Electromotive Force

Many different devices can serve as voltage sources. Batteries produce electrical energy through chemical reactions, generators convert mechanical energy into electrical energy, solar cells convert sunlight directly into electricity, and thermoelectric devices generate voltage from temperature differences. Although these devices operate in different ways, they all create a potential difference that can drive electric current through a circuit.

A few common voltage sources are shown in Figure 28.1. On the microscopic scale, the potential difference creates an electric field that exerts forces on electric charges, causing them to move. Historically, this ability to drive current gave rise to the name electromotive force, abbreviated emf.

Despite its name, electromotive force is not a force. It is a voltage. Specifically, the electromotive force (emf) of a source is the potential difference measured when no current is flowing. Like any voltage, emf is measured in volts (V).

The emf depends on the physical mechanism that produces the voltage. For a battery, it is determined by the chemical reactions occurring inside the cell. However, the voltage measured across the battery terminals while it is delivering current is generally smaller than the emf. As the battery is depleted, overloaded, or ages, this difference becomes even larger because the battery's internal resistance increases.

Internal Resistance

Not all batteries that have the same rated voltage perform equally well. For example, both a 12-V motorcycle battery and a 12-V truck battery have approximately the same emf, but the truck battery can deliver much larger currents. The main reason is that the larger battery has a much lower internal resistance, usually represented by [latex]r[/latex].

Internal resistance is the resistance to the flow of charge inside the voltage source itself. Whenever current flows, some of the electrical energy produced by the source is dissipated inside the battery before it can reach the external circuit. As a result, the voltage available at the battery terminals is smaller than the emf.

Figure 28.2 shows a simple model of a real voltage source. Every battery or generator can be represented as an ideal source of emf connected in series with an internal resistance [latex]r[/latex]. This simple model explains why batteries provide less voltage when supplying large currents.

Internal resistance is not always constant. It generally increases as a battery ages or becomes discharged, reducing the voltage and power it can deliver. It can also depend on temperature, the amount of current being drawn, and the battery's design. Rechargeable batteries, for example, often develop larger internal resistance after many charge–discharge cycles.

How Batteries Produce a Voltage

Every battery converts chemical energy into electrical energy. Inside the battery, chemical reactions separate positive and negative charges, creating a potential difference between the two terminals. Different chemical reactions produce different voltages, which is why different types of batteries have different emf values.

A lead-acid battery, commonly used in automobiles, consists of lead and lead oxide plates immersed in sulfuric acid, as shown in Figure 28.3. During the chemical reactions inside the battery, electrons accumulate on the negative terminal while electrons are removed from the positive terminal, producing a separation of charge.

The details of these chemical reactions are studied in chemistry rather than physics. For our purposes, the important point is that the reactions continuously separate charge, maintaining the emf of the battery while chemical energy remains available.

Figure 28.4 illustrates this process schematically. Chemical reactions move electrons toward the negative terminal while removing electrons from the positive terminal. When the battery is connected to a complete circuit, these electrons flow through the external circuit, delivering electrical energy to the connected devices before returning to the battery.

The amount of energy supplied to each unit of charge determines the battery's emf. Since voltage is defined as electrical potential energy per unit charge,

[latex]V=\frac{PE}{q}.[/latex]

Different chemical reactions produce different voltages because they transfer different amounts of energy to each unit of charge. For example, a lead-acid cell provides approximately 2 V, while six cells connected in series produce the familiar 12-V automobile battery.

Terminal Voltage

When a battery or other voltage source is connected to a circuit, the voltage measured across its terminals is called the terminal voltage, denoted by [latex]V[/latex]. This is the voltage actually available to operate the external circuit.

For an ideal voltage source with no internal resistance, the terminal voltage would always equal the emf. Real voltage sources, however, have internal resistance. As current flows through the source, some electrical energy is dissipated inside the battery, causing part of the emf to be unavailable to the external circuit.

