Lab 5 REDO

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Jan 9, 2024

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Table 5: Alexa Cumming ac75676 Luis Ainslie lma2862 Lab 5: Ohmic vs. Non-Ohmic Materials Part 1 Methods: In this lab we’ll test if these linear models genuinely do provide a good description of electrical effects in the real world. We will be using Ohm’s Law in which we will be attempting to show whether the resistor is an ohmic material by testing the equation ∆V = IR. V is the potential difference, I is our current that results when some of the electrons are free to move around and R is the resistance. Based on our quick check we predict that our materials tested will be ohmic due to the light bulb turning on and changing based on our actions (adjusting voltage and observing brightness increasing as voltage increases). We will be conducting our experiment on two different elements: the rheostat and the lightbulb. The rheostat is designed to be a “model system” and is expected to be ohmic, but to reduce uncertainty and increase our precision we will be conducting multiple trials. To test this model we will be considering the change in potential difference as our y value gets changed based on our change in current which is x. Our m value will be our resistance (following the y=mx linear approximation). We will test this on both of our materials and use excel to determine our experimental m and generate an equation to determine our calculated current. We will also conduct a chi-squared test to determine whether our rheostat is ohmic (this is concluded if x<1). If we get a linear slope we can conclude that the resistor is ohmic and an increase in voltage creates a linear increase in the current. If our resistor does not obey Ohm’s Law it will not be a linear slope and thus we can conclude that our material is not ohmic. Rheostat Voltage (V) 1.2 3.0 5.4 7.2 8.4 10.0 11.5 13.2 15.4 17.1 Current (A) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Experimental Resistance (Ω) 120.0 150.0 180.0 180.0 168.0 166.7 164.3 165.0 171.1 171.0 Table 1: Table showing the change in resistance based on manipulating the voltage and measuring the current. Calculated resistance was done using Excel. Rheostat Current (A) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Current based on Equation (A) 0.007 0.017 0.031 0.041 0.048 0.058 0.067 0.077 0.089 0.099 Chi-Squared Value 0.022 - indistinguishable Table 2: Table showing the current experimentally determined compared to the current determined by equation y (current) = 0.0058x (x=voltage).

Graph 1: Graph showing the relationship between current and voltage on the Rheostat. Light Bulb Voltage (V) 1.4 3.3 5.3 7.5 8.4 10.3 11.6 13.6 15.7 17.1 Current (A) 0.43 0.58 0.73 0.84 0.90 1.01 1.06 1.16 1.24 1.3 Experimental Resistance (Ω) 3.26 5.69 7.26 8.93 9.33 10.19 10.9 11.72 12.66 13.16 Table 3: Table showing the change in resistance based on manipulating the voltage and measuring the current. Calculated resistance was done using Excel. Light Bulb Current (A) 0.43 0.58 0.73 0.84 0.90 1.01 1.06 1.16 1.24 1.3 Current based on Equation 1 (A) 0.08 0.18 0.29 0.41 0.455 0.56 0.63 0.74 0.85 0.93 Chi-Squared Value 4.12 (distinguishable) meaning we will adjust the equation in an attempt below Table 4: Table showing the current experimentally determined compared to the current determined by equation y (current) = 0.0542x (x=voltage). Light Bulb Current (A) 0.43 0.58 0.73 0.84 0.90 1.01 1.06 1.16 1.24 1.3 Current based on Equation 2 (A) 0.42 0.59 0.70 0.82 0.87 0.97 1.04 1.15 1.26 1.34 Chi-Squared Value 0.05 (indistinguishable) Table 5: Table showing the current experimentally determined compared to the current determined by equation y (current) = 0.0542542 + 0.4141.(x=voltage).

