understand the effects of the internal resistance of a source of e.m.f. on the terminal potential difference

Practical Circuits – Effect of Internal Resistance on Terminal Potential Difference

Learning Objective

To understand how the internal resistance (\$r\$) of a source of e.m.f. influences the terminal potential difference (\$V\$) when a load resistance (\$R\$) is connected, and to be able to determine \$r\$ experimentally.

Key Concepts

• Source of e.m.f. (\$\mathcal{E}\$): The ideal voltage a cell would produce if it had zero internal resistance.
• Internal resistance (\$r\$): The resistance inherent to the cell or battery, causing a voltage drop when current flows.
• Terminal potential difference (\$V\$): The voltage measured across the external terminals of the source while it supplies current.
• Load resistance (\$R\$): The external resistor (or combination of resistors) connected to the source.

Theoretical Background

When a current \$I\$ flows through a source with internal resistance \$r\$, the terminal potential difference is given by

\$V = \mathcal{E} - I r\$

Ohm’s law for the external circuit gives

\$I = \frac{V}{R}\$

Combining the two equations eliminates \$I\$ and yields a linear relationship between \$V\$ and \$I\$:

\$V = \mathcal{E} - r I \quad\text{or}\quad V = \mathcal{E} - r\frac{V}{R}\$

Re‑arranging the first form shows that a plot of \$V\$ (y‑axis) against \$I\$ (x‑axis) is a straight line with:

• Slope = \$-r\$
• Y‑intercept = \$\mathcal{E}\$

Experimental Determination of Internal Resistance

1. Set up the circuit shown in the figure below.
2. Use a variable resistor (or a set of known resistors) as the load \$R\$.
3. For each value of \$R\$, measure the terminal voltage \$V\$ with a voltmeter and the current \$I\$ with an ammeter.
4. Record the data in a table.
5. Plot \$V\$ against \$I\$ and determine the gradient (slope) and intercept.
6. Calculate \$r\$ from the slope and \$\mathcal{E}\$ from the intercept.

Sample Data Table

Data Analysis Example

Using the four data points above, a linear fit gives:

\$\text{slope} = -0.5\ \Omega \quad\Rightarrow\quad r = 0.5\ \Omega\$

\$\text{intercept} = 5.2\ \text{V} \quad\Rightarrow\quad \mathcal{E} = 5.2\ \text{V}\$

Common Sources of Error

• Contact resistance at the terminals adds to the measured \$r\$.
• Voltmeters have finite internal resistance; if not much larger than \$R\$, the measured \$V\$ is reduced.
• Ammeters have internal resistance that can affect the current reading.
• Temperature changes alter \$r\$ during the experiment.

Key Points to Remember

1. The terminal potential difference falls as the current increases because of the internal voltage drop \$Ir\$.
2. A plot of \$V\$ versus \$I\$ provides a straightforward method to obtain both \$\mathcal{E}\$ and \$r\$.
3. Minimising measurement errors requires using instruments with high internal resistance (voltmeter) and low internal resistance (ammeter).
4. Internal resistance is a characteristic of the source and varies with its state of charge and temperature.

Practice Questions

1. A cell has an e.m.f. of \$12.0\ \text{V}\$ and an internal resistance of \$0.8\ \Omega\$. Calculate the terminal voltage when it supplies a current of \$3\ \text{A}\$.
2. In an experiment, the following data were obtained:

   • \$I = 0.25\ \text{A}\$, \$V = 9.8\ \text{V}\$
   • \$I = 0.40\ \text{A}\$, \$V = 9.5\ \text{V}\$
   • \$I = 0.55\ \text{A}\$, \$V = 9.2\ \text{V}\$

   Determine the internal resistance and e.m.f. of the cell from a linear fit.

3. Explain why a voltmeter must have a much larger resistance than the load resistance in this type of experiment.

Further Investigation

Explore how the internal resistance changes with:

• State of charge of a rechargeable battery.
• Temperature (heat the cell gently and repeat the measurements).
• Age of the cell (compare a new cell with an old one).