The relationship between emf and terminal voltage during discharge is

[latex]V=\text{emf}-Ir,[/latex]

where [latex]r[/latex] is the internal resistance of the source and [latex]I[/latex] is the current flowing through it. The product [latex]Ir[/latex] is the voltage drop across the battery's internal resistance.

Current is taken to be positive when it leaves the positive terminal of the source, as shown in Figure 28.2. The equation shows two important trends:

• As the current increases, the terminal voltage decreases.
• For a given current, a larger internal resistance produces a larger voltage drop inside the source.

Conceptual Insight

Think of the emf as the battery's maximum possible voltage when no current is being drawn. The internal resistance accounts for part of that voltage whenever current flows. The remaining voltage—the terminal voltage—is what is actually delivered to the external circuit.

Suppose a resistor with resistance [latex]R_{\text{load}}[/latex] is connected to a voltage source, as shown in Figure 28.5. Because the internal resistance and load resistance are connected in series, the total circuit resistance is

[latex]R_{\text{total}}=R_{\text{load}}+r.[/latex]

Applying Ohm's law to the complete circuit gives the current supplied by the source:

[latex]I=\frac{\text{emf}}{R_{\text{load}}+r}.[/latex]

This equation shows that increasing the internal resistance reduces the current supplied to the load. As batteries age or become discharged, their internal resistance typically increases. Consequently, even if the emf changes very little, the battery may deliver less current and a smaller terminal voltage under load.

Health Connection

Portable medical devices such as infusion pumps, portable ECG monitors, and defibrillators may require large bursts of current. A battery with high internal resistance may appear fully charged when measured with a voltmeter under no-load conditions, yet its terminal voltage can drop significantly during operation, preventing the device from functioning properly.

Conceptual Check

A battery is connected to the same resistor before and after it has aged. Its emf remains nearly the same, but its internal resistance has increased. What happens to the current and terminal voltage? Explain your reasoning using the equations in this section.

Battery testers, such as those shown in Figure 28.6, intentionally connect a load to the battery while measuring its terminal voltage. This test reveals whether the battery can maintain its voltage while supplying current. A battery with a large internal resistance may appear normal when no load is connected but experience a substantial voltage drop during operation.

Recharging Batteries

Rechargeable batteries work by reversing the chemical reactions that occur during discharge. To recharge a battery, an external charger applies a voltage greater than the battery's emf, forcing current to flow in the opposite direction. This restores the chemical energy stored within the battery.

Because the charging current flows opposite to the discharge current, [latex]I[/latex] is negative according to the sign convention used in

As a result, the terminal voltage during charging is greater than the battery's emf.

Multiple Voltage Sources

Many devices use more than one voltage source. Flashlights, toys, medical instruments, laptops, and vehicles often contain several cells or batteries connected together to obtain the required voltage, current, or operating time.

When voltage sources are connected in series, their emfs add algebraically and their internal resistances also add. Series connections are used when a larger total voltage is needed. For example, two 1.5-V cells connected in series provide approximately 3.0 V.

If one cell is inserted backward, its emf opposes the others and reduces the total emf. This is why a device may fail to operate if one battery is placed in the wrong direction.

If two voltage sources are connected in series with their emfs opposing one another, current flows in the direction produced by the larger emf. When no external load is present, the current is limited by the total internal resistance:

[latex]I=\frac{\text{emf}_1-\text{emf}_2}{r_1+r_2}.[/latex]

This is the basic principle behind charging a battery: the charger must have a larger emf than the battery so that current is forced through the battery in the reverse direction.

If two voltage sources are connected in series with their emfs in the same direction and connected to a load, the current is

[latex]I=\frac{\text{emf}_1+\text{emf}_2}{r_1+r_2+R_{\text{load}}}.[/latex]

Take-Home Experiment: Flashlight Batteries

Find a flashlight that uses several batteries. Predict what will happen if you use only new batteries, only old batteries, or a mixture of new and old batteries. Then test your prediction. How does the brightness change? What does this suggest about internal resistance and battery age?