Graph 2: Graph showing the relationship between current and voltage on the light bulb. Sample Calculations: - done using Excel Sheets 𝑉 = 𝐼𝑅 = 𝑅 = 𝑉 𝑅 Chi-Squared Test: ? 2 = 1 𝑁 𝑖 = 1 𝑁 ∑ (? 𝑖 − ?(? 𝑖 )) 2 δ? 𝑖 2 Rheostat: + + + … = 0.005 ? 2 = 1 10 𝑖 = 1 10 ∑ (0.01−( 0.007)) 2 0.1 2 (0.02−( 0.017)) 2 0.1 2 (0.03−( 0.031)) 2 0.1 2 (0.04−( 0.041)) 2 0.1 2 (0.1−( 0.099)) 2 0.1 2 x = 0.022 Light Bulb: + + + … = 17.01 ? 1 2 = 1 10 𝑖 = 1 10 ∑ (0.43−( 0.08)) 2 0.1 2 (0.58−( 0.18)) 2 0.1 2 (0.73−( 0.29)) 2 0.1 2 (0.84−( 0.41)) 2 0.1 2 (1.3−( 0.93)) 2 0.1 2 x = 4.12 Light Bulb: + + + … = 0.003 ? 2 2 = 1 10 𝑖 = 1 10 ∑ (0.43−( 0.42)) 2 0.1 2 (0.58−( 0.59)) 2 0.1 2 (0.73−( 0.70)) 2 0.1 2 (0.84−( 0.82)) 2 0.1 2 (1.3−( 0.1.34)) 2 0.1 2 x = 0.05 Conclusion: Based on our results we are able to conclude that the Rheostat is an Ohmic material as based on the chi-squared test we got that x <1. This also allows us to conclude that Ohm’s law follows a linear distribution and we can say with certainty that an increase in voltage creates a linear increase in current. As this was to be expected from the rheostat, we also tested a light bulb and found that it does not follow a linear relationship quite as accurately as the rheostat. In our first comparison using a chi-squared test we found that the x>3 meaning there was a significant difference between our expected and observed results. In order to minimize the ‘chi’ we added a “b” and modified the equation to follow a y = mx+b format. This we found resulted in a much more significant comparison between the experimental and calculated results and we were able to conclude that Ohm’s law will follow a linear distribution with a y-intercept. A potential reason for this might be that some of the current (in terms of electrons flowing) may get ‘lost’ and not be accounted for by the ammeter (measuring the current) when we use a material that is not an Ohmic material. Therefore, b could potentially physically represent the current loss. For our next iteration of this experiment and to further our analyses, we could test whether more than one rheostat follows the same rules in an experimental procedure as they do in a scientific theory by adding them in parallel or in series. Additionally, we could also further this same analysis and test other materials to gain more of an

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understanding on what type of objects are considered Ohmic materials and potentially give us more insight into why some materials need a modified equation. A last potential iteration would be to solely focus on understanding why some materials need the adjustment in equations and potentially investigating under what conditions are certain materials more accurate or gain a deeper understanding into what the different adjustments to the equation translate to in real life. Part 2 Methods: Based on our results from part 1 we will be testing Ohm’s law both in series and in a parallel circuit. We will be testing to see if the the resistance adds linearly (in series - R total = R 1 + R 2 ) and inversely (in parallel To test this we combined groups with Madelyn and Fran to use two 1 𝑅 ????𝑙 = 1 𝑅 1 + 1 𝑅 2 ). rheostats and connect them both in the aforementioned positions so that current passes through based on a changed voltage. We will use a t-test to determine if our expected values match with our calculated values of resistance. Additionally, we will also test a copper wire to see if this material is ohmic or not. This test will be done by placing this known conductor in series with the rheostat and calculating the chi-square value to determine its relation regarding Ohm’s Law and the graph’s linear equation (y=mx). In order to limit uncertainty we will be conducting multiple trials and attempting to be as precise as possible in our readings. This is using the same theories as part 1 but applying the scientist’s model in a different way and determining if the equation applies to the three different scenarios. Rheostat in Parallel Voltage (V) 1.3 3.9 4.7 5.7 7.3 9.0 11.5 13.0 15.0 16.4 Current (A) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Resistance (Ω) Rheostat 1 120.0 150.0 180.0 180.0 168.0 166.7 164.3 165.0 171.1 171.0 Resistance (Ω) Rheostat 2 130 185 160 160 160 151.7 151.4 158.8 167.8 173 Resistance in Parallel 62.4 82.8 84.7 84.7 82.0 79.4 78.8 80.9 84.7 86.0 Theoretical Resistance 65.0 97.5 78 71.3 73.0 75.0 82.0 81.3 83.3 82.0 T-score 0.005 0.03 0.01 0.02 0.02 0.009 0.006 0.0008 0.003 0.008 Table 6: Table showing the data gathered during the placement of two parallel Rheostats Rheostat in Series Voltage (V) 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 12.0 Current (A) 0.04 0.04 0.05 0.05 0.05 0.05 0.06 0.06 0.07 0.07 Resistance (Ω) 175.0 187.5 160.0 170.0 180.0 158.3 166.67 175.0 157.14 171.4

