# Discharge of a Capacitor

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1 Discharge of a Capacitor THEORY The charge Q on a capacitor s plate is proportional to the potential difference V across the capacitor. We express this with Q = C V (1) where C is a proportionality constant known as the capacitance. C is measured in the unit of the Farad( F), (1 F = 1 coulomb/volt). If a capacitor of capacitance C (in Farads), initially charged to a potential V0 (volts) is connected across a resistor R (in ohms), a timedependent current will flow according to Ohm s law. This situation is shown by the RC (resistorcapacitor) circuit below when the switch is closed. R Red Red C Voltage Probe V o Black Fig. 1 Black As the current flows, the charge q is depleted, reducing the potential across the capacitor, which in turn reduces the current. This process creates an exponentially decreasing current, modeled by V = V o e (t/rc) (2) The rate of the decrease is determined by the product RC, known as the time constant of the circuit. A large time constant means that the capacitor will discharge slowly. When the capacitor is charged, the potential across it approaches the final value exponentially, modeled by V = Vo (1 e (t/rc) ) (3) The same time constant RC describes the rate of charging as well as the rate of discharging. OBJECTIVES _ Measure an experimental time constant of a resistorcapacitor circuit. 45

2 _ Compare the time constant to the value predicted from the component values of the resistance and capacitance. _ Measure the potential across a capacitor as a function of time as it discharges and as it charges. _ Fit an exponential function to the data. One of the fit parameters corresponds to an experimental time constant. MATERIALS Computer Logger Pro Voltage Probe connecting wires 2Low leakage capacitor (1 to 10 µf) 2kΩ resistors (50 to 150 kω) Power supply singlepole, doublethrow switch PROCEDURE 1. Measure the values of the capacitors and resistors and record the values in the data table. The lab instructor will show you how to measure the values. 2. Connect the circuit as shown in Fig. 1 above with the first capacitor and the first resistor. 3. Connect the Voltage Probe to the Logger Pro 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. It is important that the black terminal from the logger pro is connected to the black terminal of the power supply. 3. Prepare the computer for data collection by opening Exp 27 from the Physics with Computers experiment files of Logger Pro 2.1. A graph will be displayed. The vertical axis of the graph has potential scaled from 0 to 4 V. The horizontal axis has time scaled from 0 to 10 s. 4. Charge the capacitor for 30 s or so with the switch in the position as illustrated in Fig. 1. You can watch the voltage reading at the bottom of the screen to see if the potential is still increasing. Adjust the voltage from the power supply so that the voltage lies close to three volts. Wait until the potential is constant. 5. Click collect to begin data collection. As soon as graphing starts, throw the switch to its other position to discharge the capacitor. Your data should show a constant value initially, then decreasing function. 6. 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 f(x)=, and from the function selection box, choose the Natural Exponential function, A*exp( C*x ) B. Click try fit, and inspect the fit. If the curve is a good fist, then click ok to return to the main graph window. However, if black line doesn t match the red line, try click and hold the brackets and move them around to improve the curve fit. (The black curve is the curve fit and the red curve is the data.) 7. 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 stand for capacitance. Compare the fit 46

3 equation to the mathematical model for a capacitor discharge proposed in the introduction, V = V o e (t/rc) (2) Notice that the curve fit C is equal to one over the product of resistor and the capacitor C. Find the percent error between 1/C (curve fit) and RC. 8. Print the graph of potential vs. time. 9. The capacitor is now discharged. To monitor the charging process, click collect. As soon as data collection begins, throw the switch the other way. Allow the data collection to run to completion. 10. This time you will compare your data to the mathematical model for a capacitor charging, V = Vo (1 e (t/rc) ) (3) Select the data beginning after the potential has started to increase by selecting the region of the graph to curve fit by dragging mouse across the graph and highlighting the section. Click on the f (x) = button and selecting the inverse exponent function, A*(1 exp ( C*x)) B. Click try fit and inspect the fit. Click ok to return to the main graph window. 11. Record the value of the fit parameters in your data table. Compare the fit equation to the mathematical model for a charging capacitor. 12. Now you will repeat the experiment with a different resistor and capacitor. 13. Repeat the experiment with two capacitors hooked in series. Capacitors in series add as the reciprocal. That is, 1 C = 1 1 (4) C 1 C Repeat the experiment with two capacitors hooked in parallel. Capacitors in parallel add directly. That is, C = C 1 C 2 (5) 47

4 DATA TABLE R 1 = C 1 = R 2 = C 2 = Curve Fitting Parameters Resistor Capacitor Time Constant Trials A B C 1/C R(Ω) C (F) RC (s) Percent Error Discharge 1 Charge 1 Discharge 2 Charge 2 Capacitors in Series Equivalent Capacitor = Eqn. 4 Curve Fitting Parameters Resistor Capacitor Time Constant Trials A B C 1/C R(Ω) C (F) RC (s) Percent Error Discharge 1 Charge 1 48

5 Capacitors in Parallel Equivalent Capacitor = Eqn. 5 Curve Fitting Parameters Resistor Capacitor Time Constant Trials A B C 1/C R(Ω) C (F) RC (s) Percent Error Discharge 1 Charge 1 QUESTIONS 1. A capacitor is a device that can store energy when it stores charge. The amount of energy stored is just equal to work done in moving the charge onto the capacitor. The work done increases as more charge is added to the plate. Therefore, the energy stored on the capacitor is just W= (1/2) Q V = (1/2) C V 2. Using the data from your experiment, calculate the energy stored in the capacitor for the first trial. What became of that energy when the capacitor is discharged? 49

6 2. Below shows a diagram of a charged parallel plate capacitor without a dielectric between the plates and the other diagram shows the same parallel plate capacitor that has a dielectric placed between the plates. (a) Sketch the electric field lines between the plates on each diagram. (b) If the charge on the plates remains constant while the dielectric is slipped between the plates, will the voltage on the capacitor with the dielectric increase, decrease, or remain constant when compared to the capacitor without the dielectric. 50

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