Animals as Electrical Detectors

Electricity is not only a feature of human technology—it also plays an important role in the biology of many animals. Some species can detect weak electric fields generated by other organisms, while others can produce powerful electrical discharges for hunting or defense. These remarkable adaptations illustrate many of the same physical principles discussed throughout this chapter.
Several aquatic animals, including sharks, rays, electric fish, and some species of bony fish, can sense the tiny electric fields produced by muscle contractions and nerve impulses in nearby animals. Two unusual mammals—the platypus and the echidna (spiny anteater)—also possess electroreceptors that help them locate prey hidden underwater or underground.
Electric eels generate their own voltage using specialized cells called electroplaques (or electrocytes). Each electroplaque produces only a small potential difference of about 0.15 V, but thousands of these cells are connected in series and parallel, much like the cells in a battery. Working together, they can generate voltages approaching 600 V and currents of about 1 A—more than enough to stun prey or deter predators.
The detection of external electric fields relies on the same basic mechanism responsible for nerve conduction. Small electric fields cause ions to move across specialized sensory cell membranes, producing changes in membrane potential that trigger nerve impulses. Sharks, for example, use highly sensitive electroreceptors called the ampullae of Lorenzini to detect the weak bioelectric fields produced by hidden prey. The platypus is similarly capable of detecting extremely small electric fields while foraging underwater.
This remarkable biological sensitivity highlights the importance of electric fields in living systems. The same physical principles that describe batteries, circuits, and electric potential also explain how animals communicate, navigate, hunt, and respond to their environment.

 

Biology Connection

The electric eel demonstrates an impressive application of series and parallel connections in nature. Although each electroplaque produces only about 0.15 V, connecting thousands of these cells in series increases the total voltage, while parallel arrangements allow the eel to deliver a much larger current. The same principles are used when connecting batteries in electrical devices.

Solar Cell Arrays

Solar panels provide another excellent example of multiple voltage sources connected in series and parallel. They convert sunlight directly into electrical energy through the photovoltaic (PV) effect, in which light striking a semiconductor material generates an electric current.

Most solar cells are made from silicon. An individual silicon cell typically produces a voltage of about 0.5 V. The amount of current produced depends on the intensity of the incoming sunlight: brighter sunlight generates more current, while cloudy conditions reduce the output.

Because a single cell produces only a small voltage and limited current, many cells are connected together to form a solar module (often called a solar panel). Just as with batteries, cells can be connected in series to increase the output voltage or in parallel to increase the available current. Modern residential solar panels typically contain dozens of cells and can produce several hundred watts of electrical power under full sunlight.

The electricity produced by solar cells is direct current (DC). Since homes, hospitals, and the electrical grid operate using alternating current (AC), an electronic device called an inverter converts the DC output into AC. Depending on the installation, the generated electricity may be used immediately, stored in rechargeable batteries, or supplied to the electrical grid.

Health and Technology Connection

Solar-powered electrical systems are increasingly used in healthcare. Portable vaccine refrigerators, emergency communication systems, field hospitals, and medical clinics in remote regions often rely on solar panels and battery storage to provide reliable electricity where access to the electrical grid is limited.

Guided Exploration

Imagine each solar cell produces 0.5 V and can deliver up to 2.0 A in bright sunlight.

1. How many cells connected in series are needed to produce 3.0 V?
2. Once you have a 3.0-V string, how many identical strings connected in parallel are needed to produce a total current of 6.0 A?
3. What is the total electrical power delivered by your array?
4. If the sunlight becomes weaker, which quantity changes first—the voltage, the current, or both? Explain your reasoning.

As you answer these questions, notice that connecting cells in series increases voltage, while connecting cells in parallel increases the available current. The same principles apply to batteries and many other electrical power systems.