Rheostat 1 Resistance (Ω) Rheostat 2 175.0 160.0 180.0 166.7 157.1 171.4 162.5 155.6 166.6 160.0 Resistance in Series 350.0 400.0 450.0 333.3 366.7 400.0 433.3 350.0 375.0 400.0 Theoretical Resistance 350.0 375.0 348.0 336.7 337.1 329.7 329.17 330.6 323.74 331.4 T-score 0 0.05 0.2 0.006 0.06 0.1 0.2 0.04 0.1 0.1 Table 7: Table showing the data gathered during the placement of two Rheostats in Series Sample Calculations: R in series: R total = R 1 + R 2 = 175.0 + 175.0 = 350.0 Ω R in parallel: = Ω 1 𝑅 ????𝑙 = 1 𝑅 1 + 1 𝑅 2 1 120 + 1 130 = 1 130 Expected values of resistance: V = IR = 𝑉 𝐼 = 7.0 0.04 = 175Ω (????? 𝑖? ????𝑙𝑙?𝑙 ?? ?????) = indistinguishable ? = 𝑃 1 − 𝑃 ??𝑖?ℎ??? | | δ𝑃 1 2 + δ𝑃 ??𝑖?ℎ??? 2 400− 375 | | 0.1 2 + 500 2 = 0. 05 Copper Wire Coil (30V) Voltage (V) 1.7 4.3 7.6 9.7 12.5 15 17.8 Current (A) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Experimental Resistance (Ω) 170 215 253.3 242.5 250 250 254.3 Table 8: Table showing the change in voltage based on change in current. There are only 7 trials as the voltage cannot increase past 18.6 V. Copper Wire Coil (30V) Current (A) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Current based on Equation (A) 0.006 0.02 0.03 0.04 0.05 0.06 0.07 Chi-Squared Value 0.00023 Table 9: Table showing the current experimentally determined compared to the current determined by equation y (current) = 0.0038x (x=voltage).

Graph 3: Graph showing the relationship between current and voltage on the copper wire coil. Sample Calculations: - done using Excel Sheets 𝑉 = 𝐼𝑅 = 𝑅 = 𝑉 𝑅 Chi-Squared Test: ? 2 = 1 𝑁 𝑖 = 1 𝑁 ∑ (? 𝑖 − ?(? 𝑖 )) 2 δ? 𝑖 2 Copper Wire: + + + … = ? 2 = 1 7 𝑖 = 1 7 ∑ (0.01−(0.006 )) 2 0.1 2 (0.02−(0.02)) 2 0.1 2 (0.03−(0.03)) 2 0.1 2 (0.04−(0.04)) 2 0.1 2 (0.07−( 0.07)) 2 0.1 2 0.00023 Conclusion: Overall, we would say that the scientific model was very accurate based on our results of the experiment in both Part 1 and Part 2. The linearity of the model was seen through multiple chi-squared goodness of fit tests, including multiple iterations both in series and in parallel as well as the effect on different conducting materials such as the copper wire (the light bulb did not appear to be ohmic however). Regarding the Copper Coil as our object of choice, we gathered a Chi-Square value of 0.00023, which being less than 1 is an indication that this material is Ohmic in nature. This is proven through the linear equation of y=mx, where the voltage of the system is directly proportional to the current applied, while the resistance of the circuit is kept calculated. As for the Rheostats in Series and in Parallel, both of them had T-scores that were less than 1, indicating that both instruments were indistinguishable when providing resistance to a circuit. This was proven through the summation and inverse equations for Series and Parallel resistors, respectively. Both the resistors in series and parallel showed indistinguishable results meaning we can accept the scientists model. Comparing our Rheostat results in Part 1, it seems as though despite the position, or which Rheostat is incorporated, the instrument itself is viable as an Ohmic material for future experiments regarding circuits, current, and voltage. In future iterations of this experiment we could potentially test other insulating materials and further distinguish the extent and properties of different materials and determine whether they are to be considered ohmic materials. Additionally, our scientists model proved to be very accurate in explaining

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our results so future steps to test additional models that include other components could use the same materials and procedure as this method worked very successfully for us. The accuracy and precision of our results were further supported by comparing our data and analysis with another group and realizing that we both share the same conclusion.