Direct Current Circuits

Transcription

1 PC1143 Physics III Direct Current Circuits 1 Objectives Apply Kirchhoff s rules to several circuits, solve for the currents in the circuits and compare the theoretical values predicted by Kirchhoff s rule to measured values. 2 Equipment List Breadboard and wires Digital multimeter Sources of emf and resistors 3 Theory Figure 1: Single loop circuit. Consider the circuit in Figure 1. The circuit is labeled with all of the currents. The 2 Ω resistor, 8 Ω resistor and 12 V power supply have current I 1, the 6 Ω resistor has current I 2 and the 3 Ω resistor has current I 3. This circuit is called a single-loop circuit because it can be reduced to a single resistor in series with the power supply. The 6 Ω resistor and the 3 Ω resistor are in parallel with an equivalent resistance of 2 Ω. That equivalent 2 Ω resistance is in series with the 12 V power supply and the other two resistors, reducing the circuit to a single-loop circuit. The total resistance across the 12 V power supply is 12 Ω and its current is therefore I 1 = 1 A. Applying Ohm s law to the remaining part of the circuit gives I 2 = 1/3 A and I 3 = 2/3 A. Page 1 of 8

2 Direct Current Circuits Page 2 of 8 Figure 2: Multi-loop circuit. Consider now the circuit of Figure 2. This laboratory is concerned with the fundamental difference between circuits of the type depicted in Figure 1 and circuits of the type depicted in Figure 2. The circuit in Figure 2 cannot be reduced to a single-loop circuit, but instead is called a multi-loop circuit. Before analyzing this circuit, first we will define some terms. A point at which at least three possible current paths intersect is defined as a junction. For example, points A and B in Figure 2 are junctions. A closed loop is any path that starts at some point in a circuit and passes through elements of the circuit (in this case resistors and power supplies), and then arrives back at the same point without passing through any circuit element more than once. By this definition, there are three loops in the circuit of Figure 2: (1) starting at B, going through the 10 V power supply at A and then down through the 20 V power supply back to B; (2) staring at B, up through the 20 V power supply, and then around the outside through the 10 Ω resistor and back to B; (3) completely around the outside part of the circuit. One can traverse a loop in either of two directions, but regardless of which direction is chosen, the resulting equations are equivalent. The solution for the currents in a multi-loop circuit uses two rules developed by Gustav Robert Kirchhoff. The first of these rules is called Kirchhoff s current rule (KCR). It can be stated in the following way: KCR The sum of currents into a junction = the sum of currents out of the junction. This rule actually amounts to the conservation of charge. In effect, it states that charge does not accumulate at any point in the circuit. The second rule is called Kirchhoff s voltage rule (KVR). It can be stated as: KVR The algebraic sum of the voltage changes around any closed loop is zero. This rule is essentially a statement of the conservation of energy, which recognizes that the energy provided by the power supply is absorbed by the resistors.

