Lab 02

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University of Texas *

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103N

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Electrical Engineering

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Apr 29, 2024

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Lab 2: Seeing is Believing PHY 105N - Section Number 55758 By Thomas Fung and Suszane Song Part 1: Initial Investigation Method: Goal: The goal of this lab is to test and see how optical power and lens formation are related. We are testing the formula , where P is optical power, p is the distance from the light source to the 1 ? + 1 ? = 𝑃 lens, and q is the distance from the lens to the image screen. The formulas say that the value for optical power, P, will remain constant. Our hypothesis is that the formula will be true. Equipment: PASCO Light Source, PASCO Optical Track, +100mm Convex Lenses, Meter Stick, and Dial Procedure: 1. Set the PASCO Convex lens in between the image screen and the PASCO light source on the track. 2. Turn on the PASCO light source, so that the light source shines through the Convex Lens and onto the image screen. 3. Set the image screen at the 100 centimeter mark and the PASCO light source at the 50 centimeter mark on the PASCO Optical track. 4. Move the Convex lens along the optical track until you find the point were the projected image is closest 5. Measure the distance between the PASCO light source and the Convex Lens (distance p), and the distance from the Convex Lens to the image screen (distance q) 6. Move the PASCO light source 5 centimeters back along the PASCO optical track and repeat steps 4 and 5. 7. Continue moving the light source 5 centimeters back and repeating steps 4 and 6 until you reach the 0 centimeter mark along Data: Below is a table of the data collected using the procedure above. Distance from Light Source to Image Screen (cm) Distance from Light Source to Lens (cm) Distance from Lens to Image Screen (cm) Optical Power (cm -1 ) 100.0 11.9 88.1 0.095 95.0 11.9 83.1 0.096

Results: This experiment will use the mean and propagation of uncertainty formulas to analyze the results. The calculations for mean and uncertainty are below.Values in the tables are truncated to visualization, but all calculations were done with additional decimal places to avoid inaccuracies. We found that the image quality was indistinguishable in a range of 2 ticks of the ruler, or 0.2 centimeters. Hence, 0.2 centimeters will be our uncertainty for q and p. The average optical power measurement is: 𝑃 = Σ𝑥 𝑖 ? 𝑃 = 0.095 + 0.096 + 0.096 + 0.096 + 0.096 + 0.097 + 0.097 + 0.097 + 0.098 + 0.097 + 0.098 10 𝑃 = 0. 09664 ?? −1 The formula for optical power uncertainty is: δ𝑃 = Σ( ∂𝑃 ∂𝑥 𝑖 δ𝑥 𝑖 ) 2 δ𝑃 = ( ∂𝑃 ∂? δ?) 2 + ( ∂𝑃 ∂? δ?) 2 δ𝑃 = (− 1 ? 2 δ?) 2 + (− 1 ? 2 δ?) 2 Using the above formula above, the uncertainty for each measurement was taken. Optical Power (cm -1 ) Optical Power Uncertainty (cm -1 ) 0.0954 0.00141 0.0961 0.00141 90.0 12.0 78.0 0.096 85.0 12.1 72.9 0.096 80.1 12.3 67.8 0.096 75.0 12.4 62.6 0.097 70.0 12.6 57.4 0.097 65.0 12.9 52.1 0.097 60.0 13.0 47.0 0.098 55.0 13.7 41.3 0.097 50.0 14.4 35.6 0.098

0.0962 0.00139 0.0964 0.00137 0.0961 0.00132 0.0966 0.00130 0.0968 0.00126 0.0967 0.00120 0.0982 0.00119 0.0972 0.00107 0.0975 0.00098 Next, using the average was found for the uncertainties. δ𝑃 = Σ𝑥 𝑖 ? δ𝑃 = 0.00141 + 0.00141 + 0.00139 + 0.00137 + 0.00132 + 0.00130 + 0.00126 + 0.00120 + 0.00119 + 0.00107 + 0.00098 10 δ𝑃 = 0. 00126 ?? −1 Combining the average optical power, and average uncertainty, brings our best estimate to P = 0.096 ± 0.00126 cm -1 . This gives us a range of 0.09538 cm -1 to 0.09701 cm -1 . Conclusion: The data above appears to support the claim that , where P is a constant value. The best 1 ? − 1 ? = 𝑃 estimate, P = 0.096 ± 0.00126 cm -1 , encompasses all ten measured data points. This means it is plausible that all the data points are the same true value, as they all lie in the uncertainty range. There are multiple sources of uncertainty here. First, there is a range of what can be considered the “most clear” image. Because there is solely visual inspection to determine the clearest image, it will deviate from what would be quantitatively the clearest. Secondly, there is error in the measurement itself. The ruler has a resolution of 1.0 mm, giving the measurements themselves an uncertainty of 0.5mm. To reduce the uncertainty, the experiment uses the full length of the optical track. As seen in the optical power uncertainty formula, p 2 and q 2 are in the denominator. Meaning the larger these values, the smaller the uncertainty. Thinking about it, it makes sense. When the light source, lens, and image are further apart, the same sources of error have a proportionally smaller effect. If the sources are 100 cm apart, a 0.2 centimeter tolerance on what is the “clearest” image, has a much smaller effect than if they were 10 centimeters apart.

