Physics 2 Lab 7

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Massachusetts Institute of Technology *

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8.02

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Electrical Engineering

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Oct 30, 2023

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pdf

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RC Circuits Lab Number and Title: 217 RC Circuits Name: Aaron Hsu Group ID: N/A Date of Experiment: 11/1/22 Date of Submission: 11/8/22 Course and Section Number: PHYS 121A013 Instructor’s Name: Matias Daniel de Almeida Partner’s Names: Paul Svorec, Alex Ack, and Noah Francois 1. Introduction: The goals and objectives of this lab are to observe and analyze the voltage that is going across a capacitor as a function of time(t) in a circuit that has a resistor and a capacitor 𝑉 ? that are being connected in series. When discharging a capacitor in this experiment at t < 0 the switch is in position A and the voltage on the capacitor is which means that the 𝑉 0 capacitor(C) is fully charged. Since there is no current in charging the capacitor. 𝑉 0 = 𝑉 ? When the time is 0 the witch will be moved to position B and the capacitor discharges through the resistor(R). when 𝑉 0 = 𝑉 ? −> 𝑉 ? = 𝑞 0 𝐶 = 𝑉 0 −> 𝐼 = 0 −> 𝑉 ? (𝑡) = 𝑞(𝑡) 𝐶 −> 𝑖𝑅 = ( ?𝑞 ?𝑡 )𝑅 substituting into second Kirchhoff’s law (for loops), then the solution ( ?𝑞 ?𝑡 )𝑅 + 𝑞(𝑡) 𝐶 = 0 to this equation is where is the charge on the capacitor at t=0. Since 𝑞(𝑡) = 𝑞 0 ? − 𝑡 (𝑅𝐶) 𝑞 0 the voltage across the capacitor is q(t)/C this means that you 𝑉(𝑡) = 𝑞(𝑡)/𝐶 = 𝑉 0 ? −𝑡/(𝑅𝐶) can take the logarithm of both sides so that which is a straight 𝑙𝑛[𝑉(𝑡)/𝑉 0 ] =− ( 1 𝑅𝐶 )𝑡 line with a slope of -1/(RC). The half-life is defined as the time that it takes for the 𝑡 1/2 max voltage to decrease by half which is and when you 𝑉(𝑡 1/2 ) = 𝑉 0 2 = 𝑉 0 ? −𝑡 1/2 (𝑅𝐶) solve for the half life you get . When charging the capacitor with the 𝑡 1/2 = 𝑅𝐶𝑙𝑛2 following circuit where t<0 the switch would be in position B and the voltage on the

capacitor is 0. At t=0 the switch would be moved to position A and then a voltage is 𝑉 0 applied to the RC circuit with the capacitor being charged through the resistor. and the substituting into 𝑉 ? = 0 −> 𝐼 = 0 −> 𝑉 ? (𝑡) = 𝑞(𝑡)/𝐶 −> 𝐼𝑅 = ( ?𝑞 ?𝑡 )𝑅 second Kirchhoff’s law (for loops) it is then the solution to this ( ?𝑞 ?𝑡 )𝑅 + 𝑞(𝑡) 𝐶 = 𝑉 0 equation is where is the charge on the capacitor at t= and 𝑞(𝑡) = 𝑞 0 (1 − ? −𝑡/(𝑅𝐶) ) 𝑞 0 ∞ since the voltage across the capacitor is then where is the 𝑞(𝑡) 𝐶 𝑉(𝑡) = 𝑉 0 (1 − ? −𝑡/(𝑅𝐶) ) 𝑉 0 voltage on the capacitor at t= . You can take the logarithm of both sides from the ∞ previous equation so which is just a straight line with slope 𝑙𝑛[1 − ( 𝑉(𝑡) 𝑉 0 )] =− ( 1 𝑅𝐶 )𝑡 being -1/(RC). 2. Experimental Procedure: The equipment that we used for this lab are a digital multimeter, banana cables, computer with capstone software installed, switch box, electrical circuit connection board, capacitors of 1000 F and 0.47 F, DC power supply, circuit connectors, two voltage µ µ probes, stopwatch, 850 universal interface, and resistors of 33 K and 10 K . We Ω Ω followed the same procedure as in the lab manual. For the measurement of long time constants using a digital multimeter for the discharging capacitor we first would set up the circuit and use the 1000 F capacitor and a 33,000 resistor in series. We then µ Ω would make sure that the switch on the switch box is in the A position which would be the red port and charge the capacitor so that it is reading 5 V in the voltmeter. We then would turn the switch to the B position in order to discharge the capacitor which would be the black port and as soon as we did that we would start the stopwatch. We would record the voltage change across the capacitor that is measured by the digital multimeter over time. Now with charging the capacitor we do the same thing but this time start the switch box at position B and make sure the voltmeter is reading 0 and turn on the stopwatch as soon as we switch it to position A. For the next part with measurement of short time constants we will make the circuit with 10 K and 0.47 F on the electrical Ω µ circuit connection board. We then would make sure to plug the right cables into their desired spots stated in the lab manual. We then would set up the computer and the 850

universal interface for our data. We would then start recording it and analyze the graphs along with the results that we got. 3. Results: Part A R [Ω] C [µF] t V i [V] V f [V] Charging 33,000 1000 110.37 0.006 4.992 Discharging 33,000 1000 173.37 4.96 0.188 Part B ½ Life Voltage time 0 (start) 4.9866 0 1 2.4933 0.1234 2 1.24665 0.1315 3 0.6233 0.1346

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4. Analysis and Discussion: When we measure the time it takes to charge and discharge the capacitor as well as the voltage over that time, it allows us to be able to predict the voltage. The voltages Kirchoff's law was used to predict the flow of current through the circuit and logarithms were used to predict and verify that the experimental data was the same as the theoretical. There was a very small amount of error in this lab due to the precise measurement instruments as well as there being little room to make errors. The results that we got were very accurate but the small difference between the theoretical and experimental results is from the resistance in the wires which we assumed to be zero. There was also error due to the fact that the capacitor was not ever fully charged or discharged which would be impossible since it would require an infinite amount of time to do either but if it was possible then the experimental values would even more closely match the theoretical

equations. The objectives of this lab, analyzing the voltage across a capacitor as it charges and discharges was achieved. 5. Conclusion: The objective of this lab was to explore the logarithmic relationship between voltage as a function of time in resistor-capacitor circuits (RC circuits). The time it takes to charge a capacitor is the same amount of time as it is to discharge it. Also, the slope of the logarithmic relationship is inversely proportional to the negative of the product of capacitance and resistance. The results that we received from the lab show and demonstrate these concepts and were overall successful in showing the relationships in RC circuits.