As a battery depletes, which of the following happens?

Its internal resistance increases.

Emf and Internal Resistance: Measurements & Relation Between

Emf and Internal Resistance

The electromotive force, known as emf, is the terminal potential difference of a source when there is no current flow. Internal resistance is the resistance to current flow inside the source itself. But, importantly, how do we calculate these values? Let’s find out.

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What is Emf in electric circuits?

All voltage sources create a potential difference, providing current when connected to a circuit with resistance. This potential difference produces an electric field that acts on charges as a force, causing current to flow.

Despite its name, emf is not exactly a force. In fact, it is a unique kind of potential difference and is measured in volts (V).

Emf is the potential difference of the source when there is no current flowing through it.

We can also define emf as the work W done per unit charge Q, which gives us the following equation:

\[\varepsilon = \frac{dW}{dq}\]

Think of a battery.

If the battery is supplying current, the voltage across the battery’s terminals is less than the emf. As the battery depletes, this voltage level begins to decrease.
When the battery is fully depleted and hence not supplying current, the voltage across the battery’s terminals will equal the emf.

How do we calculate emf?

We can also calculate emf (ε) with the equation below:

\[\varepsilon = \frac{E}{Q}\]

E stands for electrical energy in joules (J), and Q is the charge in coulombs (C).

In this equation, the potential difference is called the terminal potential difference. It will be equal to the emf if there is no internal resistance. However, this is not the case with real power supplies because there is always an internal resistance. Lost volts refer to the energy spent per coulomb while overcoming the internal resistance.

We know that the conservation of energy is apparent in electric circuits, and it is valid for the cases where there is internal resistance as well.

Emf and internal resistance Conservation of energy in internal resistance StudySmarter

The equation for conservation of energy with internal resistance

Lost volts is the name given to the energy spent per coulomb while overcoming the internal resistance. Also, be sure to check out our explanation on Energy Conservation.

EMF and Internal Resistance

As we've seen batteries or cells are sources of EMF, however they also have their own resistance. This resistance is known as internal resistance. We can think of real batteries or cells as being composed of an ideal EMF source connected to a resistor in series. This resistor accounts for the source's internal resistance. We already know that the load resistance (also known as external resistance) is the total resistance of the components in an external electric circuit. On the other hand, internal resistance is the resistance within the power source that resists current flow. It usually causes the power source to generate heat.

Load resistance = the total resistance of the components in an external electric circuit.
Internal resistance = the resistance within the power source that resists current flow.

Measuring the internal resistance

Ohm’s law

From Ohm’s law, we know that

\[V = I \cdot R\]

where V is the voltage in volts, I is the current in amperes, and R is the external resistance in ohms.

Internal resistance

If we include the internal resistance, the total resistance will be R+r where internal resistance is shown by r, and the voltage can be expressed as emf (ε).

\[\varepsilon = I \cdot (R + r)\]

If you expand the brackets, you will get

\[\varepsilon = I \cdot R + I \cdot r\]

where I⋅R is the terminal potential difference in volts, and I⋅r is the lost volts (also measured in volts).

Now we can rearrange the equation as

\[\varepsilon = V_R + V_r\]

where V_R is the terminal potential difference and V_r is the lost volts.

Relationship between terminal potential difference and lost volts

Here is the relationship between terminal potential difference and lost volts. You can see from the equation that if there is no internal resistance (so no lost volts), the terminal resistance will be equal to the emf.

\[V_R = \varepsilon - V_r\]

Emf and Internal Resistance Internal and load resistance diagram StudySmarter

A circuit diagram that shows the internal and load resistances

Internal resistance (r) has complex behaviour. Let’s look at our battery example again. As the battery depletes, its internal resistance rises. But what else affects the internal resistance? Here are some factors:

The size of the voltage source.
How much and how long it has been used for.
The magnitude and direction of the current through the voltage source.

