Lab 2 PHY 105N

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University of Texas *

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105N

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Electrical Engineering

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Dec 6, 2023

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Lab 2: Lenses and Uncertainty Propagation Part 1: Equations Optical Power → 𝑃 = 1 𝑓 Optical Equation → 1 𝑓 = 1 ? + 1 ? Equipment Chosen PASCO light source, PASCO optics track, meter stick, convex lens, and screen Intro + Prediction According to the model, an object placed at “p” distance away from a lens will result in an image being produced at a distance of “q” from the lens. Moreover, the p value is the distance between the object and the lens and the q value is the distance between the lens and the image formed. The optical power equation above shows that P is equal to which is equal to , and that P is a constant. Thus, in this experiment, we hypothesize 1 𝑓 1 ? + 1 ? that by placing an object at different locations and measuring the corresponding distances, p and q, our subsequent, calculated P values will all be the same. Method To test our hypothesis regarding the fact that P is a constant, we will be using a convex lens to focus the image from the PASCO light source onto the image screen on the optics track, and we will be using the included meter stick on the optics track to measure p and q . We will conduct 3 trials per each lens over varying distances of p and q to determine if the model is valid. We will then calculate P for each trial. To calculate the constant P, we will propagate the uncertainties of p and q to find the uncertainty of P, δP. The uncertainty of q will be the range in which the image begins to appear clear to where it begins to become blurry again. Our goal is to minimize this uncertainty. Data CONVEX LENS Trial P (cm) δ p +/- p (cm) δ q +/- q (cm) δ P = 1 ? + 1 ? T-Scores 1 0.002 42.0 +/- 0.05 40.8 +/- 2.7 0.048 Trial 1&2 = 0 Measured optical power (P) is indistinguishable from the original calculated value of the optical power. 2 0.008 67.4 +/- 0.05 30.0 +/- 7.3 0.048 Trial 1&3 = 0 3 0.001 35.4 +/- 0.05 50.0 +/- 2.8 0.048 Trial 2&3 = 0 Mean 0.004 0.048 cm -1

Mean P: = 0.048 cm -1 0.048 + 0.048 + 0.048 3 Standard Deviation of P: 0.004 cm -1 P = 0.048 +/- 0.004 cm -1 Propagation of Uncertainty Equation: δ𝑃 = −1 ? 2 × δ? ( ) 2 + −1 ? 2 × δ? ( ) 2 Sample Calculation (Trial 1) = 0.0016 → 0.002 cm δ𝑃 = −1 42.0 2 × 0. 05 ( ) 2 + −1 40.8 2 × 2. 7 ( ) 2 GLASS BALL Trial P (cm) δ p +/- p (cm) δ q +/- q (cm) δ P = 1 ? + 1 ? T-Scores 1 0.16 9.6 +/- 0.05 5.0 +/- 4.1 0.048 Trial 1&2 0 2 0.03 8.4 +/- 0.05 7.0 +/- 1.6 0.048 Trial 1&3 0 3 0.01 5.0 +/- 0.05 10.0 +/- 1.1 0.048 Trial 2&3 0 Mean 0.067 cm -1 0.048 cm -1 Mean P: = 0.048 cm 0.048 + 0.048 + 0.048 3 Standard Deviation of P: 0.067 cm P = 0.048 +/- 0.067 cm -1 Propagation of Uncertainty Equation: δ𝑃 = −1 ? 2 × δ? ( ) 2 + −1 ? 2 × δ? ( ) 2 Sample Calculation = 0.16 cm δ𝑃 = −1 9.6 2 × 0. 05 ( ) 2 + −1 5.0 2 × 4. 1 ( ) 2

