Resistors in Series

A series circuit provides only one path for electric charge to travel. Every electron that leaves the voltage source must pass through each resistor, one after another, before returning to the source. Because there is only one possible path, the same current flows through every component connected in series.

This idea appears in many real-world situations. For example, if a person accidentally touches an energized conductor while holding a metal screwdriver and standing on the ground, the current passes sequentially through the screwdriver, the person's body, their shoes, and finally the earth itself, which completes the circuit back to the source. In Figure 27.1(a), these four elements could be represented by resistors [latex]R_1[/latex] (the screwdriver), [latex]R_2[/latex] (the body), [latex]R_3[/latex] (the shoes), and [latex]R_4[/latex] (the earth), respectively. Since each element lies along the same path, their resistances combine in series. Wearing insulating rubber-soled shoes increases the total resistance of this path and therefore reduces the current that could flow through the body.

Figure 27.2 shows three resistors connected in series to a voltage source. Because the current must pass through each resistor in succession, it is reasonable to expect that each resistor contributes to the total opposition to current flow. We will now show that the equivalent resistance of a series circuit is simply the sum of the individual resistances.

As current flows through each resistor, part of the electrical potential energy carried by the charges is converted into thermal energy. The decrease in electric potential across a resistor is called the voltage drop.

According to Ohm's law, the voltage drop across a resistor is

where [latex]I[/latex] is the current and [latex]R[/latex] is the resistance. Since the same current flows through every resistor in a series circuit, the voltage drops across the three resistors are

Conservation of energy requires that the total electrical energy supplied by the source equals the total energy dissipated by the resistors. Because electrical potential energy is

the energy supplied by the battery is [latex]qV[/latex], while the energy dissipated in the three resistors is

Setting these equal and canceling the charge [latex]q[/latex] gives

Thus, the total voltage supplied by the source is shared among the resistors. This result is often called Kirchhoff's voltage law and follows directly from conservation of energy.

Resistors in Parallel

Resistors are connected in parallel when each resistor is connected directly across the same two points in a circuit. In this arrangement, every resistor has its own path for charge to flow between the terminals of the voltage source. Unlike a series circuit, where there is only one possible path, a parallel circuit provides multiple paths for the current.

Because each branch is connected directly across the voltage source, the voltage across every resistor is the same. However, the current is not necessarily the same. Instead, the current divides among the different branches according to their resistances. Branches with lower resistance carry larger currents, while branches with higher resistance carry smaller currents.

This arrangement is extremely common in everyday life. Household electrical wiring is connected in parallel so that every outlet and appliance receives the full line voltage and can operate independently. Turning one appliance on or off does not affect the operation of the others. Similarly, electrical systems in automobiles connect headlights, radios, and other accessories in parallel so that each device receives the full battery voltage.

Parallel circuits are also widely used in healthcare. Hospital rooms contain multiple electrical outlets connected in parallel so that patient monitors, infusion pumps, ventilators, and other equipment all receive the correct operating voltage simultaneously. If one device is disconnected or switched off, the others continue functioning normally.

Since every resistor has the same voltage across it, Ohm's law gives the current through each branch as

Conservation of charge requires that the total current supplied by the source equals the sum of the currents in the individual branches:

Substituting the expressions above gives

If the three resistors are replaced by a single equivalent resistor [latex]R_{\mathrm{p}}[/latex], Ohm's law becomes

Since both equations describe the same circuit, the expressions multiplying the voltage must be equal. Therefore, the equivalent resistance of three resistors connected in parallel satisfies

More generally, for any number of resistors connected in parallel,

Notice that the equivalent resistance of a parallel circuit is always smaller than the smallest individual resistance. Every additional branch provides another path for charge to flow, allowing more current to leave the source for the same applied voltage. Because resistance measures opposition to current, adding parallel paths reduces the overall resistance of the circuit.

Combinations of Series and Parallel

Many real circuits contain resistors connected in both series and parallel. For example, the resistance of connecting wires may be in series with a group of devices connected in parallel. This situation occurs in automobiles, household wiring, laboratory equipment, and medical devices.

