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Every real source wastes some energy inside itself, and this topic is how the exam tests whether you understand why. Expect definition marks (emf versus pd), quick calculations with ε=I(R+r), the required practical as a data-analysis question (read the emf from the intercept, the internal resistance from the gradient) or as a 6-mark method, and “explain why” questions about car batteries, EHT supplies and headlamps that dim when the engine starts.
By the end you’ll be able to
Define emf and distinguish it from terminal potential difference
Use ε = I(R + r) = V + Ir in single and multi-cell circuits
Plot terminal pd against current and read emf (intercept) and internal resistance (−gradient) — the required practical
Explain why high internal resistance matters for car batteries and EHT supplies
Show that maximum power is delivered to a load when R = r (OCR, Cambridge extension)
What the exam asks
Defining emf and distinguishing it from terminal pd in terms of energy.
Using ε=I(R+r)=V+Ir, including “lost volts”.
Cells in series and identical cells in parallel.
The required practical: plotting terminal pd against current, then reading and from the graph.
vii.Check your understanding
3 questions on EMF and internal resistance. Every option is explained once you answer.
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PromptCard 1 of 3
Define emf.
ε
r
Power in the source and the load, efficiency, and the maximum power condition R=r.
Explaining why low internal resistance matters for high-current sources and why high internal resistance makes EHT supplies safer.
Core ideas
Emf and terminal pd
The emfε of a source is the energy transferred to electrical energy per unit charge passing through the source:
ε=QE
The potential difference across a component is the energy transferred from electrical energy to other forms per unit charge passing through it. Both are measured in volts (J C⁻¹); the difference lies in the direction of the energy transfer.
A real source has internal resistancer, the resistance of its chemicals, electrodes or windings. When current I flows, some energy per coulomb is dissipated inside the source:
ε=I(R+r)=V+Ir
V=IR is the terminal pd, the pd available to the external circuit.
Ir is the lost volts, the energy per coulomb wasted inside the source.
With no current (open circuit, or an ideal voltmeter across the terminals), V=ε.
The terminal pd falls as current rises. The maximum possible (short-circuit) current is ε/r.
The V–I graph
Rearranging gives a straight-line equation:
V=−rI+ε(y=mx+c)
Plot terminal pd V (y-axis) against current I (x-axis):
y-intercept=ε
gradient=−r
x-intercept=ε/r, the short-circuit current (usually found by extrapolation)
An alternative linearisation uses the resistance of the load: R=Iε−r, so a graph of R against 1/I has gradient ε and y-intercept −r.
Combining cells
Arrangement
Total emf
Total internal resistance
n identical cells in series (same way round)
nε
nr
n identical cells in parallel
ε
r/n
One cell reversed in a series chain
Its emf subtracts
Its r still adds
Parallel cells do not raise the emf, but they lower the internal resistance and share the current, so the battery lasts longer.
Power, efficiency and maximum power
Multiplying ε=V+Ir by I gives the energy budget per second:
εI=I2R+I2r
This is the power supplied by the source, split into the power delivered to the load and the power wasted inside the source. The efficiency is
I2(R+r)I2R=R+rR=εV
The power in the load, P=(R+r)2ε2R, is a maximum when R=r, where Pmax=4rε2 and the efficiency is only 50%. A small load resistance gives a large current but most of the power is wasted in r; a large load resistance gives high efficiency but little power.
Why internal resistance matters
Car batteries must deliver 100–200 A to a starter motor, so they need a very small r (hundredths of an ohm). Even so, the terminal pd drops noticeably while the motor turns, which is why the headlamps dim.
EHT supplies (several kV) are built with a very large internal resistance (often megaohms). If someone touches the terminals, the current is limited to a safe value.
Rechargeable and lithium cells have lower r than zinc–carbon cells, so they suit high-current devices.
The required practical
Connect the cell in series with an ammeter, a variable resistor, a switch and a small fixed “protective” resistor. Connect a voltmeter directly across the cell’s terminals.
Vary the resistance and record pairs of V and I over as wide a range of currents as possible.
Open the switch between readings. A current flowing continuously heats the cell (so r changes) and runs it down (so ε falls).
The protective resistor stops the current becoming large enough to damage the cell or heat it when the variable resistor is set near zero.
Plot V against I, draw the line of best fit, then read ε from the intercept and r from minus the gradient.
Worked examples
Exam technique
Definitions: always say “energy transferred per unit charge” and state the direction: into electrical energy for emf, out of electrical energy for pd. “Voltage of the battery” earns nothing.
Graph questions: check whether the axes start at zero before reading an intercept, and use a gradient triangle covering at least half the line.
“Explain why the headlamps dim”: the large current means large lost volts Ir, so the terminal pd across the lamps falls, and their power V2/R falls.
Multi-cell problems: combine the cells into one equivalent emf and internal resistance first, then use ε=I(R+r).
6-mark method: give the circuit, what is varied and measured, the graph and how ε and r come from it, and the precautions (switch off between readings, protective resistor, wide range of readings).
Common mistakes
Quick recap
Emf = energy transferred to electrical energy per unit charge; pd = energy transferred from electrical energy per unit charge.
ε=I(R+r)=V+Ir; lost volts =Ir; short-circuit current =ε/r.
V against I: intercept =ε, gradient =−r.
Series cells: emfs and internal resistances add. Identical parallel cells: same emf, internal resistance r/n.
Efficiency =R/(R+r); maximum load power ε2/4r when R.
Low r for high-current sources (car batteries); high r for safety (EHT supplies).