T-Score Calculations ● Convex Lens = 0 0.048 − 0.048 | | 0.002 2 + 0.008 2 = 0 0.048 − 0.048 | | 0.002 2 + 0.001 2 = 0 0.048 − 0.048 | | 0.008 2 + 0.001 2 ● Glass ball = 0 0.048 − 0.048 | | 0.16 2 + 0.03 2 = 0 0.048 − 0.048 | | 0.16 2 + 0.01 2 = 0 0.048 − 0.048 | | 0.03 2 + 0.01 2 Conclusion In this lab, we measured values of p and q using both a +200mm PASCO convex lens and a glass ball to confirm if P, optical power, is truly constant. For each type of lens, three trials were conducted to find a P value, and then we calculated the mean P value and the standard deviation of the mean to determine how much the P values differed from each other between trials. Our goal was to minimize the uncertainty of our measurements in order to ensure that our measured values of P are indistinguishable. To do this, we propagated the uncertainty and calculated t-scores between our calculated P values to determine if they were indistinguishable or not. All of our T-Scores between the P values of the glass ball and the convex lens were calculated to be 0. This means that our calculated P-values are indistinguishable from one another and thus, our hypothesis was proved to be valid. Moreover, our experiment proved that P is constant. To minimize the uncertainty in our next experiment, we could implement numerous different techniques. First, when determining the range of values over which the image is clear, we could be more precise in determining this value which will lead to a smaller uncertainty in q and thus a smaller overall uncertainty for P. In addition, more trials over a larger range of distances would be sufficient to minimize our uncertainty as well in future iterations of this experiment. Part 2: Equipment Chosen PASCO optics track, PASCO light source, eye model, eye model lens set

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Method In the second iteration of this experiment, we are continuing to investigate optical power, yet this time by investigating the model P Combined = P Eye + P Lens in combination with P = . We hypothesize that the model will 1 ? + 1 ? prove to be valid in demonstrating how corrective lenses aid in correcting vision. To test the model, we will have a few different setups: normal vision, myopia, or nearsightedness, and hyperopia, or farsightedness. For each setup, we will measure p and q by finding the distance between the object and the lens and the distance between the lens and the image. We will use the same equipment as part 1, in addition to a +400mm lens used in the 2nd part of Part 2. For the first part of this section, we will find p , q , and P using the PASCO eye model bracket and a lens with a focal length of +62mm. We will measure the corresponding distances when the retina is placed in the normal position, myopia position, and hyperopia position. For the 2nd part, we will add a +400mm lens in addition to the +62mm lens to the PASCO eye bracket model. Then, we will find the p, q, and P values in both a myopia condition and a hyperopia condition. Data Part A: +62mm Lens ONLY Trial P (cm) δ Object Distance p + p (cm) δ q + q (cm) δ P = (cm -1 ) 1 ? + 1 ? T-Scores 1 (Normal) 0.118 39.3 +/- 0.05 10.5 +/- 13.0 0.121 Trial 1&2 0.044 2 (Near) 0.068 29.1 +/- 0.05 10.8 +/- 7.9 0.127 Trial 1&3 0.051 3 (Far) 0.00014 109.3 +/- 0.05 8.5 +/- 0.01 0.127 Trial 2&3 0.00 Part B: +400mm Lens and +62mm Lens Trial P (cm) δ Object Distance p + p (cm) δ q + q (cm) δ P = 1 ? + 1 ? P combined = P eye + P lens 1 (Near) 0.024 14.9 +/- 0.05 15.1 +/- 5.4 0.133 0.146 2 (Far) 0.079 31.5 +/- 0.05 11.9 +/- 11.3 0.116 0.146 PART A: Finding P Eye = 0.146 cm -1 PART B: P combined = P Eye + P Lens P Eye was calculated to be 0.125 cm -1 . P Lens is equal to = 0.025 cm -1 . 1 𝑓 = 1 40𝑐𝑚

T-Score Calculations = 0.044 0.121 − 0.127 | | 0.118 2 + 0.068 2 = 0.051 0.121 − 0.127 | | 0.118 2 + 0.00014 2 = 0.00 0.127 − 0.127 | | 0.068 2 + 0.00014 2 Conclusion In Part A of our data collection process, we used multiple eye conditions to calculate the power of the lens built into the eye model without any corrective lenses added to the model. In Part B, we found the optical power of the lenses with a corrective lens added to the model. To calculate the accuracy of our data we gathered we used the propagation of uncertainty and t-score tests. For the normal vision configuration without any corrective lenses applied we got an optical power (P) of 0.121 +/- 0.118 cm -1 . We then used corrective lenses on the eye model to determine if the eye can be corrected and obtain similar values of optical power for both far and near sighted. Our calculations determined that the new optical power was 0.146 for both near and far sightedness in the experiment so we achieved our goal in getting corrected vision in the eye model to where they are indistinguishable. The results of the experiment proves the model is accurate. The optical power of the combined lens was proved to be indistinguishable based on the calculated t-scores t < 1 for all object distances measured. For further iterations of this experiment, it would be important to increase the number of trials in order to minimize the uncertainties in the measurements. In addition, we could potentially use a longer track to increase the range of data we are able to take measurements and draw conclusions from.

Lab 2 PHY 105N

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