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Name________________________________ Date_________________ Partner(s)______________________ ______________________ Physics Laboratory The RC Series Circuit The figure below shows an RC series circuit with a battery of voltage Vo and a SPDT switch. A voltage detecting device is connected across the capacitor. up SPDT R down Red voltage C to: Vo detecting Black device At time t = 0, the switch is moved to the “up” position. In this case, the charges will be delivered to the capacitor and we say that the capacitor “is charging.” Without looking at your notes or the text, deduce and write the equation for the voltage across the capacitor as a function of time. actual equation for the voltage across the capacitor deduced voltage across the capacitor Now suppose we move the switch to the “down” position. In this case, the charges leave the capacitor and we say that the capacitor is “discharging.” If we reset the time to t = 0 when the switch is moved down, then deduce and write the equation for the voltage across the capacitor as a function of time. actual equation for the voltage across the capacitor deduced voltage across the capacitor For the capacitor charging and discharging, write the value of the voltage across the capacitor in terms of Vo. Do this at times t = 0, and after a time equal to one time constant, two time constants, three time constants, four time constants, and five time constants. Generally, we say that the voltage has reached its final value after a time equal to five time constants. t=0 charging discharging 1 2 3 4 5 In this experiment, we will graph the voltage across a 1.00 F capacitor while it is charging and discharging when connected to a resistor. The experimental time constant, found from the graph and by curve-fitting, will be compared with the theoretical time constant. Equipment and Supplies needed voltage probe Universal Laboratory Interface (ULI) Logger Pro 2.2.1 2 D cells with battery holder SPDT switch 1 Farad capacitor 1-ohm and 2 -ohm resistors connecting wires The Experiment 1. Connect the circuit as shown in the figure above with the 1.00-F capacitor and the 1.00- resistor. Record the values of your resistor and capacitor in your data table, as well as any tolerance values marked on them. 2. Connect the Voltage Probe to the ULI and across the capacitor, with the red (positive lead) to the side of the capacitor connected to the resistor. Connect the black lead to the other side of the capacitor. 3. Prepare the computer for data collection by opening “Exp 27” from the Physics with Computers experiment files of Logger Pro 2.2.1. A graph will be displayed. The vertical axis of the graph has voltage scaled from 0 to 4 V. The horizontal axis has time scaled from 0 to 10 s. 4. Click to begin data collection. As soon as graphing starts, throw the switch to its “up” position to charge the capacitor. Your data should show a constant zero value initially, and then an increasing function. Allow the data collection to run to completion. 5. To compare your data to the model, select only the data after the potential has started to increase by dragging across the graph; that is, omit the constant portion. This time you will compare your data to the mathematical model for a capacitor charging, t V (t ) Vo 1 e RC . Click the curve fit tool, , and from the function selection box, choose the Inverse Exponential function, A*(1 – exp(–C*x)) + B. Click and inspect the fit. Click to return to the main graph window. How is fit constant C related to the time constant of the circuit? 6. Record the value of the fit parameters in your data table. Notice that the C used in the curve fit is not the same as the C used to represent capacitance. Compare the fit equation to the mathematical model for a charging capacitor. 7. Print the graph of potential vs. time. 8. Now you will examine the discharging capacitor. Since the capacitor is already charged, click to begin data collection. As soon as graphing starts, throw the switch to the other position (“down”) to discharge the capacitor. Your data should show a constant value initially, then a decreasing function. 9. To compare your data to the model, select only the data after the potential has started to decrease by dragging across the graph; that is, omit the constant portion. Click the curve fit tool , and from the function selection box, choose the Natural Exponential function, A*exp(–C*x ) + B. Click , and inspect the fit. Click to return to the main graph window. 10. Record the value of the fit parameters in your data table. Notice again that the C used in the curve fit is not the same as the C used for capacitance. Compare the fit equation to the mathematical model for a capacitor discharge proposed in the introduction, V(t ) Vo e t RC . 11. Print the graph of potential vs. time. 12. Now repeat the experiment with a different resistor. How do you think this change will affect the way the capacitor discharges? Rebuild your circuit using the new resistor and repeat Steps 4 – 11. Data Table Curve Fit Values Trial A B C Resistor Capacitor R () C (F) Charging 1 Discharging 1 Charging 2 Discharging 2 Analysis 1. Calculate and enter into the results table the theoretical time constant of the circuit using your resistor and capacitor values. 2. Calculate the inverse of the fit constant C (1/C) for each trial and enter them as curve-fitting determined time constants in the results table. 3. On each of your four graphs, choose an arbitrary starting point for the charging and discharging curves and treat that point as time t = 0. From this point, find the 63% voltage position for the charging situation and the 37% position for the discharging situation, and find the time it takes to reach those voltages. Enter these time values as the graphically-determined time constants, and place them in your results table. 4. Now compare each of the graphically-determined values to the time constant with the theoretical time constant by calculating a percentage error. of your circuit by calculating a percentage error. Trial Theoretical Time Constant (s) Curve Fit Time Constant (s) Percentage Error Graphical Time Constant (s) Percentage Error Charging 1 Discharging 1 Charging 2 Discharging 2 5. Note that the resistors and capacitors are not marked with their exact values, but only approximate values with a tolerance. If the tolerance is taken into account, does the amount of error between the theoretical and experimental values of the time constant decrease.? If so, show an example;.