3 Direct Current Circuits Page 3 of 8 In a multi-loop circuit, the values of the resistors and the power supplies are known. It is necessary to determine how many independent currents are in the circuit, to label them and then to assign a direction to each current. Application of Kirchhoff s rules to the circuits, treating the assigned currents as unknowns, will produce as many independent equations as there are unknown currents. Solving those equations will determine the values of the currents. In the application of KVR to a circuit, take care to assign the proper sign to a voltage change across a particular element. The value of the voltage change across an emf ε can be either +ε or ε depending upon which direction it is traversed in the loop. If the emf is traversed from the ( ) terminal to the (+) terminal, the change in voltage is +ε. However, when going from the (+) terminal to the ( ) terminal, the change in voltage is ε. In the laboratory, we will measure the terminal voltage of the sources of emf. We will assume that those values approximate the emf. When a resistor R with an assumed current I is traversed in the loop in the same direction as the current, the voltage change is IR. If the resistor is traversed in the direction opposite that of the current, the voltage change is +IR. The sign of the voltage change across an emf is not affected by the direction of the current in the emf. The sign of voltage change across a resistor is completely determined by the current direction. Consider the application of Kirchhoff s rules to the multi-loop circuit of Figure 2. At the junction A, currents I 1 and I 2 go into the junction, current I 3 goes out of the junction, and KCR states I 1 = I 2 + I 3 (1) It might appear that applying KCR to the junction B would produce an additional useful equation, but in fact, it would result in an equation that is identical to equation (1). Applying KVR to the loop that starts at B, goes through the 10 V power supply to A, and then down through the 20 V power supply back to B, gives the following equation with values of the resistances included: R 2 I 1 + ε 1 R 1 I 1 + ε 2 + R 3 I 2 = 0 3I I I 2 = 0 (2) The signs used in equation (2) and the circuit diagrams are consistent with the description given above for determining the signs of voltage changes. Applying KVR to the loop that starts at B and goes clockwise around the right side of the circuit gives R 3 I 2 ε 2 R 4 I 3 = 0 5I I 3 = 0 (3) Equations (1), (2) and (3) are the three needed equations for the three unknowns I 1, I 2 and I 3. The solution of these equations gives values for the currents of I 1 = A, I 2 = A and I 3 = A. The currents I 2 and I 3 are negative. This indicates that the original assumption of direction for these two currents was incorrect. The interpretation of the solution is that there is a current of A in the direction indicated in the figure for I 1, a current of A in a direction opposite to that indicated in Figure 2 for I 2 and a current of A in a direction opposite to that indicated for I 3. This is a general feature of solutions using Kirchhoff s rules. Even if the original assumption of the direction of a current is wrong, the solution of the equations leads to the correct understanding of the proper direction by virtue of the sign of the current.

4 Direct Current Circuits Page 4 of 8 4 Laboratory Work Part A: Single-loop Circuits In this part of the experiment, you will investigate the behaviours of resistors in single loop circuits: series, parallel and combination circuits. Series circuit. A-1. Choose THREE unequal resistors that you have been given. We will refer to one as #1, another as #2 and the third as #3. Use the DMM to measure the resistance of each of your three resistors. Enter these values as R 1, R 2 and R 3 respectively in Data Table 1. A-2. Now connect the three resistors into the SERIES CIRCUIT as in Figure 3 with two 1.5 V AA-cell batteries as emf source. Figure 3: Three-resistor series circuit. A-3. Measure and record the resistances of the combinations R 12, R 23 and R 123 by connecting the leads of the DMM between the points at the ends of the arrows. Record your readings in Data Table 1. Note: When measuring resistance with DMM, the circuit does not need to be completed with emf source. A-4. Use the voltage function on the DMM to measure the potential differences across the individual resistors ( V 1, V 2 and V 3 ) and then across the combinations of resistors ( V 12, V 23 and V 123 ). Record your readings in Data Table 1. A-5. Now change the leads in your DMM so that they can be used to measure current. In order to measure current, the circuit must be interrupted and the current allowed to flow through the meter. Disconnect the lead wire from the positive terminal of the battery and connect it to the red (+) lead of the meter. Connect the black (-) lead to R 1, where the wire originally was connected. Record your reading in Data Table 1 as I 0. A-6. Move the DMM to the positions indicated in Figure 3, each time interrupting the circuit and carefully measuring the current in each one. Record your readings in Data Table 1.

