# Lab 5 RC Circuits: Charge Changing in Time Observing the way capacitors in RC circuits charge and discharge.

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2 When an uncharged capacitor and a resistor are initially connected in series to a voltage source, charge will flow in the circuit until the capacitor becomes fully charged, at which point the charge stored on its plates is Q = CV, where C is the capacitance (in Farads) of the capacitor, and V is the voltage. When the voltage source is removed and replaced by a simple piece of wire, the charge on the capacitor returns to zero; the capacitor discharges. A charged capacitor has equal magnitude charges of opposite sign on its two plates. One way to introduce charge onto the plates of the capacitor, i.e., to charge it, is with a resistor and capacitor in series and connected to a voltage source through a switch. Refer to Figure 1, part (a). An initially uncharged capacitor in an RC circuit in which the switch is open will remain uncharged. However, when the switch is closed, electrons from the negative terminal of the battery or power supply will move through the circuit and accumulate on the closest plate of the capacitor. This accumulation of negative charge on one plate will induce a positive charge on the other plate essentially, the electrons on the opposite plate will be repelled and will move back to the positive terminal of the battery/power supply. All motion of electrons will stop when the charge on the plates becomes Q = CV. Electrons DO NOT jump across the insulator between the two plates!! Refer to Figure 1, part (b). Once a capacitor is charged, it can be discharged (i.e., the charge can be reduced to zero on its plates) after making two changes to the RC circuit in Figure 1, part (a). First, the voltage source (the battery or power supply) must be removed from the circuit. Second, the open circuit caused by the removal of the power supply must be closed with a wire. Once these changes are made and the switch is closed, the excess electrons from one plate of the capacitor will flow through the circuit until the deficit of electrons on the other plate is removed. The excess charge on the plates of the capacitor (Q) will be zero when the capacitor is fully discharged, and no current will flow at that time. 0.3 Capacitor charge as a function of time Suppose a capacitor with capacitance C farads is given a charge Q 0 coulombs and then is made to discharge through a resistor with resitanceresistance R ohms. At time t seconds after the discharge begins, the charge Q remaining on the capacitor is given by the following formula. Q 0 Q e In the above, e is a name for a mathematical constant ( Euler number ), like is a name for The numerical value that e stands for is , so the above formula is essentially the same as the following. t RC Q Q Example of capacitor discharge as a function of time 0 In this example, the capacitanceor is C = 1000 F (1000 microfarads), and the resistor resistance is 10 k (10 kilohms). At t = 0 seconds, the charge on the capacitor is Q 0 = 0.24 C (0.24 coulombs). Find the capacitor charge Q at t = 3.0 s. Solution: First, note that the product RC = 10 seconds ( C), so t/rc = 3/10 = 0.3. If your calculator has an e x function, use it to calculate Q = 0.24 e Alternatively, you can use the raise to a power function y x to compute Q = 0.24 [ ) ]. t RC Page 2

3 Answer: Q = 0.18 C. The product RC has the units of time. Look up the fundamental units and confirm this for yourself. This product is also given a special name the time constant and is denoted by the Greek letter τ (tau). The equation above can be rewritten as: t Q 0 e Q The larger the product of RC, i.e. τ, the longer the capacitor takes to discharge. This is also true for charging the capacitor. Activity #1: Instructor demonstration of charge stored in a capacitor (This will be done in lecture.) Abstract: Your lab instructor will use a charged capacitor to light a bulb, thereby demonstrating that a capacitor stores electrical energy. Equipment: 5V, 200 ma light bulb 1 F electrolytic capacitor Variable DC Power supply, 0 to 15 VDC, 0.1 A. Voltmeter (3 digits, 20 V range) 0.5 The aim of this activity is to demonstrate the ability of a capacitor to store charge and provide a potential difference when completely charged. Your lab instructor will demonstrate this by first charging up a capacitor by connecting it across the terminals of a DC power supply. Once fully charged your lab instructor will then discharge the capacitor by connecting it across a light bulb. 0.6 In order to charge up the capacitor, connect the capacitor across the terminals of a DC power supply as shown in Figure 2. Power Supply RED BLACK Capacitor Voltmeter Figure 2: Charging a capacitor by connecting it across the terminals of a power supply. Electrons flow through the wire from the negative terminal of the power supply onto the capacitor plate on the right, making it negative. The negative charge drives electrons away from the capacitor plate on the left, making it positive. The electrons that were driven away from the left plate flow into the positive terminal of the power supply. The voltmeter does not do anything except watch the difference in voltage between the two capacitor plates. Page 3