To analyze these circuits, we simplify them step by step. First, identify a group of resistors that is clearly in series or clearly in parallel. Replace that group with its equivalent resistance. Then redraw the circuit and repeat the process until only one equivalent resistance remains.

The combination shown in Figure 27.5 is especially useful. Resistor [latex]R_1[/latex] could represent the resistance of wires connecting a battery to two devices, while [latex]R_2[/latex] and [latex]R_3[/latex] could represent devices connected in parallel, such as a starter motor and an interior light in a car.

In earlier chapters, we often assumed that wires had negligible resistance. In real circuits, wire resistance can sometimes matter. When large currents flow, even small wire resistances can produce noticeable voltage drops and reduce the power delivered to connected devices.

Practical Implications

One important implication of the previous example is that resistance in connecting wires reduces the voltage available to electrical devices. Whenever current flows through a wire with resistance, some of the source voltage is lost as an [latex]IR[/latex] drop. If the wire resistance is relatively large—as in a worn electrical cord, a very long extension cord, or undersized wiring—this voltage loss can become significant.

The effect becomes especially noticeable when a device draws a large current. As the current increases, the voltage drop across the wires also increases because

This reduces the voltage delivered to other devices connected to the same circuit, causing them to operate less effectively.

A familiar example occurs when the compressor motor in a refrigerator turns on. The interior light may briefly dim because the motor draws a large current, increasing the voltage drop in the household wiring. A similar effect is often observed when starting a car: the interior lights dim momentarily because the starter motor requires a very large current. In that case, part of the voltage drop also occurs inside the battery itself.

The process is illustrated in Figure 27.6. The device represented by [latex]R_3[/latex] has a very low resistance and therefore draws a large current when switched on. That larger current produces a greater [latex]IR[/latex] drop across the resistance of the connecting wires, represented by [latex]R_1[/latex]. As a result, the voltage across the light bulb ([latex]R_2[/latex]) decreases, making the bulb noticeably dimmer.

Health and Technology Connection

Medical equipment often requires a stable supply voltage to operate correctly. Hospitals therefore use carefully designed electrical systems with appropriately sized wiring and dedicated circuits for critical devices. Limiting voltage drops helps ensure that sensitive equipment—such as patient monitors, ventilators, and imaging systems—receives reliable electrical power.

Real-World Insight

Large currents not only reduce the voltage delivered to other devices but also increase the heating of electrical wires because the power dissipated is proportional to [latex]I^2R[/latex]. This is why electrical systems use wires with appropriate diameters and protect circuits with fuses or circuit breakers that disconnect the circuit before excessive heating can occur.

Check Your Understanding

Can every resistor network be simplified into a combination of series and parallel resistors? If not, try drawing a circuit that cannot be reduced using only the methods presented in this chapter.

No. Many practical circuits cannot be simplified using only series and parallel combinations. For example, circuits containing bridge connections or multiple interconnected loops require more advanced analysis techniques. In the next chapter, you will learn Kirchhoff's rules, which allow you to analyze these more complex circuits by applying the conservation of charge and the conservation of energy.

Problem-Solving Strategy: Series and Parallel Circuits

1. Draw a clear circuit diagram. Label every resistor, voltage source, and any known currents or voltages. Recording the given information directly on the diagram makes it easier to follow your calculations.
2. Identify the unknowns. Determine exactly what the problem is asking you to find before beginning any calculations.
3. Identify series and parallel connections. Resistors are in series when the same current must pass through each one in sequence. They are in parallel when they share the same two connection points and therefore have the same voltage across them.
4. Simplify the circuit step by step. Replace each series or parallel group with its equivalent resistance, redraw the circuit, and repeat the process until only a single equivalent resistance remains. Then use Ohm's law to determine the total current and work backward to find the voltage and current for each component as needed.
5. Be careful with parallel resistance. When calculating an equivalent parallel resistance, first add the reciprocals of the individual resistances and then take the reciprocal of the result. Forgetting this final step is one of the most common mistakes made by beginning students.
6. Check your results. Make sure your answers are physically reasonable. The equivalent resistance of resistors in series must be greater than any individual resistance, whereas the equivalent resistance of resistors in parallel must be smaller than the smallest individual resistance. Also verify that the units are correct and that your results are consistent with Ohm's law and the conservation of energy.

