Define potential difference (p.d.) as the work done by a unit charge passing through a component

4.2.3 Electromotive Force (EMF) and Potential Difference (p.d.)

Learning Objective

Students will be able to define potential difference (p.d.) as the work done by a unit charge **passing through a component**, to distinguish it from electromotive force (EMF), and to apply the concepts in calculations, measurements and simple experiments.

Key Definitions (Cambridge Syllabus wording)

• Potential Difference (p.d.): “The work done by an external agent in moving a unit positive charge through a component from one terminal to the other.”
  Unit: volt (V), where 1 V = 1 J C⁻¹.
• Electromotive Force (EMF): “The work done by a source in moving a unit charge around a complete circuit when no current flows.”
  Unit: volt (V). This is the ideal (open‑circuit) voltage of the source.

Relationship between Work, Charge and Voltage

The fundamental equation linking work, charge and voltage is

\[ V = \frac{W}{q} \]

where

• \(V\) – potential difference (volts, V)
• \(W\) – work done on the charge (joules, J)
• \(q\) – magnitude of the charge moved (coulombs, C)

For a source the electromotive force is

\[ \mathcal{E} = \frac{W_{\text{source}}}{q} \]

Terminal p.d. of a real source

A real battery has an internal resistance \(r\). When a current \(I\) flows, the terminal p.d. is reduced by the internal voltage drop:

\[ V_{\text{terminal}} = \mathcal{E} - I r \]

This equation is a core requirement of the syllabus and can be derived by applying the definition of p.d. to the two series elements (internal resistance \(r\) and external load). See the labelled diagram below.

Diagram – Battery with internal resistance

Voltmeters – Measuring Potential Difference

Both analogue and digital voltmeters are designed to measure the p.d. across a component while drawing as little current as possible.

• Analogue voltmeter
  • Uses a moving‑coil meter movement.
  • The coil itself has a relatively low resistance (a few kΩ). To obtain a usable measurement range a **shunt resistor** is placed in parallel, thereby **increasing** the overall input resistance of the meter.
  • Scale is continuous; the pointer indicates the reading.
• Digital voltmeter (DVM)
  • Displays the reading numerically on an LCD.
  • Very high input resistance (≥10 MΩ), so the current drawn from the circuit is negligible – this prevents “loading” the circuit.
  • Selectable ranges (e.g. 0‑2 V, 0‑20 V, 0‑200 V). Always start on the highest range, then reduce to the smallest range that comfortably exceeds the expected p.d. for best resolution.

When measuring p.d. the voltmeter must be connected **in parallel** with the component of interest.

V‑I Characteristic of a Real Battery (optional but useful)

Practical Determination of EMF or p.d.

Below is a concise experiment that can be performed in a school lab to find the EMF of a cell and its internal resistance.

1. Set up a circuit with the cell, a variable resistor (or a set of known resistors), an ammeter (in series) and a voltmeter (in parallel with the resistor).
2. Measure the terminal p.d. \(V\) and the current \(I\) for at least three different values of the external resistance.
3. Plot \(V\) (vertical axis) against \(I\) (horizontal axis). The straight line will have:
   • Y‑intercept = EMF \(\mathcal{E}\)
   • Negative slope = internal resistance \(r\)
4. Read the intercept and slope from the graph to obtain \(\mathcal{E}\) and \(r\).

Safety tip: Always start with the highest voltage range on the voltmeter, and never touch live terminals with bare fingers. Ensure the ammeter is correctly rated for the expected current.

Worked Example

1. Problem: A 12 V battery has an internal resistance of \(0.5\;\Omega\). The circuit draws a current of \(2\;\text{A}\). Find the terminal p.d. across the battery.
2. Solution
   • Internal‑voltage drop: \(I r = 2\;\text{A} \times 0.5\;\Omega = 1\;\text{V}\).
   • Apply \(V_{\text{terminal}} = \mathcal{E} - I r\): \[ V_{\text{terminal}} = 12\;\text{V} - 1\;\text{V} = 11\;\text{V} \]

   The external circuit therefore receives **11 V**.

Important Points to Remember

• Potential difference is defined **between two points**; it is not a property of a single point.
• EMF is a special case of p.d. measured when the circuit is open (no current).
• Both EMF and p.d. use the same unit (volt) and the same symbol “V” in most exam questions – always check the context.
• For a real source, terminal p.d. < EMF when current flows because of the internal voltage drop \(I r\).
• When a charge moves through a component, the sign of the work tells us whether the component **supplies** (+) or **consumes** (–) electrical energy.
• Voltmeters must have a much higher resistance than the component being measured to avoid loading the circuit.

Suggested Diagram (for classroom use)

Practice Questions

1. State the Cambridge syllabus definition of potential difference and give its SI unit.
2. A 9 V battery supplies a current of \(0.3\;\text{A}\) to a circuit. Its internal resistance is \(2\;\Omega\). Calculate the terminal p.d. across the battery.
3. Explain why the terminal p.d. of a real battery is always less than its EMF when current flows.
4. A resistor dissipates \(20\;\text{J}\) of energy while \(0.5\;\text{C}\) of charge passes through it. Determine the p.d. across the resistor.
5. Describe how you would use an analogue voltmeter and a digital voltmeter to measure the p.d. across a 5 Ω resistor that is part of a 12 V circuit. Include a comment on range selection and why the voltmeter must be connected in parallel.
6. Outline a simple experiment (using a voltmeter, ammeter and variable resistor) to find the EMF and internal resistance of a cell. State one safety precaution.