Kirchhoff s Voltage Law and RC Circuits

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1 Kirchhoff s oltage Law and RC Circuits Apparatus batteries 2 battery holders DC Power Supply 1 multimeter 1 capacitance meter 2 voltage probes 1 long bulb and 1 round bulb 2 sockets 1 set of alligator clips 2 pairs of red and black banana clips 3 10-Ω resistors (brown-black-black 1 51-Ω resistor (green-brown-black Ω resistor (brown-black-brown 1 39-Ω resistor (orange-white-black kΩ resistor 1 47-kΩ resistor 2 10 µf capacitors Goal In this experiment, you will: 1. learn how to measure voltage with a voltmeter and verify the loop rule (Kirchhoff s oltage Law) for resistors in series. 2. connect resistors in parallel and verify that the voltage across resistors in parallel is the same. 3. measure the voltage across the resistor and the voltage across the capacitor for a charging capacitor in an RC circuit; repeat the measurements for a discharging capacitor in an RC circuit. Measuring oltage for Resistors in Series Introduction You can measure the potential difference (or voltage) between any two locations in a circuit using the multimeter. To set the meter up to measure potential difference, plug the red lead into the socket labeled ΩHz and turn the dial to the green 20 in the section labeled DC. When set up like this the meter is referred to as a voltmeter because it will measure potential difference which has units of volts. When we measured current, we had to break the circuit open and make the ammeter part of the circuit. When measuring potential difference, we do NOT have to do this. To measure the potential difference between two locations simply touch one lead to one location and touch the other lead to the other location. The readout gives you the potential of the red lead minus the potential of the black lead ( ). For example, imagine you touch the red lead to the positive side of a battery and the black lead to the negative side and you get a reading of 1.5. This means that the potential at the positive side of the battery (where the red lead is) is 1.5 higher than the potential at the negative side of the battery (where the black lead is). Try this with your battery. What do you get? 1

2 Now, switch the leads. Put the red lead on the negative side of the battery and the black lead on the positive side. How does the reading change? Why do you think the reading is like this now? Procedure 1. Set up a 3.0 battery and a round bulb as shown below, including a picture of the circuit and the corresponding circuit diagram. Use two 1.5 batteries in series to get a 3.0 battery. Note: the terminal voltage of the battery will not be exactly 3.0 as you will see. Figure 1: 2. Measure the potential difference across the battery (across points A and B in the circuit diagram). AB = Which side of the battery is at a higher potential? 3. Measure the potential difference across the bulb (across points C and D in the circuit diagram). CD = 2

3 Which side of the bulb is at a higher potential? 4. Set up the circuit below with the round bulb and and oblong bulb. Figure 2: 5. Measure the potential difference across the battery (across points A and B in the circuit diagram). AB = 6. Measure the potential difference across the first bulb (across points C and D in the circuit diagram). CD = 7. Measure the potential difference across the second bulb (across points E and F in the circuit diagram). EF = Analysis According to the Loop Rule (Kirchhoff s oltage Law, the sum of the voltages around a closed loop is zero. Thus, if we start at point B in the circuit, BA + CD + BA = 0 Since BA = AB as you saw earlier when you switched the leads for the multimeter when measuring the voltage across the battery, then AB + CD + BA = 0 AB = CD + BA batt =

4 Using the voltages that you measured, verify that the above equation from the Loop Rule describes the voltages across the elements in your circuit. Two or more resistors in series are called a voltage divider. Why do you suppose it s called a voltage divider? The current through each resistor is the same. Using Ohm s law in your reasoning, which bulb has a higher resistance? oltage for Resistors in Parallel Procedure 1. Set up the circuit as shown below, with the round bulb and oblong bulb in parallel. Figure 3: 2. Make a prediction. Will the potential differences across each of the bulbs be less than, greater than, or equal to the potential difference across the battery? 4

5 3. Measure the potential difference across the battery (across points A and B in the circuit diagram). AB = 4. Measure the potential difference across the first bulb (across points C and D in the circuit diagram). CD = 5. Measure the potential difference across the second bulb (across points E and F in the circuit diagram). EF = Do your measurements match your predictions? If not, explain. 6. Instead of connecting clips from the second bulb to the first bulb, connect them directly to the battery as shown below. Figure 4: Are the bulbs still in parallel? Support your answer with measurements of the voltage across each bulb and the battery. 5

6 Analysis 1. Suppose that you have the circuit shown below, with a 2- battery, R 1 = 51 Ω, R 2 = 100 Ω, R 3 = 39 Ω, and R4 = 10 Ω. Apply Kirchhoff s Laws, as learned in class, to calculate the current through each resistor and the voltage across each resistor. Show all of your work. Note that you should have one node (or junction) equation and two loop equations. Figure 5: 2. Set up this circuit using your resistors and the DC power supply. Ask me to check it before you turn on the power supply. 3. Measure the voltage across each resistor with your voltmeter. 1 = 2 = 3 = 4 = 4. Compare the measured voltages to what you calculated theoretically. Comment on the significance of any differences. 6

7 RC Circuit Setting up your circuit In this experiment, you will investigate a charging capacitor and a discharging capacitor 1. Set up the circuit shown below, but do not turn on the power supply. Figure 6: Use a 100 kω resistor and a 10 µf capacitor. The capacitor should be connected with the long lead on the high potential side and the short lead on the low potential side. The power supply is like a battery, though it maintains a more nearly constant potential difference across its terminals than a battery. It is technically called a voltage source. If there are red, black, and green terminals, connect the black terminal to the green terminal. Set the voltage of the power supply to 3.0. Use a voltmeter to verify that it is Note that between the resistor and power supply, there are two alligator clips. This is so that you can disconnect the circuit at this node, and connect the resistor to the charged capacitor and thereby discharge the capacitor as shown below. 3. Study the circuit above and describe what will happen when the alligator clip is in each of the two possible positions. Sketch the direction of the current at t=0 (when the wires are first connected) in each case. Assume that in the first circuit, the capacitor has zero charge at t=0 when the circuit is connected. Then, the alligator clip is switched and the capacitor is fully charged at t=0 when the circuit is connected. We will refer to the alligator clip that you connect or disconnect as a switch. 7