5 Direct Current Circuits Page 5 of 8 Parallel circuit. A-7. Choose THREE unequal resistors that you have been given. We will refer to one as a, another as b and the third as c. Use the DMM to measure the resistance of each of your three resistors. Enter these values as R a, R b and R c respectively in Data Table 2. A-8. Construct a PARALLEL CIRCUIT using all three resistors as in Figure 4 with two D-cell batteries as emf source. Measure and record the resistances of the combination R abc in Data Table 2. Figure 4: Three-resistor parallel circuit. A-9. Use the voltage function on the DMM to measure the potential differences across the individual resistors ( V a, V b and V c ) and then across the combination of resistors ( V abc ). Record your readings in Data Table 2. A-10. Review the instruction for connecting the DMM as an ammeter. Connect it first between the positive terminal of the battery and the parallel circuit junction to measure I 0. Then interrupt the various branches of the parallel circuit and measure the individual branch currents. Record your measurements in Data Table 2. Combination circuit. A-11. Choose THREE unequal resistors that you have been given. We will refer to one as A, another as B and the third as C. Use the DMM to measure the resistance of each of your three resistors. Enter these values as R A, R B and R C respectively in Data Table 3. A-12. Connect the COMBINATION CIRCUIT as in Figure 5. Measure and record the various combinations of resistance R BC and R ABC in Data Table 3. A-13. Use the voltage function on the DMM to measure the potential differences across the individual resistors ( V A, V B and V C ) and then across the combinations of resistors ( V BC and V ABC ). Record your readings in Data Table 3.

6 Direct Current Circuits Page 6 of 8 Figure 5: Three-resistor combination circuit. A-14. Change the leads in your DMM so that they can be used to measure current. Connect it first between the positive terminal of the battery and the resistor R A to measure I 0. Then interrupt the resistor R A and the parallel circuit junction to measure I A. Interrupt the various branches of the parallel circuit and measure the individual branch currents, i.e. I B and I C. Lastly connect it between the negative terminal of the battery and the parallel circuit junction to measure I 4. Record your measurements in Data Table 3. Part B: Multi-loop Circuits In this part of the experiment, you will investigate the behaviours of resistors in multi-loop circuits. B-1. Choose values of R 1 = 500 Ω, R 2 = 750 Ω and R 3 = 1000 Ω, or values as close to those as possible. Use the digital multimeter (DMM) to measure the value of the resistors and record these values in Data Table 1. B-2. Using two battery supplies and the resistors R 1, R 2 and R 3, construct a circuit as shown in Figure 6 with ε 1 = 9.0 V and ε 2 = 3.0 V. B-3. Measure the currents I 1, I 2, and I 3 and record them in Data Table 4. Assuming that one DMM is available, the currents will have to be measured one at a time by placing the DMM in the positions shown in the circuit diagram as a circle (see Figure 6). Note: Placing the DMM in the circuit with the polarity shown in the circuit diagram will give positive readings when the current is in the direction assumed. Otherwise, the DMM will give a negative reading if the current is in the opposite direction. B-4. Measure the emfs ε 1, and ε 2 with the DMM and record those values in Data Table 4. The terminal voltages of the battery supplies are assumed to approximate the emfs.

7 Direct Current Circuits Page 7 of 8 Figure 6: Multi-loop circuit with three unknown currents. B-5. Construct the circuit of Figure 7, which also has two battery supplies but has four resistors. Choose values of ε 1 = 3.0 V, ε 2 = 9.0 V, R 1 = 1000 Ω, R 2 = 800 Ω, R 3 = 600 Ω and R 4 = 500 Ω, or values as close to those as possible. Determine and record in Data Table 5 the values of the resistors, the four current and the emfs of the battery supplies. Figure 7: Multi-loop circuit with four unknown currents.

8 Direct Current Circuits Page 8 of 8 A Appendix: Resistor colour codes Figure 8: Resistor colour code. The value of a resistor is typically identified on the component as a numeric value, or more commonly, by a series of coloured bands. The orientation of the bands can be determined by choosing the first band as the band closest to the end of the resistor body. The colour of the band makes up the first digit of the resistance value. For example, the resistance R of a resistor whose bands are yellow, violet, red and gold is R = yellow-violet red ± gold R = Ω ± (5% of Ω) R = 4700 ± 200 Ω (error rounded to one significant figure) = (4.7 ± 0.2) 10 3 Ω Last updated: Tuesday 3 rd February, :38pm (KHCM)