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To further improve this experiment, a different lens of differing strength can be used as well. This can validate the optical power formula for different situations. A longer track can be used to help further minimize uncertainties. Finally, calipers or a ruler with smaller resolution can be used to further increase the precision of the measurements. Part 2: Expanded Investigation Method Goal: The goal of this lab is to test and see how optical power and combination of lens are related. We are testing the formula , where P is optical power. The formulas say that the value for 𝑃 ????𝑖??? = 𝑃 1 + 𝑃 2 optical power, P combined , will remain constant. Our hypothesis is that the formula will be true. Equipment: PASCO Light Source, PASCO Optical Track, Eye Model, Eye Model Lenses, Meter Stick, and Dial Calipers Procedure: 1. Set the + 62mm Eye Model Lens in the “septum” slot in the eye model 2. Measure the distance from the Eye Model lens to the front surface of the image screen in the eye model for the near, normal, and far image screen slots 3. Set the PASCO Eye Model and the PASCO Light Source on the PASCO Optical Track 4. Turn on the PASCO light source, so that the light source shines through the Eye Model, onto the image screen in the Eye Model 5. Set the image screen to the “normal” distance 6. Move the Eye Model along the optical track until you find the point were the projected image is closest 7. Measure the distance from the Eye Model to the Light Source 8. Repeat the steps 6 and 7 for the image screen set to the “far” and “near” slots 9. Add a +120 mm lens to the Eye Model 10. Repeat steps 6 and 7 for each of the 3 image screen slots. Data: The below table is the data from the Eye Model containing only the +62mm lens. Image Screen Slot Distance from Light Source to Eye Model Lens (cm) Distance from Eye Model Lens to Image Screen (cm) Optical Power (cm -1 ) Far 8.10 4.07 0.369 Normal 6.50 4.90 0.358 Near 6.10 5.31 0.352

The below table is the data from the Eye Model containing the +62mm lens and the +120mm corrective lens. Image Screen Slot Distance from Light Source to Eye Model Lens (cm) Distance from Eye Model Lens to Image Screen (cm) Combined Optical Power (cm -1 ) Far 5.60 4.07 0.424 Normal 5.00 4.90 0.404 Near 4.60 5.31 0.406 Results: This experiment will use the mean and propagation of uncertainty formulas to analyze the results. The calculations for mean and uncertainty are below. Values in the tables are truncated to visualization, but all calculations were done with additional decimal places to avoid inaccuracies. It is known that the Optical Power of a corrective lens is given by the formula , where ƒ is the 𝑃 = 1 ? focal length. Knowing this, below is the calculation for the Optical Power of the +120mm lens. 𝑃 ???? = 𝑃 2 = 1 ? 𝑃 ???? = 1 12 𝑃 ???? = 𝑃 2 = 0. 0833 ?? −1 Hence, the Optical Power of the combined lens is found using the formula from the experimental goal. 𝑃 ????𝑖??? = 𝑃 1 + 𝑃 2 Here, P 1 is the Eye Model by itself, which is the first data table in the Data Section. P 2 is the Optical Power of the lens, as calculated above. This bring us to a new table Image Screen Slot P 1 (cm -1 ) P 2 (cm -1 ) Combined Optical Power (cm -1 ) Far 0.369 0.083 0.452 Normal 0.358 0.083 0.441 Near 0.352 0.083 0.436 This experiment will use the mean and propagation of uncertainty formulas to analyze the results. The calculations for mean and uncertainty are below. The average combined optical power of the measured values is:

