20.7

Parallel Wiring

Parallel wiring is another method of connecting electrical devices. Parallel wiring means that the devices are connected in such a way that the same voltage is applied across each device. Figure 20-19 shows two resistors connected in parallel between the terminals of a battery. Part a of the picture is drawn so as to emphasize that the entire voltage of the battery is applied across each resistor. Actually, parallel connections are rarely drawn in this manner; instead they are drawn as in part b, where the dots indicate the points where the wires for the two branches are joined together. Parts a and b are equivalent representations of the same circuit.

Figure 20-19 (a) When two resistors are connected in parallel, the same voltage V is applied across each resistor. (b) This circuit drawing is equivalent to that in part a. I1 and I2 are, respectively, the currents in R1 and R2.

Parallel wiring is very common. For example, when an electrical appliance is plugged into a wall socket, the appliance is connected in parallel with other appliances, as in Figure 20-20, where the entire voltage of 120 V is applied across the television, the stereo, and the light bulb. The presence of the unused socket or other devices that are turned off does not affect the operation of those devices that are turned on. Moreover, if the current in one device is interrupted (perhaps by an opened switch or a broken wire), the current in the other devices is not interrupted. In contrast, if household appliances were connected in series, there would be no current through any appliance if the current in the circuit were halted at any point.

Figure 20-20 This drawing shows some of the parallel connections found in a typical home. Each wall socket provides 120 V to the appliance connected to it. In addition, 120 V is applied to the light bulb when the switch is turned on.

When two resistors R1 and R2 are connected as in Figure 20-19, each receives current from the battery as if the other were not present. Therefore, R1 and R2 together draw more current from the battery than does either resistor alone. According to the definition of resistance, R =V/I, a larger current implies a smaller resistance. Thus, the two parallel resistors behave as a single equivalent resistance that is smaller than either R1 or R2. Figure 20-21 returns to the water-flow analogy to provide additional insight into this important feature of parallel wiring. In part a, two sections of pipe that have the same length are connected in parallel with a pump. In part b these two sections have been replaced with a single pipe of the same length, whose cross-sectional area equals the combined cross-sectional areas of section 1 and section 2. The pump (analogous to a voltage source) can push more water per second (analogous to current) through the wider pipe in part b (analogous to a wider wire) than it can through either of the narrower pipes (analogous to narrower wires) in part a. In effect, the wider pipe offers less resistance to the flow of water that either of the narrower pipes offers individually.

Figure 20-21 (a) Two equally long pipe sections, with cross-sectional areas A1 and A2, are connected in parallel to a water pump. (b) The two parallel pipe sections in part a are equivalent to a single pipe of the same length whose cross-sectional area is A1 + A2.

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 for the two resistors in Figure 20-19b, note that the total current I from the battery is the sum of I1 and I2, where I1 is the current in resistor R1 and I2 is the current in resistor . Since the same voltage V is applied across each resistor, the definition of resistance indicates that and . Therefore,

where RP is the equivalent resistance. Hence, when two resistors are connected in parallel, they are equivalent to a single resistor whose resistance RP can be obtained from . In general, for any number of resistors wired in parallel, the total current from the voltage source is the sum of the currents in the individual resistors. Thus, a similar line of reasoning reveals that the equivalent resistance is

| (20.17)

The next example deals with a parallel combination of resistors that occurs in a stereo system.

| Example 10 | | | Main and Remote Stereo Speakers

Most receivers allow the user to connect “remote” speakers (to play music in another room, for instance) in addition to the main speakers. Figure 20-22 shows that the remote speaker and the main speaker for the right stereo channel are connected to the receiver in parallel (for clarity, the speakers for the left channel are not shown). At the instant represented in the picture, the ac voltage across the speakers is 6.00 V. The main-speaker resistance is , and the remote-speaker resistance is .* Determine (a) the equivalent resistance of the two speakers, (b) the total current supplied by the receiver, (c) the current in each speaker, and (d) the power dissipated in each speaker.

Figure 20-22 (a) The main and remote speakers in a stereo system are connected in parallel to the receiver. (b) The circuit schematic shows the situation when the ac voltage across the speakers is 6.00 V.

