# Experiment 9 ~ RC Circuits

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1 Experiment 9 ~ RC Circuits Objective: This experiment will introduce you to the properties of circuits that contain both resistors AND capacitors. Equipment: 18 volt power supply, two capacitors (8 µf and unknown capacitor), USB voltmeter, computer with Data Studio. Theory: In this experiment, you will work with a circuit where a resistor is combined in series with a capacitor. This is called an RC circuit. Remember from lecture that capacitors are devices which store charge. In the lecture, we talked about a simple, idealized kind of capacitor, a parallel plate capacitor, which consists of two metal sheets separated by a dielectric. In practice, real capacitors are usually made up of thin metal sheets ( plates ), separated by a thin plastic insulator and rolled up. The two plates are not in electrical contact, and charge can be stored on them with the + charge on one, and the - charge on the other. There is a voltage difference between the two plates of a capacitor, and the capacitance C (with units of Farads) is defined as the amount of charge (Q) stored on the plates per unit potential difference (V): Figure 1: An RC circuit with power supply showing a "grounding" wire. = / (1) In practice, you will never see a capacitance as large as 1 F. Most capacitances you might encounter in real life can be measured in microfarads (1 µf = 1 x 10-6 F). Therefore, as you can see from the above equation, charges found in the laboratory will usually be of the order of microcoulombs (µc) in order to have voltages in the range of a few volts. An RC circuit is shown in Figure 1. The grounding wire shown in the figure is equivalent to having a switch in the circuit. When the wire is connected to ground, no current will flow in the capacitor. What happens when the grounding wire is removed and current is allowed to flow in the circuit? Current will flow from the power supply through the resistor onto the + plate of the capacitor (the top plate in the figure), and a net + charge will collect on the plate. An equal - charge will collect on the other plate (the bottom one in the figure).

2 Why does this happen? Essentially, positive charge will flow through the circuit, trying to get from the positive terminal of the battery to the negative terminal (flowing from positive to negative, as we have said in class). But it can t cross the capacitor, because there is an insulator between the plates, so positive charge will collect on the top plate of the capacitor. Like passengers collecting at Lambert Airport when flights are cancelled: more and more charges waiting, with no place to go. The same thing will happen at the negative plate (imagine more and more people waiting to pick up their relatives at O Hare, but no planes are landing). As the charge accumulates on the plates of the capacitor, the voltage across the capacitor, V c, slowly builds up, approaching a maximum value, the voltage of the power supply, V. Once the voltage has reached its maximum, no more current can flow (the airport terminals are full to bursting). Why is V c = V? Recall Kirchhoff s first law. V c = V because the sum of the potential differences around a closed circuit has to equal zero, (there is no current flowing through the resistor). In lecture, we used calculus to show that V c changes with time according to the equation below: = 1 / (2) Where e is , and t is the time since the circuit switch was closed (or the grounding wire removed), i.e., the time since current began to flow. To review what we discussed in class, you can get an idea of what this function should look like by thinking what will happen in the limits of t 0 and t. As t 0, the exponential term approaches 1 (since e 0 = 1). Thus, V c = 0 when t = 0. Likewise, when t, the exponential term is zero, since e - = 0. Thus V c = V when t =. The rate at which this buildup occurs is governed by the time constant RC. Note that the exponent in equation (2) must be dimensionless. Therefore the product RC has units of seconds when R is in Ohms and C in Farads. One final thing to review from lecture: a charged capacitor can store an amount of energy equal to = 22 =12 =12 2 (3) The Experiment: Discharging of a Capacitor All voltmeters have a high resistance to keep too much current from passing through the meter. Most meters have an equivalent resistance of greater than a MΩ. For the experimental section of the lab you will be determining the equivalent resistance of the meter with a known capacitor, and then using the known resistance to determine the capacitance of an unknown capacitor. The equation for the discharging of the capacitor is: = /, (4) where R is the internal resistance of the meter and C is the capacitance. For this part of the lab you will use the laptop connected to your set up. Save the Data Studio file to the desktop. The file can be downloaded from the Physics lab site at:

3 Once you have the laptop on and the sensors plugged in you can double click on the saved file to open the Data Studio program. If you need to find it later the program can be found in the Education folder under the programs in the start menu. To start taking measurements, click on the run button on the upper tool bar. The lab TA will provide more instruction. If you make a mistake with the program you can start over by closing the program without saving and opening it again from the Desktop. V + - C Figure 2: An RC circuit with power supply showing a charged capacitor. + USB Voltmeter - 1. Set the power supply voltage to 10 volts D.C. 2. Connect the power supply, meter and the 8 µf capacitor as shown in Figure In Data Studio an graph should appear; the curve shown is similar to the data you should expect to see. Delete the last data run and start the new measurement by pressing the green play button at the top of the screen. 4. Remove both power supply leads at the same time to discharge the capacitor through the meter. 5. The voltage will decrease according to Equation 4. This will allow you to plot V c as a function of time. 6. Highlight the data from the run which corresponds to the discharging of the capacitor. 7. Use the natural exponential trendline fit to find a value for the exponential part of the trendline equation, that is find the time constant RC and record these values in the appropriate Data Table. 8. Repeat the experiment using the unknown capacitor.

4 Data Table 1 R: (MΩ) C 1 : 8 x 10-6 (F) V (power supply) : ( V) Time Constant (=RC 1 ) I c (Amps) V c (Volts) I c (Amps) V c (Volts)

5 Data Table 2 R: (MΩ) C 2 : (F) V (power supply) : ( V) Time Constant (=RC 2 ) I c (Amps) V c (Volts) I c (Amps) V c (Volts)

6 Analysis and Questions 1. For experiment 1 determine the internal resistance of the meter using the known capacitance of 8 µf. 2. For experiment 2 determine the unknown capacitance using the internal resistance of the meter that you found in the previous step. 3. For each data set; use Excel and the above results to calculate the current and voltage at 5 second increments. Use 10 V for the initial voltage. Insert these values in Data Table 1 and For each data set, plot time on the x axis and your values of V c on the y axis. Show these plots on the same graph. 5. For each data set, plot time on the x axis and your values of I c on the y axis. Show these plots on the same graph. 6. Do the time constants make sense with your data? Explain. 7. How many time points do you need to take before V c reaches a constant value? A typical rule is that you should record data until t=3rc. Why? 8. For this experiment, what would a plot of Q vs. t look like, where Q is the charge on the capacitor? Draw a sketch by hand. How would this plot be different for the 8 µf vs. the 20 µf capacitor? 9. List some possible sources of error in your measurements and calculations, and classify them as random or systematic errors.

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