Section Summary

• Every voltage source consists of two essential components: an electromotive force (emf), which provides electrical energy, and an internal resistance, r, which limits the current the source can deliver.
• The emf of a source is its potential difference when no current is flowing.
• When current flows, the measured output is the terminal voltage, given by
  [latex]V=\text{emf}-Ir[/latex]

  where I is the current supplied by the source.

• As the current or the internal resistance increases, the terminal voltage decreases.
• When voltage sources are connected in series, their emfs add algebraically and their internal resistances also add.
• Solar cells and batteries can be connected in series to increase voltage or in parallel to increase the available current.

Conceptual Questions

13. Every emf is a potential difference, but is every potential difference an emf? Explain the difference in your own words.
14. Study Figure 28.14. Which battery is charging the other? Explain your reasoning using the relative values of their emfs.

15. Suppose you have a battery, several resistors, a voltmeter, and an ammeter. Describe an experimental procedure for determining the battery's internal resistance.
16. Two 12-V automobile batteries are rated at 600 and 850 cold-cranking amps (CCA). Which battery likely has the smaller internal resistance? Explain your reasoning.
17. What are the advantages and disadvantages of connecting batteries in series? What are the advantages and disadvantages of connecting identical batteries in parallel?
18. A heavy-duty truck uses four 12-V batteries. The starter motor requires 24 V, while most of the vehicle's electrical systems operate at 12 V. Suggest how the batteries could be connected to provide each voltage. Why is a 24-V system advantageous for starting a large engine?

