define and use the electromotive force (e.m.f.) of a source as energy transferred per unit charge in driving charge around a complete circuit

Practical Circuits – Electromotive Force (e.m.f.)

Learning Objective

Define the electromotive force (e.m.f.) of a source as the energy transferred per unit charge inside the source, and use this definition to analyse, calculate and experimentally determine e.m.f. and internal resistance in complete circuits.

Cambridge AS & A Level Physics 9702 (2025‑2027) Alignment

Topic Checklist (AS Level)

This set of notes focuses on Topic 10 – D.C. circuits. The remaining AS‑level topics (1‑9, 11) are covered in separate modules. For a full AS‑level revision package, see the accompanying “Topic Index”.

Bridging to A‑Level Extensions (Topics 12‑25)

Understanding e.m.f. is a foundation for later A‑level topics such as:

• Electric fields and potential (Topic 12) – e.m.f. is the line integral of the electric field around the source.
• Capacitance and energy storage (Topic 13) – the work done by a source in charging a capacitor is \(\mathcal{E}Q\).
• Alternating currents (Topic 15) – the rms value of an AC source is analogous to the e.m.f. of a DC source.
• Quantum physics (Topic 24) – photoelectric emission is described using the e.m.f. of a photovoltaic cell.

When moving to these topics, keep the definition \(\displaystyle\mathcal{E}= \frac{W_{\text{source}}}{Q}\) in mind; the “work done by the source” is the same physical quantity, only the mechanism of charge movement changes.

Key Concepts

• Electromotive force (e.m.f.) \(\mathcal{E}\) – work done by the source per coulomb of charge as the charge moves inside the source; unit: volt (V).
• Internal resistance \(r\) – the inherent resistance of the source; causes a voltage drop \(Ir\) when current \(I\) flows.
• Terminal voltage \(V\) – the potential difference measured across the external terminals while a current is flowing.
• Kirchhoff’s first law (current law) – the algebraic sum of currents at a junction is zero; in a single‑source series circuit it reduces to the relation \(V=\mathcal{E}-Ir\).
• Potential‑divider rule – in a series circuit the voltage across each element is proportional to its resistance; the e.m.f. acts as the total voltage of the series combination \((r+R)\).

Definition and Rigorous Derivation of e.m.f.

For a charge \(Q\) that moves through a source, the source does work \(W_{\text{source}}\). By definition,

\[

\mathcal{E}= \frac{W_{\text{source}}}{Q}\quad\text{(J C}^{-1}\text{ = V)}.

\]

When a current \(I\) flows, the charge experiences two voltage drops:

1. Across the external circuit – the terminal voltage \(V\).
2. Across the internal resistance – a drop of magnitude \(Ir\).

The total energy supplied per coulomb is therefore the sum of these two drops:

\[

\mathcal{E}=V+Ir\qquad\Longrightarrow\qquad V=\mathcal{E}-Ir .

\]

The minus sign appears because the internal resistance opposes the current supplied by the source, reducing the voltage that appears at the terminals.

Experimental Determination of e.m.f. and Internal Resistance

Method 1 – V‑I (Linear‑fit) Method

1. Construct the circuit shown in the figure below: a source with internal resistance \(r\), a variable external resistor \(R\), an ammeter in series, and a voltmeter across the source terminals.
2. Vary \(R\) to obtain at least four distinct current values. Record the corresponding terminal voltage \(V\) and current \(I\).
3. Plot \(V\) (vertical axis) against \(I\) (horizontal axis). The points should lie on a straight line described by \(V=\mathcal{E}-Ir\).
4. Use a calculator, spreadsheet, or graph‑plotting software to perform a least‑squares regression. The regression yields:

   • Intercept \(V_{0}\) = \(\mathcal{E}\) (e.m.f.)
   • Slope \(m\) = \(-r\) (internal resistance)

5. From the regression output obtain the standard errors of the intercept (\(\sigma{\mathcal{E}}\)) and slope (\(\sigma{r}\)). These give the uncertainties in \(\mathcal{E}\) and \(r\). (For AO3 you need only quote the uncertainties; full propagation is not required.)

Method 2 – Potentiometer / Null‑Method

1. Set up a uniform potentiometer wire of total length \(L\) carrying a known current \(I{\text{p}}\). The potential gradient is \(\displaystyle \frac{V{\text{p}}}{L}\) (V m\(^{-1}\)).
2. Connect the unknown source in series with a standard cell of known e.m.f. \(\mathcal{E}_{\text{std}}\) and a galvanometer.
3. Adjust the sliding contact along the potentiometer until the galvanometer reads zero (null condition). At this point the potential drop along the wire equals the e.m.f. of the unknown source because no current flows through the galvanometer.
4. Measure the length \(l\) from the start of the wire to the null point. The e.m.f. of the unknown source is then

\[

\mathcal{E}= \frac{V_{\text{p}}}{L}\;l .

\]

To obtain the internal resistance \(r\):

• Repeat the null measurement with a known external resistance placed in series with the source; the change in the null length gives the voltage drop across the internal resistance, from which \(r\) can be calculated.
• Alternatively, combine the potentiometer result for \(\mathcal{E}\) with a V‑I measurement (as in Method 1) to determine \(r\) from the slope.

Data Analysis – Worked Example (V‑I Method)

Using Excel’s LINEST function (or a scientific calculator) the best‑fit line is

\[

V = (1.60\ \text{V})\;-\;(1.6\ \Omega)I .

\]

• \(\mathcal{E}=1.60\ \text{V}\) with an uncertainty of \(\pm0.02\ \text{V}\) (standard error of the intercept).
• \(r = 1.6\ \Omega\) with an uncertainty of \(\pm0.10\ \Omega\) (standard error of the slope).

These values agree with the manufacturer’s specification of 1.60 V ± 0.05 V for the e.m.f. and 1.5 Ω ± 0.2 Ω for the internal resistance, demonstrating good experimental technique.

Common Sources of Error (AO3)

• Contact resistance at the voltmeter or ammeter terminals – adds to the apparent internal resistance.
• Parallax error when reading analogue meters – reduces the precision of \(V\) and \(I\) values.
• Unaccounted wire resistance in the external circuit – contributes to the measured slope.
• Temperature rise of the source during the experiment – changes both \(\mathcal{E}\) and \(r\).
• Linear‑fit assumptions – using only two data points ignores random errors; a minimum of four points and regression is recommended.
• Potentiometer gradient drift (Method 2) – caused by fluctuations in the current through the potentiometer wire.

Safety Precautions

• Check that all connections are tight before energising the circuit.
• Do not exceed the rated voltage or current of the source; overheating can permanently alter the internal resistance.
• Use measuring instruments within their specified ranges to avoid overload.
• Always disconnect the power before changing resistors or adjusting the potentiometer contact.
• Wear safety glasses when handling batteries or high‑current circuits.

Summary of Symbols

Suggested Diagram

Exam‑style Question (Practice)

“A battery of unknown e.m.f. and internal resistance is connected to a variable resistor. The following V‑I data are obtained:

Using a linear‑fit, determine the e.m.f. and internal resistance, including uncertainties. Comment on two possible sources of error and how they could be minimised.

*(This question tests AO1, AO2 and AO3 – definition, calculation and experimental evaluation.)*