8 Figure 7: Charging capacitor In this part of the experiment, you will determine the mathematical function that describes a charging capacitor. 1. Discharge the capacitor by changing the position of the switch such that the capacitor is connected to the resistor (and no current will flow through the power supply). 2. Connect a voltmeter across the terminals of the battery to check the potential difference across the power supply. It should be slightly less than 3 volts. 3. Insert a differential voltage probe into Channel 1 of the LabPro. 4. Connect the leads of the voltage probe across the capacitor with the red lead on the high potential side of the capacitor and the black lead on the low potential side of the capacitor. 5. Connect the second differential voltage probe to Channel 2 of the LabPro and connect it across the resistor to measure the resistor s voltage. 6. Open the Logger Pro software. 7. Just to be sure that it s working, click the Collect button on the top toolbar. You should see a graph of the potential difference across the capacitor and resistor plotted in real-time. You may click Stop to stop data collection prematurely. 8. It should read zero; if not, then zero the probes. 9. When you are ready to collect data, click the Collect button. Shortly after data collection begins, switch the alligator clip, thus connecting the power supply to the resistor and capacitor. 10. You should see the potential difference across the capacitor increase. 11. Once the capacitor is fully charged, stop collecting data by clicking the Stop button. 12. Note that the time when the capacitor started to charge is not zero. Record what time t the capacitor started to charge. 13. Now, make a new calculated column in Logger Pro called new time (or whatever you want to call it) and calculate the data for this column as t t 0 where t 0 is the initial time. This will give you a column for time that starts at t = 0. Be sure to name this variable with a unique short name, like tn for new time. You will use this variable when fitting a curve to the data. 14. Graph the potential difference across the capacitor as a function of new time. 15. Go to Analyze >Curve Fit.... 8

9 16. Enter an equation of the form A (1 exp( t/b)). Be sure to use the correct name for the time variable t that appears in the menu of functions. 17. Write down the mathematical function and values of the coefficients that fit the curve. 18. From this function, what is the final (maximum) voltage across the capacitor? 19. What is the time constant (the units are seconds). 20. What is the theoretical value for the time constant for this circuit? (τ = RC) 21. What is the maximum charge stored on the capacitor? (Q = C C ) 22. Use a 47 kω resistor and repeat the experiment. Make a prediction. Will the time constant be greater or less than with the 100 kω resistor. 9

10 What is the time constant in this case? By what factor did it change from the previous case? Discharging capacitor In this part of the experiment, you will determine the mathematical function that describes a discharging capacitor. 1. With the capacitor fully charged, begin collecting data for the potential difference across the capacitor. Shortly after starting data collection, throw the switch so that the capacitor is connected directly to the resistor (and the power supply is no longer part of the circuit). 2. Recalibrate the time scale just as before so that t=0 corresponds to the moment that you throw the switch. 3. Do a curve fit, this time using an exponential decay A exp( t/b). Again, use the correct variable for time. 4. Write the mathematical function that fits the curve. 5. What is the time constant for the discharging capacitor? Other graphs Based on analytical reasoning and your experience in this lab, sketch what you expect for the following graphs: Magnitude of the charge Q on each plate as a function of time for a charging capacitor Magnitude of the charge Q on each plate as a function of time for a discharging capacitor 10

11 Current as a function of time for a charging capacitor Current as a function of time for a discharging capacitor oltage across the resistor as a function of time for a charging capacitor oltage across the resistor as a function of time for a discharging capacitor Application 1. If you were to connect two 10 µf capacitors in series with each other, what would be the equivalent capacitance? How would it affect the time constant? Connect two capacitors in series and measure the time constant for charging or discharging capacitors and verify your prediction. 2. If you were to connect two 10 µf capacitors in parallel with each other, what would be the equivalent capacitance? How would it affect the time constant? Connect two capacitors in series and measure the time constant for charging or discharging capacitors and verify your prediction. 11

12 Lab Report Name: Lab Partners: Date: Description of experiment: 1. State Kirchhoff s voltage law. 2. What were your measured values and what were the theoretical values of the voltages across the resistors for the circuit in Figure 5? 3. What was the measured time constant for the RC circuit when you used R = 100 kω and C = 10 µf? What is the theoretical value of the time constant? (Measure R and C with a multimeter and use these values in the theoretical calculation.) 4. What was the measured time constant for the RC circuit when you used R = 47 kω and C = 10 µf? What is the theoretical value of the time constant? (Measure R and C with a multimeter and use these values in the theoretical calculation.) 5. When you placed two capacitors in series, what was the total capacitance? Explain the observations or measurements that led to your conclusion. 6. When you placed two capacitors in parallel, what was the total capacitance? Explain the observations or measurements that led to your conclusion. 7. For a charging capacitor, at an instant t, the voltage across the capacitor is C and the voltage across the battery is bat. Write an equation for the voltage across the resistor R in terms of these variables. 8. For a discharging capacitor, at an instant t, the voltage across the capacitor is C. Write an equation for the voltage across the resistor R in terms of the voltage across the capacitor. 12

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