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RC Circuits In today’s lab, you will construct a simple RC circuit using a battery, a resistor, a capacitor, and a switch. You will determine the voltage across the capacitor as it charges and discharges, and measure the time constant of the RC circuit. Background The figure to the right shows the circuit you will be constructing (Serway, Fig. 28.16). Suppose the switch is moved to position ‘a’. Charge will flow and a current will be present in the circuit until the potential difference across the capacitor equals the potential difference of the battery. When the switch is moved to position ‘b’ the capacitor discharges through the resistor. The potential difference between the capacitor plates and current in the circuit while the capacitor is charging are given by the following equations: −t ⎛ RC V (t ) = ε ⎜⎜1 − e ⎝ I (t ) = ε ⎛⎜ e R ⎜⎝ −t RC ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ Where ε is the emf of the battery, R is the equivalent resistance, C is the capacitance, and t is the time, in seconds. The potential difference between the capacitor plates and the current in the circuit while the capacitor is discharging are given by: −t ⎛ RC V (t ) = V0 ⎜⎜ e ⎝ I (t ) = ε ⎛⎜ e R ⎜⎝ −t RC ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ Where V0 is the potential difference between the plates of the fully charged capacitor, ε is the emf of the battery, R is the equivalent resistance, C is the capacitance, and t is the time, in seconds. The quantity RC which appears in the exponent of the above equations is called the time constant of the circuit. τ = RC The time constant is a good indicator of how fast the capacitor in the RC circuit charges and discharges. For instance, when t = τ for a discharging capacitor in an RC circuit, the potential difference across the capacitor plates is V = V0*e-1 = 0.368V0. In other words, after one time constant has elapsed, the potential difference has dropped to approximately 37% of its maximum value. In this experiment, you will use this fact to measure the time constant of an RC circuit. Preparation Complete the following steps and include them in the “Introduction” section of your lab report, along with the usual required material that should be in your introduction. 1. Draw the RC circuit with the switch in position ‘a’ as shown above, and use Kirchoff’s loop rule (ΣΔV = 0 around any closed loop) to derive a simple relationship expressing the voltage across the capacitor while it is charging in terms of the emf of the battery and the voltage across the resistor. Call this “Equation 1”. 2. Draw the RC circuit with the switch in position ‘b’ as shown above, and once again use Kirchoff’s loop rule to derive a simple relationship expressing the voltage across the capacitor while it is discharging in terms of the voltage across the resistor. Call this “Equation 2”. Remember, the battery is not part of the circuit while the capacitor is discharging. 3. In this experiment, you will be measuring the voltage across the resistor as the capacitor charges and discharges. Draw a schematic of the RC circuit in terms of the equipment at your lab station. Show how the resistor and capacitor are connected to the breadboard, how the voltmeter is connected, and how the switch will be wired up to charge and discharge the battery. 4. Calculate the value of the time constant of your circuit. Setting up the Experiment and Collecting Data 1. Set up the RC circuit you laid out in #3 above. Be careful to connect the capacitor in the correct direction or it will not charge correctly. (For the capacitors you will be using in this experiment, the negative side has a light blue stripe running down it.) You will use the PasPort Voltage/Current sensor with the Data Studio software to measure the voltage across the resistor in your circuit as the capacitor charges and discharges. 2. Set up Data Studio. The program should open automatically when you plug in the sensor, but if it doesn’t, open it manually and start a new experiment. Set up the program to display both a graph of voltage as a function of time, and a table of the data. Also, change the sampling rate of the voltage sensor to 2 Hz by selecting the voltage sensor on the left side of the screen and clicking ‘Setup’. 3. With the switch in the middle position (between points ‘a’ and ‘b’), turn on the power supply and set the terminal voltage to 9.0 V. Use the digital multimeter to check. You are now ready to start collecting data. 4. Click “Start” in Data Studio to start collecting data. Then flip the switch to position ‘a’ to charge the capacitor. Once the capacitor is fully charged, flip the switch to position ‘b’ to discharge it. Once the capacitor is fully discharged, click “Stop” in Data Studio to end data collection. Analyzing your data 1. Export the data you collected to a text file. In Data Studio, click Display → Export Data, and save your data as a .txt file on the Windows Desktop. 2. Open your data file in Microsoft Excel. You will have to do some work to get the formatting just right. Ultimately you want a spreadsheet with two columns. The first column should have the time, in seconds. You may have to create this column yourself if Data Studio did not export it. (The values should start at t = 0 and increase in increments of 0.5 seconds). The second column will be VR, the voltage measured across the resistor. 3. Create a third column in your spreadsheet and calculate the voltage across the capacitor as a function of time. Use your “Equation 1” to calculate the voltage while the capacitor is charging and your “Equation 2” to calculate the voltage while the capacitor is discharging. 4. Make a plot of Vc vs. time. You should print out a copy and include it with your report. Also, print out the table with your data. Make sure all the columns are labeled correctly. 5. Use your data to calculate the time constant of your circuit. Compare your measured value to the value you calculated earlier and compute a percent difference, and comment on how your measured value compares to your predicted value. Questions 1. In this experiment, you measured the voltage across the resistor in an RC circuit, then computed the voltage across the capacitor using the loop rule. Why couldn’t you have just measured the voltage across the capacitor directly? (Hint: an ideal voltmeter has infinite resistance. The ones we use are not ideal. They have a very high resistance, but it is not infinite. For instance, the digital multimeter has a resistance of 10 MΩ.) 2. You are given 3 resistors, each with a resistance of 1000-Ω, and two 20,000 μF capacitors. Design an RC circuit using all three resistors and both capacitors that has a time constant of 60 seconds. 3. Suppose you have an RC circuit with a time constant of 2.0 X 10-3 seconds. How long will it take for the potential difference between the capacitor plates to drop to 50% of its maximum value?