𝑃 ???????? = Σ𝑥 𝑖 ? 𝑃 ???????? = 0.424+0.404+0.406 3 𝑃 ???????? = 0. 411 ?? −1 The uncertainty for the measured combined optical power is: δ𝑃 ???????? = Σ( ∂𝑃 ∂𝑥 𝑖 δ𝑥 𝑖 ) 2 δ𝑃 ???????? = ( ∂𝑃 ∂? δ?) 2 + ( ∂𝑃 ∂? δ?) 2 δ𝑃 ???????? = (− 1 ? 2 δ?) 2 + (− 1 ? 2 δ?) 2 The above propagation of uncertainty formula gives us the uncertainty for the measured data points. The results of applying this formula are below. The same 0.2 centimeter uncertainty, for the indistinguishable clearness, is used. Image Screen Slot Measured Combined Optical Power (cm -1 ) Measured Combined Optical Power Uncertainty (cm -1 ) Far 0.424 0.014 Normal 0.404 0.012 Near 0.406 0.012 The average of the measured combined optical powers uncertainty is δ𝑃 ???????? = Σ𝑥 𝑖 ? δ𝑃 ???????? = 0.014 + 0.012 + 0.012 10 δ𝑃 ???????? = 0. 0123 This brings the best estimate of the measured Combined Optical Power of 0.411 ± 0.0123 cm -1 . This gives us a range of 0.0399 cm -1 to 0.424 cm -1 . The average combined optical power of the calculated values is: 𝑃 ?????????? = Σ𝑥 𝑖 ? 𝑃 ?????????? = 0.452+0.441+0.436 3

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𝑃 ?????????? = 0. 443 ?? −1 The uncertainty for the calculated combined optical power is: 𝑃 ?????????? = 𝑃 1 + 𝑃 2 𝑃 ?????????? = 1 ? + 1 ? + 1 ? δ𝑃 ?????????? = Σ( ∂𝑃 ∂𝑥 𝑖 δ𝑥 𝑖 ) 2 δ𝑃 ?????????? = ( ∂𝑃 ????𝑖??? ∂? δ?) 2 + ( ∂𝑃 ????𝑖??? ∂? δ?) 2 + ( ∂𝑃 ????𝑖??? ∂? δ?) 2 δ𝑃 ?????????? = (− 1 ? 2 δ?) 2 + (− 1 ? 2 δ?) 2 + (− 1 ? 2 δ?) 2 The formula was applied to the data, giving us the following uncertainties. The same 0.2 centimeter uncertainty, for the indistinguishable clearness, is used. Image Screen Slot Measured Combined Optical Power (cm -1 ) Measured Combined Optical Power Uncertainty (cm -1 ) Far 0.452 0.084 Normal 0.441 0.084 Near 0.436 0.084 δ𝑃 ?????????? = Σ𝑥 𝑖 ? δ𝑃 ?????????? = 0.084+0.084+0.084 3 δ𝑃 ?????????? = 0. 0840 ?? −1 This brings the best estimate of the measured Combined Optical Power of 0.443 ± 0.0840 cm -1 . This gives us a range of 0.359 cm -1 to 0.527 cm -1 . Finally, we can do a t-score to compare the two data sets. We have the following data: 𝑃 ???????? = 0. 411 ?? −1 δ𝑃 ???????? = 0. 0123 𝑃 ?????????? = 0. 443 ?? −1 δ𝑃 ?????????? = 0. 0840 ?? −1

The T-Score is found using the following formula: ? = |𝑃 ???????? −𝑃 ?????????? | δ𝑃 ???????? 2 +δ𝑃 ?????????? 2 ? = |0.411−0.443| 0.0123 2 +0.084 2 ? = |0.411−0.443| 0.0123 2 +0.084 2 ? = 0. 376 Conclusion: The calculated t-score of 0.376 falls below the critical value of 1. This suggests that the measured and calculated values are from the same statistical population, supporting the hypothesis that the formula . accurately represents the combination of optical powers. Both the results from Part 𝑃 ????𝑖??? = 𝑃 1 + 𝑃 2 1 and Part 2 are consistent with this theoretical prediction. However, several sources of uncertainty must be acknowledged. The measurements of p and q are subject to the precision limitations of the ruler used, which has a resolution of 1.0mm and an inherent uncertainty of ±0.5mm. Additionally, subjectivity in determining the point of clearest image formation introduces a potential source of error. This experiment has elucidated the additive nature of optical powers when combining lenses. By selecting lenses that cumulatively provide the necessary optical power, we were able to achieve a desired focus, as demonstrated by the improved clarity of the image when a +120mm lens was introduced. This principle is directly applicable to corrective vision practices, indicating that prescriptive lenses can be tailored to enable individuals to see clearly both at distance and close range, by adjusting the total optical power to the needs of the user’s vision.

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