Ph February, Kirchhoff's Rules Author: John Adams, I. Theory

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1 Ph February, 2006 I. Theory Kirchhoff's Rules Author: John Adams, 1996 quark%/~bland/docs/manuals/ph122/elstat/elstat.doc This experiment seeks to determine if the currents and voltage drops in a two-loop circuit obey Kirchhoff s rules. A two-loop circuit is a circuit that has two distinct paths through which current can flow. The currents and voltage drops in such a circuit containing multiple resistors and power supplies will be measured. An diagram of the two-loop circuit we will study today is shown in Figure 1 below. One can use a simple I1 I2 algebraic method to calculate the voltage drop across and the current R2 + R through each resistor. The algebraic R I method involves the application of ε Kirchhoff's two rules. loop 1 - loop 2 Loop Rule: When any closed circuit loop is R traversed, the algebraic sum of 1 R4 the voltage drops around that closed loop must equal zero. Figure 1. Two-loop circuit. Junction Rule: At any junction point in a circuit where the current can divide (such as where two or more wires connect), the sum of the currents into the junction must equal the sum of the currents out of the junction. Now, we ll demonstrate how to calculate these quantities. During this laboratory experiment, however, we will experimentally measure each current and voltage drop. Your instructor will explain how to apply these rules to the circuit you will study today. For this discussion, we will only outline how to determine the current in each part of the circuit. Once we've calculated the current in each part of the circuit, calculating the voltage drops is trivial. When we analyze the circuit shown above using Kirchhoff's rules we obtain the following three equations: (a) ε 1 = V 2 + V 3 + V 1. (b) ε 2 = V 5 + V 3 + V 4. (c) I 3 = I 1 +I 2 We can rewrite equations (a) and (b) in terms of I 1 and I 2 and then solve the three equations for I 1 and I 2. Therefore upon solving we get: (1) I 2 = (1/R 3 )(ε 1 -I 1 (R 1 +R 2 +R 3 )) (2) I 1 {(1/R 3 )(R 1 +R 2 +R 3 )(R 3 +R 4 +R 5 )-R 3 } = ε 1 {(1/R 3 )(R 3 +R 4 +R 5 )}-ε ε Kirchhoff - 1

2 The first circuit you will study will contain resistors of approximately 100 Ω and power supplies, ε 1 and ε 2, set at approximately 10 V and 5 V, respectively. Substituting 100 Ω for all resistances, 10 V for ε 1, and 5 V for ε 2 into equations (1), (2), and (c) we obtain the following values: I 1 = ma I 2 = 6.25 ma I 3 = 37.5 ma We will use these values for I 1, I 2, and I 3 to benchmark our results. Important Note: In the following procedures, the initial meter s scale settings are given as a general starting place for you to begin making your measurements. You should, however, use a meter scale, according to the value being measured, so that you obtain the most accurate reading possible. For example, if you are measuring 1 ma, a full-scale setting of 4 ma gives a much more accurate measurement than a full-scale setting of 400 ma. Newer meters such as ours may autorange to the most sensitive setting possible. II. Experimental Procedure A. Verifying Kirchhoff s Rules The resistor boards should be pre-connected with the 100Ω resistors as shown Figure 1. Assume the resistors on the circuit board are all exactly 100Ω. In this part you will use the values of the currents calculated above, using equations (1), (2), and (c) and assuming the resistors are exactly 100 Ω. Then you will actually measure the currents and from your measurements decide whether or not the equations you used to calculate the theoretical values are accurate. Record all measurements on the Results Sheet. 1. We will be using two power sources in this experiment. Adjust one to 10V; this is ε Adjust the other power supply to 5 V; this is ε Connect the power supplies to the circuit as shown in Figure Turn the ammeter to the "400mA" setting. Measure the current I 1 by measuring the current through resistor R 2. Record I Measure the current I 2 by measuring the current through resistor R 5. Record I Measure the current I 3 by measuring the current through resistor R 3. Record I Also measure and record the currents through resistors R 1 and R Turn the multimeter to the "V" setting. Measure the voltage drop across each resistor. Record V 1, V 2, V 3, V 4, and V Answer the questions on the Results Sheet part A. B. Detailed Measurements on a Two Loop Circuit The two-loop circuit in this part of the experiment will be constructed using approximately 75 Ω resistors (some other value may possibly be supplied). In this section, once again, you will measure the currents and voltage drops, and from your measurements you will decide if the currents and voltage drops in the two-loop circuit obey Kirchoff s rules. Also, you Kirchhoff - 2

