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Lesson Explainer: Electromotive Force and Internal Resistance Physics

In this explainer, we will learn how to relate the electromotive force (emf) of a battery to its terminal voltage and its internal resistance.

Batteries are usually thought of as supplying a potential difference to other components of a circuit in order to produce a current in those components. This is correct. It is also true, however, that a battery produces a potential difference across itself in order to produce a current through itself.

Consider a battery that produces a potential difference 𝑉 across its terminals. When a conducting wire connects the positive and negative terminals of the battery, a circuit is closed. A direct current 𝐼 is produced in the circuit. The current in the wire is given by 𝐼=𝑉𝑅, where 𝑅 is the resistance of the circuit.

The direction of the current is from the positive terminal to the negative terminal. In a series circuit, the current at all points in the circuit is equal. This means that there must be equal currents out from the positive terminal and into the negative terminal. This is shown in the following figure.

From this we see that there must also be a current in the battery equal to the current at its terminals. This is shown in the following figure.

We have seen that 𝐼=𝑉𝑅.

For two resistances in series 𝑅 and 𝑅, their combined resistance 𝑅total is given by 𝑅=𝑅+𝑅.total

We see then that 𝑅total for the circuit consisting of a wire and a battery must be the sum of the resistance of the wire and of the battery. We can call the resistance of the wire 𝑅 and the resistance of the battery 𝑟.

The equation 𝐼=𝑉𝑅,total can be rearranged to have 𝑉 as its subject, which gives us 𝑉=𝐼𝑅.total

The current in the circuit can therefore be expressed as 𝑉=𝐼(𝑅+𝑟).

𝑅 is called the external resistance (and is also called the load) while 𝑟 is called the internal resistance.

A potential difference 𝑉 can be expressed as 𝑉=𝑊𝑄, where 𝑊 is the work done by the potential difference on a charge 𝑄 across the potential difference.

The difference in potential across the ends of a wire is the decrease in potential across the wire. This is shown in the following figure.

The potential difference produced by a battery across a wire equals the work done per coulomb of charge on charges that move through the wire from one battery terminal to the other. The potential decreases along the length of the wire.

As well as when moving through the wire, work must be done to move charges through the battery. When this happens, the potential energy of the charges increases rather than decreases. The potential must then increase along the length of the battery. This is shown in the following figure.

For many purposes, a circuit containing a battery is modeled as having purely external resistance. The potential difference across such an external circuit can be measured using a voltmeter connected in parallel to the resistance of the circuit as shown in the following figure.

It is important to note that the wires connecting the battery, resistor, and voltmeter are modeled as having negligible resistance in this diagram.

It might be expected that a voltmeter could also measure the potential difference across a battery by connecting the voltmeter to the battery, as shown in the following figure.

This circuit would not, however, measure the potential difference across the battery. Both voltmeters in the circuit would measure the same value, which is the potential difference across the external circuit.

If we want to measure the potential difference across the battery terminals for charges moving inside the battery, a voltmeter would have to measure the work done on charges moving through the battery rather than through the external circuit.

We see then that a voltmeter in a circuit cannot measure the potential difference across the battery. This appears to show that there is no way to know the value of the internal resistance of a battery or the potential difference across it. In fact, it is possible to determine these values by using multiple measurements.

Considering a battery as a component of a circuit that has an external resistance 𝑅, we see that there must be a decrease of potential, 𝑉battery, across the battery. This is given by 𝑉=𝐼𝑟,battery where 𝐼 is the current in the circuit.

Consider now the equation 𝑉=𝐼(𝑅+𝑟), where 𝑉 is the potential difference across the external resistance, which can be measured using a voltmeter.

This can be written as 𝑉=𝐼𝑅+𝐼𝑟𝑉=𝐼𝑅+𝑉.battery

To use a voltmeter to measure the full potential difference that the battery can produce, 𝑉battery must be zero.

𝑉battery must be zero if the value of 𝐼 is zero. This would give us the equation 𝑉=𝐼𝑅+0𝑉=𝐼𝑅.

This value of 𝑉 corresponds to all the potential difference of the battery doing work on the external circuit.

