Series and Parallel Wiring


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1 Series and Parallel Wiring Thus far, we have dealt with circuits that include only a single device, such as a light bulb. There are, however, many circuits in which more than one device is connected to a voltage source. This lesson introduces two methods by which such connections may be made, namely, series and parallel wiring. Series Wiring Series wiring means that the devices are connected in such a way that there is the same electric current through each device. The diagram below shows a circuit in which two different devices, represented by resistors R 1 and R 2, are connected in series with a battery. Note the following characteristics of devices connected in series: they are connected along the same current pathway if the current in one device is interrupted, the current in the other is also interrupted Because of the series wiring, the voltage V supplied by the battery is divided between the two resistors. The diagram indicates that the portion of the voltage across R 1 is V 1, while the portion across R 2 is V 2, therefore, V = V 1 + V 2 Applying Ohm s law to each resistor shows that V = V 1 + V 2 = IR 1 + IR 2 ( ) IR s = I R 1 + R 2 where R s is called the equivalent resistance of the series circuit. Thus, two resistors in series are equivalent to a single resistor whose resistance is R s = R 1 + R 2
2 This line of reasoning can be extended to any number of resistors in series, such that: R s = R 1 + R 2 + R Example 1 A 6.00 Ω resistor and a 3.00 Ω resistor are connected in series with a 12.0 V battery. Find a) the equivalent resistance. b) the current. c) the power dissipated in each resistor. d) the total power supplied by the battery.
3 Parallel Wiring Parallel wiring means that the devices are connected in such a way that the same voltage is applied across each device. The diagram below shows two resistors connected in parallel between the terminals of a battery. Note the following characteristics of devices connected in parallel: they are connected along different current pathways if the current in one device is interrupted, the current in the other is not As in a series circuit, it is possible to replace a parallel combination of resistors with an equivalent resistor that results in the same total current and power for a given voltage as the original combination. To determine the equivalent resistance, note that I = I 1 + I 2 Where I 1 is the current in resistor R 1, and I 2 is the current in resistor R 2. Since the same voltage V is applied across each resistor, Ohm s law states that: I 1 = V R 1 and I 2 = V R 2 Therefore, V I = I 1 + I 2 V = V + V R p R 1 R 2 1 R p = V 1 R + 1 R 1 2 where R p is the equivalent resistance of the parallel circuit.
4 Thus, two resistors in parallel are equivalent to a single resistor whose resistance is 1 R p = 1 R R 2 This line of reasoning can be extended to any number of resistors in series, such that: 1 R p = 1 R R R It is interesting to note that the two parallel resistors behave as a single equivalent resistance that is smaller than either R 1 or R 2. Example 2 A 4.00 Ω and an 8.00 Ω resistor are connected in parallel across a 6.00 V battery. Determine a) the equivalent resistance of the circuit. b) the total current supplied by the battery.
5 c) the current in each resistor. d) the power dissipated in each resistor. e) the total power delivered by the battery.
6
7 Circuits Worksheet #4 1. The current in a 47 Ω resistor is 0.12 A. This resistor is in series with a 28 Ω resistor, and the series combination is connected across a battery. What is the battery voltage? (9 V ) 2. Three resistors, 25, 45, and 75 Ω, are connected in series, and a 0.51 A current passes through them. What is (a) the equivalent resistance and (b) the potential difference across the three resistors? (145 Ω, 74 V ) 3. A 36 Ω resistor and an 18 Ω resistor are connected in series across a 15 V battery. What is the voltage across (a) the 36 Ω resistor and (b) the 18 Ω resistor? (10 V, 5 V ) 4. A battery dissipates 2.50 W of power in each of two 47 Ω resistors connected in series. What is the voltage of the battery? (21.7 V ) 5. Three resistors, 9.0, 5.0, and 1.0 Ω, are connected in series across a 24 V battery. Find (a) the current in, (b) the voltage across, and (c) the power dissipated in each resistor. ((a) 1.6 A (b) 14.4 V, 8 V, 1.6 V (c) W, 12.8 W, 2.56 W ) 6. The current in a series circuit is 15 A. When an additional 8 Ω resistor is inserted in series, the current drops to 12 A. What is the resistance in the original circuit? ( 32 Ω) 7. A 16 Ω loudspeaker and an 8 Ω loudspeaker are connected in parallel across the terminals of an amplifier. Assuming the speakers behave as resistors, determine the equivalent resistance of the two speakers. (5.3 Ω) 8. What resistance must be placed in parallel with a 155 Ω resistor to make the equivalent resistance 115 Ω? ( Ω) 9. How many 4 Ω resistors must be connected in parallel to create an equivalent resistance of onesixteenth of an ohm? (64 resistors) 10. Two resistors, 42 and 64 Ω, are connected in parallel. The current through the 64 Ω resistor is 3 A. (a) Determine the current in the other resistor. (b) What is the total power consumed by the two resistors? ( 4.57 A, 1450 W ) 11. A coffee cup heater and a lamp are connected in parallel to the same 120 V outlet. Together, they use a total of 84 W of power. The resistance of the heater is Ω. Find the resistance of the lamp. (240 Ω )
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