3 will determine if the labeling scheme for the currents used in your circuit diagram is consistent with the values measured during the experiment. 1. Disassemble the entire circuit from part I. You should now have a completely bare resistor board. 2. Now construct the two loop circuit shown in figure 1 using the 75 Ω resistors. Measure the resistance of each resistor as you add it to the circuit, and record on the Results Sheet. I suggest that you begin constructing the circuit at R 1. Measure the resistance of R 1 and then attach that resistor to the circuit board in the R 1 position according to the circuit diagram. 3. Connect the two power supplies to the circuit as shown in figure Turn the ammeter to the "400mA" setting. Measure the currents through the five resistors and record on the Results Sheet. 5. Turn the voltmeter to the "V" setting. Measure the voltage difference provided by each power supply and the voltage drop across each resistor. 6. Answer the questions in part B on the Results Sheet. III. Equipment proto-board five 100 Ω resistors (band coding: brown/black/brown) five 75 Ω resistors (band coding: violet/green/black) multimeter (Metex M-38500) dual power supply with +5V and 0-15 V (HY ) wire leads: 3 short red and three short black, and one red and one black wire with banana plug on one end and mini-grabber on the other. IV. Appendix: The Resistor Code One of the important components in an electric circuit is the resistor. The most common kind is made from a thin carbon film. You should have some at your table. Their resistance can vary from less than one ohm to 20 million ohms or so. Each one is marked with the value of its resistance, using the resistor color code. (See Table I.) There are four colored bands on a resistor. The first three colors represent numbers: a, b, and c. The value of the resistance in ohms is then given by the number c R = abx10 Ω Kirchhoff - 3 a b c d TABLE I. RESISTOR Figure 2. A color-coded resistor. COLOR CODE color number black 0 brown 1 red 2 orange 3 yellow 4 green 5 blue 6 violet 7 gray 8 white 9 silver 10% precision gold 5% precision

4 For example, if a=6, b=8, and c=3, R = 68 x 10 3 ohms. The fourth band indicates the precision with which the resistance is known. NOTE: you may be supplied with high-precision resistors, which have five bands, with the extra band used to give another significant figure to the resistance value. Kirchhoff - 4

5 Phys 122 Name: Date: Lab Section: Results Sheet Kirchhoff s Rules A. Verifying Kirchhoff s Rules Current Through Voltage Drop R 1 V 1 = R 2 I 1 = V 2 = R 3 I 3 = V 3 = R 4 V 4 = R 5 I 2 = V 5 = 1. Compare the current through R 1 with the current through R 2 and compare the current through R 4 with the current through R 5. What do these values tell you? 2. Compare the values for I 1, I 2, and I 3 that you measured to the values that were calculated using equations (1), (2), and (c). Do the calculated values agree closely enough with the measured values to trust that equations (1), (2), and (c) are correct expressions for the currents? 3. Using your measured values, does I 3 = I 1 + I 2? Can you assert with confidence that the measured currents obey Kirchoff s Junction Rule? 4. Using your measured values, are equations (a) and (b) valid, within reason? Explain your answer. Kirchhoff - 5

6 B. Detailed Measurements on a Two Loop Circuit Measured with Ohmmeter Current Voltage Drop R 1 R 1 = R 2 R 2 = R 3 R 3 = R 4 R 4 = R 5 R 5 = 1. Which of the five resistors should carry the same current and which currents, in terms of I 1, I 2, and I 3, are they? 2. How does your experimental data support your answer to 1) above? Explain your answer in detail, using the experimental values as proof for your answer. 3. Explain in detail if I 3 = I 1 + I 2 for your complete set of measured current values (at both of the nodes in the circuit). What does this mean, in terms of conservation of electric charge? 4. Discuss, using your experimental data as support for your answer, whether or not the equations (a) and (b) are satisfied, within reasonable margins. Explain your answer. 5 Why are the voltages in Part II the same as in Part I, while the currents are different? A qualitative argument will be good enough. Kirchhoff - 6