The physics of main and remote stereo speakers.

Reasoning The total current supplied to the two speakers by the receiver can be calculated as , where RP is the equivalent resistance of the two speakers in parallel and can be obtained from . The current in each speaker is different, however, since the speakers have different resistances. The average power delivered to a given speaker is the product of its current and voltage. In the parallel connection the same voltage is applied to each speaker.
Solution

a. | According to Equation 20.17, the equivalent resistance of the two speakers is given by

This result is illustrated in part b of the drawing.
b. | Using the equivalent resistance in Ohm’s law shows that the total current is

| (20.14)

c. | Applying Ohm’s law to each speaker gives the individual speaker currents:

The sum of these currents is equal to the total current obtained in part (b).
d. | The average power dissipated in each speaker can be calculated using with the individual currents obtained in part (c):

| (20.15a)

| (20.15a)

Problem solving insight The equivalent resistance RP of a number of resistors in parallel has a reciprocal given by , where R1, R2, and R3 are the individual resistances. After adding together the reciprocals , , and , do not forget to take the reciprocal of the result to find RP.

In Example 10, the total power delivered by the receiver is the sum of the individual values that were found in part (d), . Alternatively, the total power can be obtained from the equivalent resistance and the total current in part (b):

| (20.15b)

In general, the total power delivered to any number of resistors in parallel is equal to the power delivered to the equivalent resistor.

Problem solving insight

In a parallel combination of resistances, it is the smallest resistance that has the largest impact in determining the equivalent resistance. In fact, if one resistance approaches zero, then according to Equation 20.17, the equivalent resistance also approaches zero. In such a case, the near-zero resistance is said to short out the other resistances by providing a near-zero resistance path for the current to follow as a shortcut around the other resistances.
An interesting application of parallel wiring occurs in a three-way light bulb, as Conceptual Example 11 discusses.

| Conceptual Example 11 | | | A Three-Way Light Bulb and Parallel Wiring

Three-way light bulbs are popular because they can provide three levels of illumination (e.g., 50 W, 100 W, and 150 W) using a 120-V socket. The socket, however, must be equipped with a special three-way switch that enables one to select the illumination level. This switch does not select different voltages, because a three-way bulb uses a single voltage of 120 V. Within the bulb are two separate filaments. When one of these burns out (i.e., vaporizes), the bulb can produce only one level of illumination (either its lowest or its intermediate level) but not the highest level. How are the two filaments connected, in series or in parallel? And how can two filaments be used to produce three different illumination levels?

The physics of a three-way light bulb.

Reasoning and Solution If the filaments were wired in series and one of them burned out, no current would pass through the bulb and none of the illumination levels would be available, contrary to what is observed. Therefore, the filaments must be connected in parallel, as in Figure 20-23.

Figure 20-23 A three-way light bulb uses two filaments connected in parallel. The filaments can be turned on one at a time or both together.

As to how two filaments can be used to produce three illumination levels, we note that the power dissipated in a resistance R is , according to Equation 20.15c. With a single value of 120 V for the voltage Vrms, three different wattage ratings for the bulb can be obtained only if three different values for the resistance R are available. In a 50-W/100-W/150-W three-way bulb, for example, one resistance R50 is provided by the 50-W filament, and the second resistance R100 comes from the 100-W filament. The third resistance R150 is the parallel combination of the other two and can be obtained from . Figure 20-23 illustrates a simplified version of how the three-way switch operates to provide these three alternatives. In the first position of the switch, contact A is closed and contact B is open, so current passes only through the 50-W filament. In the second position, contact A is open and contact B is closed, so current passes only through the 100-W filament. In the third position, both contacts A and B are closed, and both filaments light up, giving 150 W of illumination.
Related Homework: Problem 50

Test your understanding of the material in Sections 20.1, 20.2, 20.3, 20.4, 20.5, 20.6 and 20.7:

· | Electromotive Force and Current
· | Ohm’s Law
· | Resistance and Resistivity
· | Electric Power
· | Alternating Current
· | Series Wiring
· | Parallel Wiring

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