Electromotive force and internal resistance

IB Physics SL· 12 min read

1. Core Definitions of emf and Internal Resistance★★☆☆☆⏱ 8 min

Ideal voltage sources have zero internal resistance, so their terminal voltage never drops no matter how much current is drawn. All real cells, batteries, and generators have a small but non-zero internal resistance that causes a voltage drop inside the source itself when current flows.

📘 Definition

Electromotive Force (emf)

$ε$

The work done per unit charge by non-electrical forces inside the source to separate positive and negative charges, creating a potential difference across the open terminals of the cell.

Example:

A standard 1.5 V AA alkaline cell has an emf of 1.5 V when no current is drawn.

✓ Quick check

Test your understanding of the definitions below:

When a cell is on open circuit (no current flowing), what is the relationship between its terminal voltage and its emf?
- Terminal voltage is zero
- Terminal voltage equals emf
- Terminal voltage is less than emf
- Terminal voltage is greater than emf
Reveal answer
Terminal voltage equals emf —
No current flows so there is no voltage drop across the internal resistance, so the full emf appears across the terminals.

2. Derivation of the emf Circuit Relationship★★★☆☆⏱ 10 min

🔬 Derivation

Goal:

Derive the relationship between emf, terminal voltage, current and internal resistance

Starting from:

Conservation of energy for a complete circuit with a cell of emf $ε$ , internal resistance $r$ , connected to an external load resistor $R$

1
Total energy supplied per unit charge by the cell is equal to the sum of energy dissipated per unit charge across all resistances in the circuit
2
$\varepsilon = V_{\text{load}} + V_{\text{internal}}$
3
Apply Ohm's Law to the internal resistance: the voltage drop across internal resistance is $V_{internal} = I r$
4
Apply Ohm's Law to the external load: $V_{load} = I R = V$ , the terminal voltage across the cell
5
$\varepsilon = V + Ir$

Result:

This rearranges to the standard formula for terminal voltage: $V = ε - I r$ . As current $I$ drawn from the cell increases, the terminal voltage $V$ decreases linearly.

📐 Worked Example

A cell of emf 12 V has internal resistance 0.5 Ω. It is connected to a 5.5 Ω external resistor. Calculate the terminal voltage across the cell.

1
First calculate total circuit resistance: $R_{total} = R + r = 5.5 + 0.5 = 6.0 Ω$
2
$Calculate total current in the circuit using Ohm's Law: $I = \frac{\varepsilon}{R_{\text{total}}} = \frac{12}{6} = 2.0\ \text{A}$$
3
Use the emf formula to find terminal voltage: $V = ε - I r$
4
$V = 12 - (2.0 \times 0.5) = 12 - 1 = 11\ \text{V}$
5
Verify by calculating voltage across external resistor: $V = I R = 2.0 \times 5.5 = 11 V$ , which matches.

3. V-I Graph for a Discharging Cell★★★☆☆⏱ 7 min

When you measure terminal voltage $V$ across a cell for different values of drawn current $I$ , you get a straight line graph with negative gradient, following the equation $V = - r I + ε$ , which matches the standard straight line form $y = m x + c$ .

Graph Feature	Physical Quantity
Y-intercept (at I = 0)	Emf $ε$ of the cell
Gradient (slope magnitude)	Internal resistance $r$ of the cell
X-intercept (at V = 0)	Short-circuit current of the cell

4. Common Pitfalls

Wrong move:

Confusing emf with terminal voltage for a cell that is supplying current

Why:

Students often use the emf value as the terminal voltage when current is flowing, ignoring the internal voltage drop

Correct move:

Always use $V = ε - I r$ to calculate terminal voltage when the cell is not on open circuit

Wrong move:

Forgetting that the gradient of the V-I graph is negative and taking the sign into account for internal resistance

Why:

Students sometimes calculate a negative value for internal resistance, which is physically impossible

Correct move:

Take the magnitude of the negative gradient to get the positive internal resistance value

Wrong move:

Treating internal resistance as an external resistor in series when calculating the terminal voltage

Why:

This leads to double-counting the voltage drop across internal resistance

Correct move:

The internal resistance is inside the cell, so the terminal voltage is the voltage across the external load only

Wrong move:

Stating that emf is the force that pushes charges around the circuit

Why:

Emf is a potential difference, measured in volts, not a force measured in newtons, this is a mark-losing definition error

Correct move:

Define emf as work done per unit charge by non-electrical forces inside the source

Wrong move:

Calculating short circuit current as $εR$ instead of $ε / r$

Why:

Short circuit means external load resistance R = 0, so total resistance is only r

Correct move:

Short circuit current $I_{sc} = ε / r$ , no external resistance is present

5. Quick Reference Cheatsheet

Quantity	Formula	Units
Electromotive force	$ε = V + I r$	Volts (V)
Terminal voltage	$V = ε - I r$	Volts (V)
Internal resistance	$r = \frac{ε - V}{I}$	Ohms ( $Ω$ )
Short circuit current	$I_{sc} = \frac{ε}{r}$	Amperes (A)
Maximum load power	$P_{max} = \frac{ε ^{2}}{4 r}$	Watts (W)

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 2
emf and internal resistance calculation
2022 · 1
multiple choice on terminal voltage
2021 · 2
V-I graph analysis for a cell

What's Next

You will frequently combine these concepts to solve multi-loop circuit problems that appear on Paper 2 section A and extended response questions. Understanding how real batteries behave under load also helps you connect theory to practical lab work, where you will perform a investigation you might design for your IA to measure internal resistance by varying external resistance and plotting a V-I graph. This practical is commonly assessed in your IA, so make sure you can explain sources of uncertainty and error for this experiment clearly.