Topic B.5 — Current & CircuitsPaper 1 & 2ε = I(R + r)~7 min read
Electromotive Force & Internal Resistance
A brand new 9 V battery sounds like it should give exactly 9 volts every time. In reality, it almost never quite does. Let’s find out why — and it turns out the answer is something we already know a lot about.
📘 What You Need to Know
Electromotive force (emf, ε) is the total energy given to every coulomb of charge by the source, measured in volts
Every real cell has some internal resistance (r) of its own — it isn’t a perfect, resistance-free source
As current passes through the cell, some energy is lost heating up the cell itself — these are the “lost volts”
ε = I(R + r), where R is everything else in the circuit and r is the cell’s own resistance
The terminal p.d. — what you’d actually measure across the cell’s terminals — is always a little less than the full emf
What emf really means
Emf is the maximum “push” a source can possibly give — the total energy handed to every coulomb of charge before anything gets used up along the way. You can only measure the true emf when no current is flowing at all, using a voltmeter with an enormous resistance connected straight across the cell’s terminals, with nothing else in the circuit.
Why don’t we get the full emf?
Here’s the twist: a cell isn’t just a source of energy, it’s also made of the same kind of material as any other conductor — and we already know conductors resist current a little. So as charge passes through the cell itself, it bumps into ions inside the cell, just like it does everywhere else, and hands over a little energy as heat. That’s internal resistance, and it means a bit of the emf gets “used up” before the charge even leaves the cell.
Picture the cell as a perfect emf source with a small internal resistance built right in. Some of the emf is “spent” inside the cell as lost volts, and the rest reaches the rest of the circuit as the terminal p.d.
Putting a number on it
The full emf always splits into two parts: the p.d. “lost” pushing charge through the cell’s own internal resistance, and the p.d. left over for the rest of the circuit — the terminal p.d. Since current is the same all the way round a single loop, we can write:
EMF equation
ε = I(R + r)
where ε is emf in volts (V), I is the current in amps (A), R is the resistance of everything else in the circuit, and r is the cell’s internal resistance. Expanding the brackets splits it neatly into the two parts we just described:
Two parts of the emf
ε = IR + Ir = terminal p.d. + lost volts
Quick recap: ε = I(R + r); the “lost volts” (Ir) heat up the cell itself; the terminal p.d. (IR) is all that’s left for the rest of the circuit; a bigger internal resistance means a bigger gap between emf and terminal p.d.
WE 1
A cell has an emf of 9.0 V and an internal resistance of 0.50 Ω. It is connected to an external resistor of 8.5 Ω. Find the current in the circuit and the lost volts.
Find the current first:
ε = I(R + r)
I = 9.0 ÷ (8.5 + 0.50) = 9.0 ÷ 9.0I = 1.0 ANow find the lost volts:lost volts = Ir = 1.0 × 0.50Lost volts = 0.5 V
WE 2
A battery of emf 6.0 V and internal resistance 1.5 Ω is connected to a 4.5 Ω lamp. Find the current and the terminal potential difference.
Find the current:I = 6.0 ÷ (4.5 + 1.5) = 6.0 ÷ 6.0I = 1.0 ANow find the terminal p.d.:V = IRV = 1.0 × 4.5Terminal p.d. = 4.5 VThat leaves 1.5 V as lost volts, since 4.5 + 1.5 = 6.0 V, exactly the emf.
💡 Top Tips
Only use the emf equation for the resistance actually available to the rest of the circuit — if a question adds or removes resistors, R changes, but r never does
Remember the internal resistance is a fixed property of the cell — it can’t be changed by anything you do to the external circuit
If a question gives you an “ideal” cell, that simply means r = 0, so terminal p.d. equals emf exactly
⚠ Common Mistakes
Forgetting internal resistance completely and treating terminal p.d. as if it always equals emf
Mixing up which resistance is “R” and which is “r” when substituting into the equation
Assuming a cell’s internal resistance changes when you add resistors to the circuit — it’s the external resistance that changes, not r
Up next: Variable Resistance — now we understand real cells, let’s look at components whose own resistance can be adjusted or changes with conditions, like thermistors and LDRs.
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