Measuring the EMF and Internal Resistance of a Battery

Calculating the internal resistance of a source is an important factor in achieving optimum efficiency and getting the source to provide maximum power to the electric circuit. Here are some examples of calculating different quantities with internal resistance.

Remember that R is for load resistance and r is for internal resistance.

A battery has an emf of 0.28V and an internal resistance of 0.65Ω. Calculate the terminal potential difference when the current flowing through the battery is 7.8mA.

Solution

Emf (ε), internal resistance (r), and the current (I) flowing through the battery are given in the question. Let’s put these into the terminal potential difference (V_R) equation.

\[V_R = \varepsilon - V_r = 0.28V - (0.65 \Omega \cdot 7.8 \cdot 10^{-3} A)\]

\[V_R = 0.275 V\]

A cell has 0.45A flowing through it with an internal resistance of 0.25Ω. Find the energy wasted per second on the internal resistance in joules.

Solution

We know that

\[P = I^2 \cdot R\]

where P is the power in watts, I is the current in amperes, and R is the resistance in ohms.

Since the question asks for the energy wasted per second, we use the power equation because power is energy per second. We can also put the internal resistance r for resistance in the equation.

\[P = I^2 \cdot r\]

\[P = 0.45^2 A \cdot 0.25 \Omega = 0.05 W\]

A battery has an emf of 0.35V. The current flowing through the battery is 0.03A, and the load resistance is 1.2Ω. Find the internal resistance of the battery.

Solution

The emf value (ε) of the battery, the current (I) flowing through the battery, and the load resistance (R) are all given in the question. This is the right equation to use to find the internal resistance (r):

\[\varepsilon = I \cdot R + I \cdot r\]

Let’s put the given variables into the equation:

\[0.35V = 0.03 A \cdot 1.2 \Omega + 0.03 A \cdot r\]

If we solve the equation for r, we will get \(r = 10.47 \Omega\)

Emf and Internal Resistance - Key takeaways

Electromotive force is not exactly a force: it is a unique kind of potential difference and is measured in volts.
If there is no current, the voltage across the terminals of the voltage source will be equal to the emf.
Lost volts is the given name for the energy spent per coulomb while overcoming the internal resistance.
Internal resistance is the resistance within the power source that resists current flow and generally causes the power source to generate heat.
The internal resistance of a voltage source depends on a variety of conditions, including how much it has been used, the size of the voltage source, the magnitude, and the direction of the current flowing through the voltage source.

Frequently Asked Questions about Emf and Internal Resistance

How do determine the emf and internal resistance of an electrical cell?

By using the following equation, you can determine the emf and internal resistance of an electrical cell. The equation that describes the relation between emf, terminal voltage, and internal resistance is ε = V_R+ V_r, where ε is emf in volts, V_R is the terminal voltage in volts, I is current in amperes, and r is the internal resistance in ohms.

How do you calculate efficiency with emf and internal resistance?

Calculating the internal resistance of a source is an important factor in achieving optimum efficiency and getting the source to provide maximum power to the electric circuit. By using the following equation, you can calculate efficiency with emf and internal resistance. The equation that describes the relation between emf, terminal voltage, and internal resistance is ε = V_R + V_r, where ε is emf in volts, V_R is the terminal voltage in volts, I is current in amperes, and r is the internal resistance in ohms.

How do you draw the emf and internal resistance graph gradient?

If you draw a graph that has the terminal potential difference on the y-axis and the circuit's current on the x-axis, you will obtain a straight line that has a negative gradient. The emf is then the intercept on the y-axis and the gradient represents r, the internal resistance.

What are the emf and internal resistance of a battery?

Emf is the potential difference of the source when there is no current flowing through it, and internal resistance is the resistance within the power source that resists current flow.

Why is measuring the emf and internal resistance of a source important?

It is important to know the emf and internal resistance values of a source in order to determine how to get the source to provide maximum power to an electric circuit.

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Emf and Internal Resistance

StudySmarter Editorial Team