7.8: Circuits

Learning Objectives

• Draw a circuit with resistors in parallel and in series.
• Explain why total resistance of a parallel circuit is less than the smallest resistance of any of the resistors in that circuit.

Most circuits have more than one component, called a resistor that limits the flow of charge in the circuit. A measure of this limit on charge flow is called resistance. The simplest combinations of resistors are the series and parallel connections illustrated in Figure \(\PageIndex{1}\). The total resistance of a combination of resistors depends on both their individual values and how they are connected.

Figure \(\PageIndex{1}\): (a) A series connection of resistors. (b) A parallel connection of resistors.

Resistors in Series

When are resistors in series? Resistors are in series whenever the flow of charge, called the current, must flow through devices sequentially. For example, if current flows through a person holding a screwdriver and into the Earth, then \(R_{1}\) in Figure \(\PageIndex{1}\)(a) could be the resistance of the screwdriver’s shaft, \(R_{2}\) the resistance of its handle, \(R_{3}\) the person’s body resistance, and \(R_{4}\) the resistance of her shoes.

Figure \(\PageIndex{2}\) shows resistors in series connected to a voltage source. It seems reasonable that the total resistance is the sum of the individual resistances, considering that the current has to pass through each resistor in sequence. (This fact would be an advantage to a person wishing to avoid an electrical shock, who could reduce the current by wearing high-resistance rubber-soled shoes. It could be a disadvantage if one of the resistances were a faulty high-resistance cord to an appliance that would reduce the operating current.)

Figure \(\PageIndex{2}\): Three resistors connected in series to a battery (left) and the equivalent single or series resistance (right).

MAJOR FEATURES OF RESISTORS IN SERIES

1. Series resistances add: \(R_{\mathrm{s}}=R_{1}+R_{2}+R_{3}+\ldots\)
2. The same current flows through each resistor in series.
3. Individual resistors in series do not get the total source voltage, but divide it.

Resistors in Parallel

Figure \(\PageIndex{3}\) 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). For example, an automobile’s headlights, radio, and so on, are wired in parallel, so that they utilize the full voltage of the source and can operate completely independently. The same is true in your house, or any building. (See Figure \(\PageIndex{3}\)(b).)

Figure \(\PageIndex{3}\): (a) Three resistors connected in parallel to a battery and the equivalent single or parallel resistance. (b) Electrical power setup in a house. (credit: Dmitry G, Wikimedia Commons)

MAJOR FEATURES OF RESISTORS IN PARALLEL

1. Parallel resistance is found from \(\frac{1}{R_{\mathrm{p}}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}+\ldots\), and it is smaller than any individual resistance in the combination.
2. Each resistor in parallel has the same full voltage of the source applied to it. (Power distribution systems most often use parallel connections to supply the myriad devices served with the same voltage and to allow them to operate independently.)
3. Parallel resistors do not each get the total current; they divide it.

Combinations of Series and Parallel

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 \(\PageIndex{4}\). 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 \(\PageIndex{4}\): 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 \(\PageIndex{5}\), 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.

CONNECTIONS: CONSERVATION LAWS

The derivations of the expressions for series and parallel resistance are based on the laws of conservation of energy and conservation of charge, which state that total charge and total energy are constant in any process. These two laws are directly involved in all electrical phenomena and will be invoked repeatedly to explain both specific effects and the general behavior of electricity.

Practical Implications

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 \(I R\) 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 \(\PageIndex{6}\). 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 \(I R\) drop in the wires represented by \(R_{1}\), reducing the voltage across the light bulb (which is \(R_{2}\)), which then dims noticeably.

Figure \(\PageIndex{6}\): 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 \(I R\) drop in the wires and reduces the voltage across the light.

Whether something is wired in parallel or in series will also affect the performance of the devices involved, as we will see in a couple of examples. You are probably familiar with the custom of decorating with strands of light for winter holidays (in fact, these are sometimes called Christmas lights even if they are used for a different purpose). These are typically wired in series, as the number of wires necessary to wiring them in parallel would be prohibitive to their function. Because they are wired in series, this means that the current has to pass through each bulb before going to the next bulb. If one of the bulbs stops functioning, then current no longer can flow through any of them, and none of the lights will function. Modern string lights are often designed with a bypass system so that they can continue to function even if one of them goes out.

Another important application to understand is the power strip. These are useful tools to have when there are lots of lower power devices that need to be plugged in but not enough plugs in the area of need. However, they can be quite dangerous if used incorrectly. Because a power strip is wired in parallel for each of the outlets, they can each draw the full voltage of the line the power strip is plugged into. If the current to each of the plugs is low, then it is safe. But if there is a high current, perhaps because the power is being used to heat something, then there might be more power passing through the power strip than it can reasonably handle. If used unsafely, this can result in a fire.

Additional safety issues for other electrical applications are addressed later in this chapter.

Section Summary

• The total resistance of an electrical circuit with resistors wired in a series is the sum of the individual resistances.
• Each resistor in a series circuit has the same amount of current flowing through it.
• The voltage drop, or power dissipation, across each individual resistor in a series is different, and their combined total adds up to the power source input.
• The total resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components.
• Each resistor in a parallel circuit has the same full voltage of the source applied to it.
• The current flowing through each resistor in a parallel circuit is different, depending on the resistance.
• If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.

Glossary

series

a sequence of resistors or other components wired into a circuit one after the other

resistor

a component that provides resistance to the current flowing through an electrical circuit

resistance

causing a loss of electrical power in a circuit

Ohm’s law

the relationship between current, voltage, and resistance within an electrical circuit: \(V=IR\)

voltage

the electrical potential energy per unit charge; electric pressure created by a power source, such as a battery

voltage drop

the loss of electrical power as a current travels through a resistor, wire or other component

current

the flow of charge through an electric circuit past a given point of measurement

Joule’s law

the relationship between potential electrical power, voltage, and resistance in an electrical circuit, given by: \(P_{e}=I V\)

parallel

the wiring of resistors or other components in an electrical circuit such that each component receives an equal voltage from the power source; often pictured in a ladder-shaped diagram, with each component on a rung of the ladder