Unfortunately, if the value of 𝐼 is zero, then the equation 𝑉=𝐼𝑅 must have the values 𝑉=0×𝑅.

This seems to tell us that the only way that a voltmeter can measure the full potential difference of a battery is if that potential difference is zero. This seems inevitable, as a battery with a nonzero potential difference would produce a nonzero current and so a nonzero value of 𝑉battery.

This conclusion is incorrect, however. The reason why the conclusion is incorrect is deduced later in this explainer. However, understanding why the conclusion is incorrect first requires us again to consider the equation 𝑉=𝐼𝑅+𝐼𝑟.

We have seen that the reading shown by a voltmeter equals 𝐼𝑅. As we know that the potential difference across the external circuit plus the potential difference across the battery sum to give a total potential difference, we can make the following equation: 𝑉=𝑉+𝑉.totalvoltmeterbattery

There are specific names for the quantities in this equation. 𝑉battery is called the lost volts, 𝑉voltmeter is called the terminal voltage, and 𝑉total is called the electromotive force or emf.

All these quantities have the unit of volts. emf is denoted by the symbol 𝜀. emf is, despite its name, not a force but a potential difference.

Formula: Electromotive Force of a Battery

The electromotive force 𝜀 of a battery that has a terminal voltage 𝑉 is given by 𝜀=𝑉+𝐼𝑟, where 𝐼 is the current in the battery and 𝑟 is the internal resistance of the battery.

Let us now look at an example in which the emf of a battery is determined.

Example 1: Determining the emf of a Battery

A circuit is powered by a battery with a terminal voltage of 2.5 V. The circuit has a resistance of 3.5 Ω and the battery has an internal resistance of 0.65 Ω. What is the electromotive force of the battery? Give your answer to one decimal place.

Answer

The emf 𝜀 of a battery is given by the equation 𝜀=𝑉+𝐼𝑟, where 𝑉 is the terminal voltage of the battery, 𝑟 is the internal resistance of the battery, and 𝐼 is the current in the circuit.

Substituting the known values into this equation, we see that 𝜀=2.5+0.65𝐼.

We see that as 𝐼 is not given, we are not yet able to determine 𝜀.

We recall, though, that the terminal voltage is given by 𝑉=𝐼𝑅.

This equation can be rearranged to make 𝐼 the subject, giving 𝐼=𝑉𝑅.

Substituting the known values into this equation, we see that 𝐼=2.53.5=2.53.5.VΩA

The value of 𝐼 can be substituted into 𝜀=2.5+0.65𝐼 to give 𝜀=2.5+0.65×2.53.5.

Rounding the value of 𝜀 to one decimal place, we have 𝜀=3.0V.

Let us now look at an example in which the internal resistance of a battery is determined.

Example 2: Determining the Internal Resistance of a Battery

A battery with an electromotive force of 4.50 V is connected to a circuit with a resistance of 2.75 Ω. The current in the circuit is 1.36 A. What is the internal resistance of the battery? Give your answer to two decimal places.

Answer

The emf 𝜀 of a battery is given by the equation 𝜀=𝑉+𝐼𝑟, where 𝑉 is the terminal voltage of the battery, 𝑟 is the internal resistance of the battery, and 𝐼 is the current in the circuit.

This equation can be rearranged as follows to make 𝑟 the subject. 𝜀=𝑉+𝐼𝑟𝜀𝑉=𝐼𝑟𝜀𝑉𝐼=𝑟𝑟=𝜀𝑉𝐼.

Substituting the known values into this equation, we see that 𝑟=4.50𝑉1.36.

We see that as 𝑉 is not given, we are not yet able to determine 𝑟.

We recall, though, that the terminal voltage is given by 𝑉=𝐼𝑅.

Substituting the known values into this equation, we see that 𝑉=1.36×2.75=3.74.AΩV

Substituting this value of 𝑉 into 𝑟=4.503.741.36, then rounding the value of 𝑟 to two decimal places, we have 𝑟=0.56Ω.

A circuit like the one in the following figure can be used to determine the emf and internal resistance of a battery.

The voltmeter in the circuit measures the terminal voltage of the battery. The variable resistor in the circuit allows the resistance of the circuit to be changed. Changing the resistance of the circuit changes the current in the circuit. The value of terminal voltage for different values of current can therefore be measured.

