curvature needed to produce a focal point of 6 at an index of refraction of the lens of 1.17 Figure 2 Screen shot of the required radius of curvature needed to produce a focal point of 6 at an index of refraction of the lens of 1.92

The following table illustrates the required index of refraction to produce a focal point at six for varying values of the radius of curvature for the two lenses. Index of Refraction of the Lens ( n lens ) Radius of Curvature ( r L ) Radius of Curvature ( r R ) 1.17 2 -2 1.25 3 -3 1.33 4 -4 1.42 5 -5 1.5 6 -6 1.58 7 -7 1.67 8 -8 1.75 9 -9 1.83 10 -10 1.92 11 -11 The data indicates a rise in the required index of refraction as the radii of curvature are increased. The following graph illustrates the required index of refraction to produce a focal point at six for the varying values of the radius of curvature. The data indicates a rise in radius of curvature will cause a rise in the index of refraction. The shape of the graph is due to the negative values representing a lens that is facing the other direction. Discussion : The experiment found that the required index of refraction to maintain the original focal point position increased as the radius of curvature for each lens was increased. These results clearly indicate that the relationship between focal length, index of refraction, and radius of curvature for the lens are consistent with the Lens Maker’s Equation. These results are significant because it allows a lens to be designed based on either desired materials, desired focal length, or desired size (radius of curvature). In personal electronic devices where a small

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spherical lens would be desirable, a small radius of curvature would be preferrable for a smaller profile. Since a long focal length and a short focal length have different uses, the designer will need to determine the desired focal length. Once the focal length and desired radius are established, the required index of refraction can be calculated and the material with that index of refraction can be found. These results raise questions such as what is the most limiting factor in designing a lens (i.e., material, size, focal point)? At what point would a change in the radius of curvature have a negligible change in index of refraction? __________________________________________________________________________ Conclusion: The purpose of this lab was to observe the effects of the radius of curvature on the required index of refraction of the lens. I began with chosen desired constant focal length. Then, I varied the radius of curvature of the two lenses and measured the required index of refraction to maintain the focal point at the constant focal length. It was found that an increase in the radius of curvature resulted in an increase in the required index of refraction. This result proved the relationship between the focal length of the lens, the index of refraction of the lenses’ material, and the radii of curvature show in the Lens-Maker’s Equation. __________________________________________________________________________ References: Lumen Learning. (n.d.) Lenses. Retrieved April 23, 2022, from https://courses.lumenlearning.com/boundless-physics/chapter/lenses/

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