A battery labelled “1.5 V” never quite delivers 1.5 V to your circuit. Some of its energy gets wasted inside the battery itself, heating it up. This is why a phone or a torch gets warm as it works. The full push a cell can give is its EMF; the wasted part is down to its internal resistance. Get these two straight and one tidy equation ties the whole thing together.
📘 What you need to know
EMF (ε) is the energy given to each coulomb of charge by the source — its full “push”, in volts
Every real cell has internal resistance (r) — resistance inside the cell itself
This internal resistance wastes some energy as heat: the lost volts = Ir
The terminal p.d. is what’s actually left for the circuit: V = IR
The key equation is ε = I(R + r), i.e. EMF = terminal p.d. + lost volts
EMF equals the terminal p.d. only when no current flows (an open circuit)
What is EMF?
When a charge passes through a source like a battery, it gains energy. The electromotive force (EMF), symbol ε (epsilon), is the amount of energy given to each coulomb of charge by the source:
EMF — definition
The energy converted from other forms into electrical energy, per unit charge, as charge passes through the source
Notice this is just like potential difference — energy per coulomb — so EMF is also measured in volts. Despite the name, it is not a force. Think of EMF as the total voltage a source could deliver if nothing were wasted. It’s the number printed on the battery.
Quick myth-buster: “electromotive force” is a historical name, and a slightly misleading one. It is not a force and isn’t measured in newtons. It’s an energy-per-charge, measured in volts, exactly like p.d. If a question ever tempts you to treat EMF as a force, don’t — it’s a voltage.
Internal resistance
Here’s the catch. The materials inside a real cell — the chemicals, the electrodes — aren’t perfect conductors. They have their own resistance, called the internal resistance (r). As charge flows through the cell, it has to push through this internal resistance too, and that wastes some energy as heat inside the cell.
That’s exactly why a battery gets warm in use, and why its voltage sags a little when it’s working hard. So we picture a real cell as two things in series: a perfect source of EMF, plus a small internal resistance r tucked inside it.
We model a real battery as a perfect EMF source (ε) in series with its internal resistance (r). The load R is the rest of the circuit.
Lost volts and terminal p.d.
Now follow the energy. The source hands each coulomb an amount ε (the EMF). But before that charge even leaves the battery, some of its energy is spent pushing through the internal resistance r. The voltage used up there is called the lost volts:
Lost volts
lost volts = Ir
What’s left over is delivered to the external circuit — the terminal potential difference, or terminal p.d. This is the voltage you’d actually measure across the battery’s terminals:
Terminal p.d.V = IR (the voltage across the load R)
Since energy is conserved, the full EMF must equal what’s delivered plus what’s wasted:
EMF is shared outε = terminal p.d. + lost volts = IR + Ir
The whole bar is the EMF. Most of it (teal) reaches the circuit as terminal p.d.; a small slice (red) is lost inside the cell.
The key equation
Factor out the current I from IR + Ir and you get the equation you’ll use again and again:
EMF and internal resistanceε = I(R + r)
Where:
ε = EMF, in volts (V)
I = current, in amperes (A)
R = resistance of the external circuit (the load), in ohms (Ω)
r = internal resistance of the cell, in ohms (Ω)
A warning that catches people out: R is the resistance of everything outside the cell — the load — and it changes if you add or remove components. The internal resistance r belongs to the cell itself and stays fixed no matter what you connect. When you substitute, make sure R is the external resistance, not the total.
WE 1
A cell of EMF 1.5 V and internal resistance 0.50 Ω is connected to a 7.0 Ω resistor. Calculate the current in the circuit.
Step 1 — use ε = I(R + r)
rearrange: I = ε / (R + r)
Step 2 — substituteI = 1.5 ÷ (7.0 + 0.50) = 1.5 ÷ 7.5I = 0.20 AAdd the internal resistance to the external one — the current sees the total.
WE 2
For the same circuit (ε = 1.5 V, r = 0.50 Ω, R = 7.0 Ω, I = 0.20 A), calculate the lost volts and the terminal p.d.
Step 1 — lost volts = Irlost volts = 0.20 × 0.50 = 0.10 VStep 2 — terminal p.d. = IR (or ε − lost volts)V = 0.20 × 7.0 = 1.4 Vlost volts = 0.10 V, terminal p.d. = 1.4 VCheck: 1.4 + 0.10 = 1.5 V, the full EMF. The circuit only “sees” 1.4 V, not the labelled 1.5 V.
Measuring the EMF
Here’s a clever consequence. If no current flows (an open circuit), then I = 0, so the lost volts (Ir) become zero. With nothing wasted inside, the terminal p.d. rises to equal the full EMF.
So to measure a cell’s EMF, you connect a high-resistance voltmeter straight across it with nothing else drawing current. Because an ideal voltmeter draws virtually no current, the reading is (near enough) the EMF itself.
EMF on open circuit
When I = 0: lost volts = 0, so terminal p.d. = ε
WE 3
A battery has an EMF of 6.0 V. When it drives a current of 1.5 A, its terminal p.d. drops to 5.4 V. Calculate the internal resistance.
Step 1 — find the lost volts
lost volts = ε − terminal p.d. = 6.0 − 5.4 = 0.60 VStep 2 — lost volts = Ir, so r = lost volts / Ir = 0.60 ÷ 1.5r = 0.40 ΩThe bigger the current, the bigger the voltage drop — that’s the internal resistance showing itself.
Full push EMF ε
minus lost volts (Ir, inside cell)
Terminal p.d. V = IR, to circuit
💡 Top tips
ε = I(R + r) is the master equation — EMF drives current through the total resistance.
EMF = terminal p.d. + lost volts. Terminal p.d. = IR; lost volts = Ir.
R is external, r is internal. R can change; r is a fixed property of the cell.
No current → terminal p.d. = EMF. That’s how you measure EMF (high-resistance voltmeter, open circuit).
Bigger current → bigger lost volts → the terminal p.d. sags more.
⚠ Common mistakes
Treating EMF as a force — it’s an energy per charge, measured in volts
Using the total resistance as R in the equation — R is the external load only
Forgetting the internal resistance entirely and assuming terminal p.d. = EMF while current flows
Mixing up lost volts (Ir) with terminal p.d. (IR)
Thinking r changes when you swap the load — r belongs to the cell and stays constant
Quick recap: EMF (ε) is the full energy per coulomb a source gives — measured in volts, not a force. Every real cell has internal resistance r, which wastes energy as lost volts (Ir). What reaches the circuit is the terminal p.d. (IR). Energy conservation gives ε = I(R + r). With no current flowing, terminal p.d. equals the EMF.
That completes the core of the electricity in this unit — you can now handle current, voltage, resistance, power and even what goes on inside a real battery. The last piece is the family of components whose resistance changes with their surroundings: thermistors and light-dependent resistors. In Variable Resistance & Sensors we’ll see how they let circuits respond to heat and light.
Want internal resistance to finally click?
Book a free meeting and we’ll work through lost volts, terminal p.d. and the EMF equation together.