Problems & Exercises

14. Standard automobile batteries have six lead-acid cells in series, creating a total emf of 12.0 V. What is the emf of an individual lead-acid cell?
15. Carbon-zinc dry cells (sometimes referred to as non-alkaline cells) have an emf of 1.54 V, and they are produced as single cells or in various combinations to form other voltages.
    1. How many 1.54-V cells are needed to make the common 9-V battery used in many small electronic devices?
    2. What is the actual emf of the approximately 9-V battery?
    3. Discuss how internal resistance in the series connection of cells will affect the terminal voltage of this approximately 9-V battery.
16. What is the output voltage of a 3.0000-V lithium cell in a digital wristwatch that draws 0.300 mA, if the cell's internal resistance is [latex]2.00~\Omega[/latex]?
17. A large 1.54-V carbon-zinc dry cell used in a physics lab supplies 2.00 A to a circuit and has an internal resistance of [latex]0.100~\Omega[/latex].
    1. What is its terminal voltage?
    2. How much electrical power does the cell produce?
    3. What power is delivered to the load?
18. What is the internal resistance of an automobile battery that has an emf of 12.0 V and a terminal voltage of 15.0 V while a current of 8.00 A is charging it?
19. A 12.0-V motorcycle battery has an internal resistance of [latex]0.600~\Omega[/latex] and is being charged with a current of 10.0 A.
    1. Find the terminal voltage of the battery.
    2. What is the output voltage of the battery charger?
20. A car battery with a 12-V emf and an internal resistance of [latex]0.050~\Omega[/latex] is being charged with a current of 60 A.
    1. What is the potential difference across its terminals?
    2. At what rate is thermal energy dissipated in the battery?
    3. At what rate is electrical energy converted into chemical energy?
    4. Repeat parts (a) and (b) when the battery supplies 60 A to the starter motor.
21. The hot resistance of a flashlight bulb is [latex]2.30~\Omega[/latex], and it is powered by a 1.58-V alkaline cell with an internal resistance of [latex]0.100~\Omega[/latex].
    1. What current flows?
    2. Calculate the power supplied to the bulb using [latex]P=I^2R[/latex].
    3. Is this the same power obtained from [latex]P=\frac{V^2}{R}[/latex]?
22. The label on a portable radio recommends rechargeable nickel-cadmium cells (1.25 V each), although alkaline cells provide 1.58 V. The radio has a resistance of [latex]3.20~\Omega[/latex].
    1. Draw a circuit diagram of the radio and its batteries.
    2. Calculate the power delivered when using NiCd cells with an internal resistance of [latex]0.0400~\Omega[/latex] each.
    3. Repeat using alkaline cells with an internal resistance of [latex]0.200~\Omega[/latex] each.
    4. Does the difference become more significant when the radio volume is increased? Explain.
23. An automobile starter motor has an equivalent resistance of [latex]0.0500~\Omega[/latex] and is powered by a 12.0-V battery with an internal resistance of [latex]0.0100~\Omega[/latex].
    1. Find the current supplied to the motor.
    2. Find the voltage across the motor.
    3. Find the power delivered to the motor.
    4. Repeat the calculations if corrosion adds [latex]0.0900~\Omega[/latex] to the circuit.
24. A child's electronic toy is powered by three 1.58-V alkaline cells (each with an internal resistance of [latex]0.0200~\Omega[/latex]) in series with a 1.53-V carbon-zinc cell having an internal resistance of [latex]0.100~\Omega[/latex]. The load resistance is [latex]10.0~\Omega[/latex].
    1. Draw the circuit diagram.
    2. What current flows?
    3. How much power is supplied to the load?
    4. If the dry cell deteriorates so that only 0.500 W is delivered to the load, what is its internal resistance?
25. A voltage source experiences a 2.00-V drop in terminal voltage when the current increases by 5.00 A.
    1. What is its internal resistance?
    2. Can its emf be determined from this information alone?
26. A person whose body resistance between the hands is [latex]10.0~\text{k}\Omega[/latex] accidentally grasps the terminals of a 20.0-kV power supply.
    1. Draw a circuit diagram.
    2. If the power supply has an internal resistance of [latex]2000~\Omega[/latex], what current flows through the person's body?
    3. How much power is dissipated in the body?
    4. How large must the internal resistance be to limit the current to 1.00 mA?
    5. Would increasing the internal resistance reduce the effectiveness of the power supply for low-resistance loads? Explain.
27. Electric fish generate current using biological cells called electroplaques. In the South American electric eel, the electroplaques are arranged in 140 rows, each containing 5000 electroplaques. Each electroplaque has an emf of 0.15 V and an internal resistance of [latex]0.25~\Omega[/latex]. If the surrounding water has a resistance of [latex]800~\Omega[/latex], what current can the eel produce between its head and tail?
28. Integrated Concepts
    1. A 12.0-V automobile battery has a terminal voltage of 16.0 V while being charged by a current of 10.0 A. What is the battery's internal resistance?
    2. What power is dissipated inside the battery?
    3. Assuming no heat escapes, at what rate (in [latex]^\circ\text{C}/\text{min}[/latex]) will the battery's temperature increase if its mass is 20.0 kg and its specific heat capacity is [latex]0.300~\text{kcal}/(\text{kg}\cdot^\circ\text{C})[/latex]?
29. Unreasonable Results
    1. A 1.58-V alkaline cell with an internal resistance of [latex]0.200~\Omega[/latex] supplies 8.50 A to a load. What is its terminal voltage?
    2. What is the load resistance?
    3. What is unreasonable about the result?
    4. Which assumptions are unreasonable or inconsistent?
30. Unreasonable Results
    1. A 1.54-V dry cell supplies 1.00 W to a [latex]15.0~\Omega[/latex] bulb. What is the cell's internal resistance?
    2. What is unreasonable about the result?
    3. Which assumptions are unreasonable or inconsistent?

Glossary

electromotive force (emf)

the maximum potential difference provided by a voltage source when no current is flowing; measured in volts

internal resistance

the effective resistance inside a voltage source that reduces the terminal voltage when current flows

potential difference

the difference in electric potential energy per unit charge between two points; measured in volts

terminal voltage

the actual voltage measured across the terminals of a voltage source while it is supplying or receiving current