Name: ______Row # ______
Partners:
Lab: RC Circuits
Research Problem: To investigate the way capacitors charge and discharge.
Background and Theory: When a capacitor is connected to a battery, charge builds up on the capacitor plates and the potential difference (or voltage) across the plates increases until it equals the voltage of the battery. At any time, the charge Q of the capacitor is directly related to the voltage V across the capacitor plates: Q = CV , where C is the capacitance of the capacitor in farads (F). The rate of voltage ‘rise’ depends on the capacitance of the capacitor and the resistance in the circuit. Similarly, when a charged capacitor is discharged, the rate of voltage ‘decay’ depends on the same quantities.
Both the charging and discharging times of a capacitor are characterized by a quantity called the time constant τ , which is the product of the capacitance C and the resistance R, that is, τ = RC.
When a capacitor is charged by connecting it to a battery, it will almost instantaneously reach its maximum charge. To slow this process down, we connect it in series with a resistor. Closing the switch with a resistor in the circuit (see Figure 1a) will cause current to flow in the circuit more slowly and hence charge the capacitor at a rate that can more easily be studied.
When the switch is first closed, the current in the circuit will flow rapidly, charging up the capacitor plates, but then slow down to almost zero when the capacitor is almost completely charged. Thus we say that the current “decays” from its initial maximum value to almost zero. This state of nearly zero current is called the “steady state” of the circuit.
Also, when the switch is first closed, there is no voltage drop across the capacitor (VC = 0) but as the current flows and the charge builds up on the capacitor plates, the voltage across the capacitor increases to a maximum value which occurs when the circuit is in the steady state. At this point, the voltage drop across the capacitor will be nearly equal to the voltage of the battery (VC ≈ V0). (Since almost no current is flowing in the circuit, almost no voltage will drop across the resistor.) Thus we say that the voltage across the capacitor “rises” from zero to a maximum value.
To quickly discharge the capacitor, connect a wire directly between the two legs of the capacitor or touch the two legs together (called “shorting” the capacitor) and the stored charge will rapidly flow from one side of the capacitor to the other side until the parallel plates are neutral.
Since the voltage drop across the capacitor (VC) is not constant, it varies with time. The voltage, VC, as a function of time, t, is given by
Equation (1) serves as the mathematical model for the relationship between voltage and time for the capacitor. (e = 2.718 . . . is the base of natural logarithms.)
The curve of the exponential rise of the voltage with time during the charging process is illustrated in Figure 2.
The quantity τ = RC is called the time constant of the circuit. It represents the amount of time necessary for the voltage to rise to a value VC = (1 – 1/e)V0 ≈ 63% of its final steady state value V0. A capacitor is said to be fully charged when it has been charging for a length of time equal to five time constants (5t).
Materials: capacitor, resistor, voltage probe, connecting wires, 3 batteries, battery holder, switch, computer, interface cable (Go-Link), LoggerPro software
Procedure and Data Collection:
1. Record the resistance of the resistor (R) and the capacitance of the capacitor (C). (Read the capacitance off the label of the capacitor and the resistance from the color codes on the resistor [see appendix].) Then calculate the time constant of the circuit.
Resistance color code = ______
Resistance value = ______
Capacitance = ______
Time constant (t) = ______
2. Inspect the circuit provided on the breadboard. Use the appendix for learning to use a breadboard to verify that the circuit is set up as in Figure 1a. Connect the battery to the breadboard terminals to power the circuit. BE SURE to connect the battery in the proper orientation with the positive (+) end connected to the red terminal and the negative end (-) connected to the black terminal. If they are backward, they could damage or destroy the capacitor.
3. Open up LoggerPro on the computer and plug the voltage probe into the interface cable and then plug the interface cable into a USB port of the computer. LoggerPro should auto-detect the voltage probe.
4. First you will measure the Vo of the battery. Push/pull back the protective plastic casings of the read and black leads to reveal the metal tips. Hook/hold the probe tips against the ends of the battery pack until you get a steady voltage reading. Record the battery voltage.
Vo = ______
5. Now you are going to measure the voltage drop across the capacitor as it charges. Connect the voltage probe to the capacitor legs as shown in Figure 1b. Hook the red probe to the positive leg of the capacitor and the black probe to the negative end of the capacitor. The negative end of the capacitor is the shorter leg and is usually marked with a negative sign.
6. Set up the experiment by choosing Experiment/Data Collection… from the top toolbar and then setting the Duration to 60 seconds. Set up the graph window by double-clicking on the graph. Choose the Axes Options tab and set the Y-axis to Top: 5 and Bottom: 0. This sets the maximum and minimum voltage that will be recorded.
7. To collect data hit the green ENTER button and then close the switch once you see the data on the screen. You should see a real-time graph of the charging of the capacitor. If it looks good, you can proceed with the next step. If not, try again but be sure to fully discharge the capacitor by touching its two legs together or connecting them briefly with a wire. Open the switch when your capacitor is charged.
8. On the axes at right, sketch what you see on the screen. Be sure to label the axes appropriately.
9. Now you are going to fir an experimental curve to your data. By clicking and dragging, highlight the section of the graph from when the switch was closed until the experiment ended. Then choose Analyze/Curve fit. From the list of functions, choose Inverse Exponent A*(1 – EXP(-Ct)) + B and then Try Fit. Click OK if the fit looks good. Write the experimental relationship below.
Experimental Relationship:
10. Use your math model and your experimental relationship to determine Vo and t. Write those values below.
Vo = ______t = ______
Discharging the Capacitor
11. With the switch open, remove the wires from the ends of the battery pack and connect them to each other so that the batteries are no longer in the circuit. Then, choose Experiment/Clear Latest Run to get a new graph. Once this is done, hit the green COLLECT button and close the switch to discharge the capacitor.
12. On the axes at right, sketch what you see on the screen. Be sure to label the axes appropriately.
Data Analysis
13. Based on the data you have collected and your understanding of the circuit used in this experiment, sketch smooth curves for each of the following and write a math model for each:
a) Charging the capacitor
b) Discharging the capacitor
APPENDIX: Breadboard Connections
The “breadboard” is a solderless circuit board so a circuit can be made, tested, and revised without having to solder the parts permanently together. It is a very useful tool for developing electronic circuits. The lines above show how the holes on the board are connected underneath so that if two electric components have their legs stuck into holes in the same line they are connected to each other. Of special note are the four long lines, two above and two below, called “power rails” or “power busses,” that are usually used for power, as one is made positive and one is made negative. Connecting a component to one of these rails is just like connecting it to the battery.
APPENDIX: Reading Resistor Color Codes
Look at the four colored bands and notice that they are placed closer to one end of the resistor than the other. Turn the resistor so that the end they are closer to is on the left, then read the colors from left to right.
First band = first digit
Second band = second digit
Third band = multiplier (x 10 raised to this power)
Fourth band = tolerance range (uncertainty interval)
The color-coding system is:
Black / 0 / Blue / 6Brown / 1 / Violet / 7
Red / 2 / Gray / 8
Orange / 3 / White / 9
Yellow / 4 / Gold / ± 5%
Green / 5 / Silver / ± 10%
Adapted from General Physics Lab Handbook by D.D.Venable, A.P.Batra, T.H¨ubsch, D.Walton & M.Kamal
http://www.physics1.howard.edu/MSIP/GenLab2/GL2-07.pdf