4 0.7 Once fully charged, the capacitor is now disconnected from the terminals of the power supply and connected across a light bulb as shown in Figure Carefully observe what happens to the light bulb when it is connected across the terminals of the capacitor. Light bulb Capacitor Voltmeter Figure 3: Discharging a capacitor through a light bulb. Electrons on the negative capacitor plate can move to the positive plate to which they are attracted by traveling through the light bulb. In passing through the light bulb, they make the bulb glow. As with the previous circuit, the voltmeter does not do anything except watch the difference in voltage between the two capacitor plates. Equipment: Computer running Logger Pro 3.8.x Lab Quest Interface Voltage Probe plugged into Ch 1 of the Lab Quest interface Voltmeter (3 digits, 20 V or 30 V range) Low-voltage power supply set for 5 volts Banana-plug circuit board with 1000 F & 470 F capacitors and two 1 k resistors See Figure 4Figure 4. RC_Circuits.cmbl (Logger Pro initialization file, optional) Figure 4 Banana-plug circuit board holding the resistor and the capacitors Page 4

6 1.6.2 Remove the switch s wire plug (free end) from the connector on the side of the power supply (the banana plug where the black alligator clip is attached) Start data collection using the Collect button on the screen As soon as the data collection trace becomes visible on the screen, plug the switch s wire plug into the connector (resistor side; see Figure 6 on page 12). Data will be recorded as the capacitor discharges, ending with zero voltage across the capacitor. 1.7 Save the graph you have just obtained in the following manner: From the tool bar, select Data, and then Store Latest Run. The graph just obtained will remain on your screen, but will appear lighter than it was. Q 2 Describe in words the way in which voltage across the capacitor (and so also the charge on its plates) varies during the discharge process. Q 3 You already (in Q 1) calculated the initial charge on the capacitor plates. Now use Logger Pro s Analyze-Examine tool to find the voltage 3 seconds after the discharge began, and then use Q = CV to calculate the charge on the capacitor plates 3 seconds after the discharge began. t RC Q 4 Starting with: Q Q 0 e, divide both sides by C, the constant value of the capacitance. The value Q0/C is a constant, so let s call this V0, while Q/C is just the voltage V on the capacitor. Write the equation in the space below. Q5 Now take the natural log of each side, keeping in mind that ln(ab)=ln(a) ln(b). Write this below. Page 6

7 Q6 Compare this equation to those you frequently use in physics classes, such as a straight line or a parabola. Which form does your new equation look like? Please write the general form (y= ) for the equation below, copy your new formula from above, and match up the variables and the constants. Q7 Now create a graph of the discharge portion of the curve only. Use either Excel or LoggerPro to create this graph. Perform the appropriate fit on this graph and determine the value of C. Q8 Perform an exponential curve fit (y=a*exp(-cx) B) on your original LoggerPro data by selecting only the discharge region. Q9 How do these two values of C compare? 2. Activity #3: Observing the charging of a capacitor in an RC circuit 2.1 Now you are to reverse the process and monitor a charging capacitor. The idea is to convert the circuit of Figure 6 back into the circuit of Figure 5 by moving exactly one connection. Proceed as follows, referring to Figure 5 on page Unplug the wire from the left end of the resistor Click the Collect button to start data collection As soon as the trace appears on the screen, plug the same wire plug into the banana plug socket that is connected to the minus side of the power supply New data will be collected while the previous run remains faintly visible on the screen. 2.2 Print a copy of the Logger Pro graph (containing two curves) for everyone in your group. 2.3 By hand, label the discharge graph Discharge, and label the charging graph Charging. Page 7

