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Power is energy per unit time watts , and so conservation of energy requires the power output of the source to be equal to the total power dissipated by the resistors. Figure 3 shows resistors in parallel , wired to a voltage source. Resistors are in parallel when each resistor is connected directly to the voltage source by connecting wires having negligible resistance. Each resistor thus has the full voltage of the source applied to it.
Each resistor draws the same current it would if it alone were connected to the voltage source provided the voltage source is not overloaded. The same is true in your house, or any building. See Figure 3 b. Figure 3. To find an expression for the equivalent parallel resistance R p , let us consider the currents that flow and how they are related to resistance. Conservation of charge implies that the total current I produced by the source is the sum of these currents:.
The terms inside the parentheses in the last two equations must be equal. Generalizing to any number of resistors, the total resistance R p of a parallel connection is related to the individual resistances by.
This relationship results in a total resistance R p that is less than the smallest of the individual resistances. This is seen in the next example. When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, and so the total resistance is lower.
The total resistance for a parallel combination of resistors is found using the equation below. Entering known values gives. Note that in these calculations, each intermediate answer is shown with an extra digit. We must invert this to find the total resistance R p.
This yields. Current I for each device is much larger than for the same devices connected in series see the previous example. A circuit with parallel connections has a smaller total resistance than the resistors connected in series. The power dissipated by each resistor can be found using any of the equations relating power to current, voltage, and resistance, since all three are known.
The power dissipated by each resistor is considerably higher in parallel than when connected in series to the same voltage source. The total power can also be calculated in several ways. Note that both the currents and powers in parallel connections are greater than for the same devices in series. More complex connections of resistors are sometimes just combinations of series and parallel.
These are commonly encountered, especially when wire resistance is considered. In that case, wire resistance is in series with other resistances that are in parallel. Combinations of series and parallel can be reduced to a single equivalent resistance using the technique illustrated in Figure 4.
Various parts are identified as either series or parallel, reduced to their equivalents, and further reduced until a single resistance is left. The process is more time consuming than difficult. Figure 4. This combination of seven resistors has both series and parallel parts.
Each is identified and reduced to an equivalent resistance, and these are further reduced until a single equivalent resistance is reached.
The simplest combination of series and parallel resistance, shown in Figure 4, is also the most instructive, since it is found in many applications. For example, R 1 could be the resistance of wires from a car battery to its electrical devices, which are in parallel.
R 2 and R 3 could be the starter motor and a passenger compartment light. We have previously assumed that wire resistance is negligible, but, when it is not, it has important effects, as the next example indicates. Figure 5 shows the resistors from the previous two examples wired in a different way—a combination of series and parallel. We can consider R 1 to be the resistance of wires leading to R 2 and R 3.
Figure 5. These three resistors are connected to a voltage source so that R 2 and R 3 are in parallel with one another and that combination is in series with R 1. To find the total resistance, we note that R 2 and R 3 are in parallel and their combination R p is in series with R 1.
Thus the total equivalent resistance of this combination is. First, we find R p using the equation for resistors in parallel and entering known values:. The total resistance of this combination is intermediate between the pure series and pure parallel values Thus its IR drop is.
We must find I before we can calculate V 1. The voltage applied to R 2 and R 3 is less than the total voltage by an amount V 1. When wire resistance is large, it can significantly affect the operation of the devices represented by R 2 and R 3. To find the current through R 2 , we must first find the voltage applied to it. We call this voltage V p , because it is applied to a parallel combination of resistors. The voltage applied to both R 2 and R 3 is reduced by the amount V 1 , and so it is.
The current is less than the 2. The power is less than the One implication of this last example is that resistance in wires reduces the current and power delivered to a resistor. If wire resistance is relatively large, as in a worn or a very long extension cord, then this loss can be significant. If a large current is drawn, the IR drop in the wires can also be significant. For example, when you are rummaging in the refrigerator and the motor comes on, the refrigerator light dims momentarily.
Similarly, you can see the passenger compartment light dim when you start the engine of your car although this may be due to resistance inside the battery itself. What is happening in these high-current situations is illustrated in Figure 6. The device represented by R 3 has a very low resistance, and so when it is switched on, a large current flows. This increased current causes a larger IR drop in the wires represented by R 1 , reducing the voltage across the light bulb which is R 2 , which then dims noticeably.
Figure 6. Why do lights dim when a large appliance is switched on? The answer is that the large current the appliance motor draws causes a significant drop in the wires and reduces the voltage across the light.
A switch has a variable resistance that is nearly zero when closed and extremely large when open, and it is placed in series with the device it controls. Explain the effect the switch in Figure 7 has on current when open and when closed. Figure 7. A switch is ordinarily in series with a resistance and voltage source.
Ideally, the switch has nearly zero resistance when closed but has an extremely large resistance when open. Note that in this diagram, the script E represents the voltage or electromotive force of the battery.
There is a voltage across an open switch, such as in Figure 7. Why, then, is the power dissipated by the open switch small? A student in a physics lab mistakenly wired a light bulb, battery, and switch as shown in Figure 8.
Explain why the bulb is on when the switch is open, and off when the switch is closed. Do not try this—it is hard on the battery!
Figure 8. Knowing that the severity of a shock depends on the magnitude of the current through your body, would you prefer to be in series or parallel with a resistance, such as the heating element of a toaster, if shocked by it?
Some strings of holiday lights are wired in series to save wiring costs. An old version utilized bulbs that break the electrical connection, like an open switch, when they burn out. If one such bulb burns out, what happens to the others? If such a string operates on V and has 40 identical bulbs, what is the normal operating voltage of each?
Newer versions use bulbs that short circuit, like a closed switch, when they burn out. If such a string operates on V and has 39 remaining identical bulbs, what is then the operating voltage of each? If two household lightbulbs rated 60 W and W are connected in series to household power, which will be brighter? Suppose you are doing a physics lab that asks you to put a resistor into a circuit, but all the resistors supplied have a larger resistance than the requested value. How would you connect the available resistances to attempt to get the smaller value asked for?
Explain why resistance cords become warm and waste energy when the radio is on.
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