The measured values can be plotted on a graph.

As the value of 𝐼 decreases, the value of 𝑉 increases. The value of 𝐼 for 𝑉=0 cannot be found from a voltmeter reading but can be estimated using voltmeter readings where 𝐼>0. This is shown in the following figure.

This graph is the graph of a straight line that has a 𝑦-axis intercept of 𝜀. The graph has a negative slope.

The graph of a straight line can be written as 𝑦=𝑚𝑥+𝑐, where 𝑚 is the slope of the graph and 𝑐 is the 𝑦-axis intercept.

The graph used to estimate 𝜀 has values of 𝐼 on its 𝑥-axis and values of 𝑉 on its 𝑦-axis. This shows us that the equation of the line of this graph is 𝑉=𝑚𝐼+𝜀𝑉=𝜀+𝑚𝐼𝑉=𝜀+𝐼𝑚.

We can rearrange the equation 𝜀=𝑉+𝐼𝑟 as 𝜀𝐼𝑟=𝑉𝑉=𝜀𝐼𝑟.

Comparing 𝑉=𝜀𝐼𝑟 to the equation for the line of the graph used to estimate 𝜀, 𝑉=𝜀+𝐼𝑚, we see that 𝑚=𝑟.

The graph can therefore be used to determine 𝑟 as well as 𝜀. Both the emf and internal resistance of a battery can be determined.

Let us now look at an example in which the emf and internal resistance of a battery are determined from measurements.

Example 3: Determining the emf and Internal Resistance of a Battery Using Multiple Measurements

The graph shows the change in the current in a circuit with the terminal voltage of the battery that produces the current.

  1. What is the electromotive force of the battery?
  2. What is the internal resistance of the battery?

Answer

Part 1

The emf 𝜀 of a battery is given by the equation 𝜀=𝑉+𝐼𝑟, where 𝑉 is the terminal voltage of the battery, 𝑟 is the internal resistance of the battery, and 𝐼 is the current in the circuit.

The value of the emf 𝜀 of the battery is equal to the 𝑦-axis intercept of the line of best fit for the points plotted on the graph. This is shown in the following figure.

The emf of the battery is 6 V.

Part 2

The internal resistance 𝑟 of the battery is determined using the equation 𝜀=𝑉+𝐼𝑟.

This equation can be rearranged as 𝑉=𝜀𝐼𝑟 and expressed as 𝑉=𝜀+(𝑟)𝐼.

This equation can be compared to the equation for the line of best fit 𝑦=𝑐+𝑚𝑥, where 𝑦 is the terminal voltage, 𝑥 is the current, 𝑐 is the emf, and 𝑚 is the slope of the line.

We see then that 𝑚=𝑟, and so 𝑟=𝑚.

The slope of the line of best fit is given by 𝑚=Δ𝑦Δ𝑥=Δ𝑉Δ𝐼.

We can take two very clear values of 𝑉 and 𝐼 from the graph: 𝑉=5.90V and 𝐼=0.80A, and 𝑉=5.85V and 𝐼=1.20A.

This gives us a value of Δ𝑉 as follows: Δ𝑉=5.905.85=0.05.VVV

This gives us a value of Δ𝐼 as follows: Δ𝐼=0.801.20=0.40.AAA

This gives a value of 𝑚 as follows: 𝑚=0.050.40=0.125.VAΩ

We know that 𝑟=𝑚, and so, 𝑟 equals 0.125 Ω.

Let us now summarize what has been learned in this explainer.

Key Points

  • A battery has a resistance called its internal resistance.
  • Some of the potential difference that a battery produces does work to move charges through the battery. This potential difference is not available to move charges through a circuit that the battery is connected to.
  • The full potential difference that a battery produces is called the emf of the battery.
  • The potential difference that a battery provides to a circuit connected to the battery is called the terminal voltage of the battery.
  • The emf 𝜀, terminal voltage 𝑉, and internal resistance 𝑟 of a battery connected to a circuit carrying a current 𝐼 are related by the equation 𝜀=𝑉+𝐼𝑟.
  • The emf and internal resistance of a battery cannot be directly measured but can be indirectly estimated.

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