8 Q 10 Describe the graph of the data for charging a capacitor. How does this graph look similar to and different from the graph obtained when discharging the capacitor? Are the charging and discharging rates constant? How can you tell? Q 11 When the capacitor is discharging (as it did in Activity #2), where does the charge on each plate of the capacitor go? 3. Activity #4: Discharging different capacitors through the same resistor 3.1 In this Activity, we will compare the discharge curve of a circuit with a 1000 F capacitor to the discharge curve of the same circuit with a 470 F capacitor. 3.2 Following the directions in Section (and refer to Figure 5 on page 11), charge the 1000 F capacitor again. 3.3 You will discharge the capacitor by switching out the power supply as in section 1.6 (refer to Figure 6 on page 12). However, with the intent of comparing the two different capacitors, use the following procedure instead of the procedure in section When the capacitor is fully charged, you may unplug the switch s wire plug, holding it ready for the next step. It is best to have one person in charge of the wire plug It is also helpful to have a second person responsible for monitoring the data collection. Have this person start data collection by clicking on the Collect button. When the trace on the screen reaches the 5 seconds mark, this person should say Now, so that the first person the one in charge of the wire plug - can close the switch by plugging the wire into the plug connected to the resistor. See Figure 6 on page Save the data from this run as you did in Section Modify the circuit so that the 470 F capacitor will be charged and reconnect the leads of the Voltage Probe across the new capacitor, as in Figure 7 on page While referring to Figure 7 (page 1313) and Figure 8 (page 1414), repeat the procedures in Sections with the 470 F capacitor. Page 8

9 3.6 Print out copies of the graph (containing two curves) for everyone in your group. 3.7 By hand, label the curve for the 1000 F capacitor 1000 F, and label the curve for the 470 F capacitor 470 F. Q 12 Compare the two curves on your graph, i.e. do both capacitors discharge at the same rate? Q 13 Use Q = CV and the data on the graph to calculate the charges on the two capacitors at a time 3 seconds after the discharges began. 4. Activity #5: Discharging a capacitor through different resistors 4.1 In this Activity, we will compare the discharge curve of a circuit with a 1000 F capacitor with a 1 kω resistor in series, to that with a 2 kω resistor in series. 4.2 As before, charge the 1000 F capacitor to 5 V and discharge it though the 1 kω resistor while collecting data 4.3 Now, modify your circuit so the 1000 F capacitor will be charged and discharged through a 2 kω resistor. You do not have a 2 kω resistor, but can effectively create one with available parts at your lab station. Please sketch your charging circuit below. 4.4 Display on a single graph, the discharge characteristics of the 1000 F capacitor discharging through the 1 kω resistor and the 2 kω resistor. 4.5 Print and label your graph showing which data represents each resistor. Q 14 What effect does increasing the resistance have on the discharge characteristics of an RC circuit? Page 9

11 Voltmeter (GND) Power Supply () COM Connect voltmeter directly to power supply Switch 1 k 1000 F To LQ Port F Figure 5 Circuit diagram for charging the 1000 F capacitor through resistor R. Page 11

12 Voltmeter (GND) Power Supply () COM Connect voltmeter directly to power supply Switch 2.2 k 1000 F To LQ Port F Figure 6 Circuit diagram for discharging the 1000 F capacitor through resistor R. Current must always flow in a closed loop, and the only closed loop in the circuit is through the resistor and back to the capacitor. Page 12

13 Voltmeter (GND) Power Supply () COM Connect voltmeter directly to power supply Alligator Clips 1 k 1000 F 470 F To LQ Port 1 Figure 7 Circuit diagram for charging the 470 F capacitor. Page 13

14 Voltmeter (GND) Power Supply () COM Connect voltmeter directly to power supply Alligator Clips 1 k 1000 F 470 F To LQ Port 1 Figure 8 Circuit diagram for discharging the 470 F capacitor through the resistor R. Page 14

16 6. A 1000 F capacitor discharges through a 2200 resistor. Assume the capacitor initially held 5.0 mc (5 millicoulombs, or C) of charge. Calculate the amount of charge on the capacitor at t = 1 s, 2 s, 3 s, 4 s, and 5 s. Graph the initial charge and the calculated charges on the graph below. Show all calculation in the space under the graph Charge (Coulombs) Time (seconds) t RC 7. The charge Q on a capacitor is Q Q 0 e. The current is simply dq/dt. Please differentiate Q to determine the current in the RC circuit at any time t. What is the current at time t = 0? What is the current at a time t equal to RC? Page 16

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