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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