Section Summary

• For resistors connected in series, the equivalent resistance is the sum of the individual resistances:
  [latex]R_{\text{s}}=R_1+R_2+R_3+\cdots[/latex]
• In a series circuit, the same current flows through every resistor because there is only one path for charge to travel.
• The source voltage is divided among the resistors. The individual voltage drops add to the source voltage, consistent with the conservation of energy.
• For resistors connected in parallel, the equivalent resistance is found from
  [latex]\frac{1}{R_{\text{p}}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\cdots[/latex]

  The equivalent resistance is always less than the smallest individual resistance.

• In a parallel circuit, every branch has the same voltage as the source.
• The current divides among the parallel branches according to their resistances. The sum of the branch currents equals the total current supplied by the source, consistent with the conservation of charge.
• Circuits containing both series and parallel resistors can often be simplified by replacing groups of resistors with their equivalent resistance one step at a time until a single equivalent resistance remains.

Conceptual Questions

1. 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 27.7 has on the current when it is open and when it is closed.

2. What is the voltage across the open switch in Figure 27.7?
3. There is a voltage across an open switch, such as the one in Figure 27.7. Why is the power dissipated by the open switch nevertheless very small?
4. Why is the power dissipated by a closed switch, such as the one in Figure 27.7, also very small?
5. A student in a physics laboratory mistakenly wires a light bulb, battery, and switch as shown in Figure 27.8. Explain why the bulb is on when the switch is open and off when the switch is closed. Do not try this configuration because it can damage the battery.

6. Knowing that the severity of an electric shock depends on the magnitude of the current through your body, would you prefer your body to be in series or in parallel with a resistance, such as the heating element of a toaster, if you were shocked by it? Explain.
7. Would automobile headlights dim when the engine is started if the wires in the automobile were superconductors? Do not neglect the battery's internal resistance. Explain.
8. Some strings of holiday lights are wired in series to reduce wiring costs.
   1. In an older design, each bulb opens the circuit when it burns out. What happens to the other bulbs when one bulb burns out?
   2. If the string operates on 120 V and contains 40 identical bulbs, what is the normal operating voltage across each bulb?
   3. In a newer design, a burned-out bulb creates a short circuit. What happens to the other bulbs when one bulb burns out?
   4. If the string operates on 120 V and 39 identical bulbs remain in operation, what is the voltage across each remaining bulb?
9. If two household light bulbs rated 60 W and 100 W are connected in series to household power, which bulb will be brighter? Explain.
10. Suppose you are completing a physics laboratory activity that asks you to place a resistor of a particular value in a circuit, but every available resistor has a resistance larger than the requested value. How could you connect the available resistors to obtain a smaller equivalent resistance?
11. Before World War II, some radios received power through a resistance cord with significant electrical resistance. The cord reduced the voltage supplied to the radio's vacuum tubes and other components, avoiding the expense of a transformer. Explain why the cord became warm and wasted energy while the radio operated.
12. Some light bulbs have three nonzero power settings produced by multiple filaments that can be switched individually and are wired in parallel. What is the minimum number of filaments required to produce three power settings?

Problems & Exercises

Note: Data taken from figures may be assumed to be accurate to three significant digits.

1. 1. What is the equivalent resistance of ten [latex]275~\Omega[/latex] resistors connected in series?
   2. What is their equivalent resistance when connected in parallel?
2. 1. What is the equivalent resistance of [latex]1.00\times10^{2}~\Omega[/latex], [latex]2.50~\text{k}\Omega[/latex], and [latex]4.00~\text{k}\Omega[/latex] resistors connected in series?
   2. What is their equivalent resistance when connected in parallel?
3. What are the largest and smallest equivalent resistances you can obtain by connecting [latex]36.0~\Omega[/latex], [latex]50.0~\Omega[/latex], and [latex]700~\Omega[/latex] resistors?
4. An 1800-W toaster, a 1400-W electric frying pan, and a 75-W lamp are plugged into the same outlet on a 15-A, 120-V circuit. The devices are connected in parallel.
   1. What current is drawn by each device?
   2. Will this combination blow the 15-A fuse?
5. A car's 30.0-W headlight and 2.40-kW starter are ordinarily connected in parallel in a 12.0-V electrical system. What power would one headlight and the starter consume if they were connected in series to a 12.0-V battery? Neglect all other resistance in the circuit and any change in the resistance of the two devices.
6. Given a 48.0-V battery and [latex]24.0~\Omega[/latex] and [latex]96.0~\Omega[/latex] resistors:
   1. Find the current and power for each resistor when they are connected in series.
   2. Repeat the calculation when the resistors are connected in parallel.
7. Referring to the example that combines series and parallel circuits and to Figure 27.5, calculate [latex]I_3[/latex] in two ways:
   1. Use the known values of [latex]I[/latex] and [latex]I_2[/latex].
   2. Use Ohm's law for [latex]R_3[/latex].

