Lab Report 3

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School

University of Notre Dame *

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10001

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Electrical Engineering

Date

Dec 6, 2023

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What is the Relationship Between Resistance and the Length of a Wire? Katrina Chin, Alyssa Davis, Allysyn Pajek University of Notre Dame Introduction As an electron travels through wires and loads of the external circuit, it encounters resistance that hinders the flow of charge. Three variables that affect the total amount of resistance to charge flow within a wire of an electric circuit are the total length of the wires, the cross-sectional area of the wires, and the material the wire is made of. These three variables respective effects on a wire’s resistance are demonstrated in the following equation, where R is resistance, ⍴ is resistivity, L is length, and A is cross-sectional area of the wire. (1) 𝑅 = ρ𝐿 𝐴 Resistance can also be related to voltage and current via conductivity, the inverse of resistivity. In most conductors, the electric current that flows through them is directly proportional to the voltage applied. Consequently, the resistance can be determined by finding the ratio of voltage to current. This concept is expressed as what is known as Ohm’s Law, where I is the electric current, V is the voltage, and R is the resistance. (2) 𝐼 = 𝑉 𝑅 In this experiment, the relationship between a wire’s resistance and the total length of the wire is investigated. This experiment aims to accomplish three main objectives: (1) to understand the behavior of resistance, (2) to observe the relationship between resistance and length of wire, and (3) to connect geometry of wires to the physics of resistivity in wires. As the total length of the wires increases, the voltage is expected to decrease, the resistance is expected to increase, and the current is expected to remain constant, as demonstrated in the two equations aforementioned. Procedure In this experiment, a circuit board was set up as indicated below (Figure 1). Figure 1: Circuit Apparatus. After measuring the current using the digital multimeter (DMM), the switch was removed and replaced by the black and red wires before recording the voltage at various lengths along the outer dots. To measure the current, a battery was connected to one side of the circuit and a resistor to the other. A switch was then placed adjacent to the battery and turned on. After attaching two wires (one black and one red) to a digital multimeter (DMM), the DMM was used to measure and record the voltage and current. Next, the switch was removed and replaced by the black and red wires. After recording the voltage at these points, the red wire was moved to © 2022, K Chin, A Davis, A Pajek - Page 1

the adjacent dot (increasing its distance from the black wire), and the new resulting voltage was recorded. This process was repeated until a total of 24 data points were recorded. Derivation-based analysis using equation 2, the recorded voltages, and current was used to calculate resistance, R followed by error propagation to calculate the uncertainty of the resistance incurred through measuring voltage, V, and current, I. Regression analysis (fitting a line to a plot of R vs. L) was used to calculate ⍴ /A and the uncertainty of this value. Finally, this calculated ⍴ /A value was compared to the theoretical value found in the circuit manual to calculate percent error. Experimental Observations/Data For this experiment, the voltage of a circuit was measured at 24 equidistant data points, with subsequent points increasing in distance by 2.8 cm. A single measurement for the current of the circuit was also taken at the beginning of the experiment. Using these data and Equation 2, the resistance at each point was calculated and plotted against the length of the wire in order to determine a constant value for ohms per centimeter. Figure 2 shows the result of this analysis. Figure 2: Graph of resistance in ohms versus length in centimeters of a circuit. The plot contains 24 data points. Regression analysis yields a slope of m = 0.7701 ohm/cm. A digital multimeter (DMM) was used to measure both the current of the circuit and its voltage at any given length. A plastic ruler was used to measure the length of the wire at each data point. Tool errors for the voltmeter, ammeter, and ruler are summarized in Table 1. Because measurements for each data point were taken only once, all process errors are assumed to be zero. Thus, the total error for all variables (voltage, current, and length) were calculated and reported in Table 1 using the following equation: (3) σ ???𝑎𝑙 2 = σ ???𝑙 2 +σ ???𝑐𝑒?? 2 Table 1: Tool, process, and total error for voltage, current, and length measurements. Variables σ ???𝑙 σ ???𝑐𝑒?? σ ???𝑎𝑙 Voltage ± 4.404 mV 0 ± 4.404 mV Current ± 5.8768 μA 0 ± 5.8768 μA Length ± 0.05 cm 0 ± 0.05 cm Results and Conclusions For this experiment, the voltage at different distances in a circuit was measured at a constant current. This left the variables of resistivity (ρ) and resistance (R) unknown. Resistance was calculated by dividing the voltage at each point by the current. The graph in the previous section (Figure 2) shows a positive trend, indicating that resistance increases with distance. The error of resistance was calculated using the equation © 2022, K Chin, A Davis, A Pajek - Page 2

σ 𝑅 = ( δ𝑅 δ𝐼 ) 2 •σ 𝐼 2 + ( δ𝑅 δ𝑉 ) 2 •σ 𝑉 2 where and . By using our values δ𝑅 δ𝐼 = −𝑉 𝐼 2 δ𝑅 δ𝑉 = 1 𝐼 found for current, voltage, and error for each, σ 𝑅 was found to be ± 30.45 ohms. Since A, the cross sectional area of the wire, was too small to measure, the ratio of resistivity over area was calculated for each distance. Upon taking the average of the values, it was calculated that ρ/A was approximately 7.90 Ω • cm. Since each value was only measured once, error in this experiment can be attributed to the tools and their functions alone. To improve this experiment in the future, measurements should be taken multiple times to ensure accuracy. References [1] Halliday, David, Robert Resnick, and Kenneth S. Krane. Physics. New York: Wiley, 2002. Print. Appendix Lab 3 Data © 2022, K Chin, A Davis, A Pajek - Page 3

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