   In both parts, explicitly show how you follow the steps in the Problem-Solving Strategy: Series and Parallel Circuits.

8. Referring to Figure 27.5:
   1. Calculate [latex]P_3[/latex] and compare it with the value of [latex]P_3[/latex] found in the first two worked examples in this chapter.
   2. Find the total power supplied by the source and compare it with the sum of the powers dissipated by the resistors.
9. Refer to Figure 27.6 and the discussion of lights dimming when a high-power appliance starts.
   1. Given that the voltage source is 120 V, the wire resistance is [latex]0.400~\Omega[/latex], and the bulb is rated at 75.0 W, what power will the bulb dissipate if a total current of 15.0 A flows through the wires when the motor starts? Assume that the bulb's resistance does not change significantly.
   2. What power is consumed by the motor?
10. A 240-kV power transmission line carrying [latex]5.00\times10^{2}~\text{A}[/latex] is supported by grounded metal towers using ceramic insulators. Each insulator has a resistance of [latex]1.00\times10^{9}~\Omega[/latex], as shown in Figure 27.9.
    1. What is the equivalent resistance to ground of 100 of these insulators?
    2. Calculate the total power dissipated by the 100 insulators.
    3. What fraction of the power carried by the transmission line is dissipated through the insulators?

    Explicitly show how you follow the steps in the Problem-Solving Strategy: Series and Parallel Circuits.

11. Show that if two resistors [latex]R_1[/latex] and [latex]R_2[/latex] are combined and [latex]R_1\gg R_2[/latex]:
    1. Their series resistance is approximately equal to the larger resistance, [latex]R_1[/latex].
    2. Their parallel resistance is approximately equal to the smaller resistance, [latex]R_2[/latex].
12. Unreasonable Results. Two resistors, one having a resistance of [latex]145~\Omega[/latex], are connected in parallel to produce an equivalent resistance of [latex]150~\Omega[/latex].
    1. What is the resistance of the second resistor?
    2. What is unreasonable about the result?
    3. Which assumption or premise is inconsistent?
13. Unreasonable Results. Two resistors, one having a resistance of [latex]900~\text{k}\Omega[/latex], are connected in series to produce an equivalent resistance of [latex]0.500~\text{M}\Omega[/latex].
    1. What is the resistance of the second resistor?
    2. What is unreasonable about the result?
    3. Which assumption or premise is inconsistent?

Glossary

series connection

a circuit configuration in which components are connected one after another so that the same current flows through each component

resistor

an electrical component designed to oppose the flow of electric current by providing a specified resistance

resistance

the opposition a material or circuit component provides to the flow of electric current, measured in ohms ([latex]\Omega[/latex])

Ohm's law

the relationship among voltage, current, and resistance in an electrical circuit:

[latex]V=IR[/latex]

voltage

the electric potential difference between two points; the electrical energy available per unit charge that can drive current through a circuit

voltage drop

the decrease in electric potential across a circuit element as electrical energy is transferred or converted

current

the rate at which electric charge flows through a circuit, measured in amperes (A)

Joule's law

the relationships describing the electrical power dissipated in a circuit:

[latex]P=IV,\qquad P=I^2R,\qquad P=\frac{V^2}{R}[/latex]

parallel connection

a circuit configuration in which components are connected across the same two points, so that each component experiences the same voltage while the current divides among the branches