EMF vs. Potential Difference: The Difference That Actually Matters

Ask a student to define EMF and you’ll usually get “the work done per unit charge,” and ask for potential difference and you’ll get the same phrase back. Both definitions are technically correct, which is exactly the problem — they sound identical, and the textbook that gave you both of them rarely explains what’s different about the work being counted. That difference is the entire point, and it’s the thing a multimeter actually has to discriminate between every time you use one.

EMF counts work the battery does against its own resistance too

A real battery isn’t a pure source of potential difference; it’s a source of EMF, ε, in series with an unavoidable internal resistance, r, built from the chemistry and the electrode material inside it. ε is defined as the total work done per unit charge by whatever non-electrostatic process — chemical reactions inside a cell, changing flux in a generator — is driving charge around the circuit, including the work spent pushing that charge through the battery’s own internal resistance. It’s a property of the source alone, and it doesn’t depend on what’s connected outside.

That’s why EMF is what you read on a voltmeter connected directly across the battery’s terminals with no external circuit completed — the open-circuit reading. With no current flowing, there’s no voltage drop across r to subtract, so the terminal reading and the EMF coincide. That coincidence disappears the instant you close the circuit.

Terminal p.d. is what’s left after the internal drop

Close the circuit and current I flows, which means that current has to push through the internal resistance r as well as the external circuit. By Ohm’s law, that internal passage costs a voltage drop of Ir — energy per unit charge spent inside the battery, never delivered to the outside circuit at all. What remains, available to the external resistor, lamp, or wire, is the terminal potential difference:

V = ε − Ir

This is the actual, derivable relationship, not a rule to memorise separately: EMF is the total work per unit charge the source supplies, Ir is the share of that work consumed internally, and V is the remainder — exactly what a voltmeter across the external circuit reads while current is actually flowing. It is always less than ε whenever I > 0, and the gap grows with higher current, which is also why a battery under heavy load (starting a car, for instance) shows a noticeably lower terminal voltage than its rated EMF.

Run the battery in reverse — charging it, rather than discharging it — and the sign on the Ir term flips, so V = ε + Ir: the charger has to supply more than the EMF, because now it’s paying for the internal resistance instead of being paid out of it. Same equation, same internal resistance, opposite direction of current, and the relationship between V and ε reverses accordingly — which is a good check that the mechanism, not just the formula, has actually been understood.

This also explains a result that otherwise looks contradictory on the formula sheet: how two separate quantities, ε and V, can both legitimately be called “the voltage of the battery” depending on who’s asking. A battery’s label — “1.5 V,” “9 V” — is its EMF, measured or defined under open-circuit conditions. The number that actually determines how brightly a bulb connected to it glows is the terminal p.d., which is always slightly less, and by an amount that depends on what else is drawing current from the same source. Two fresh AA batteries with identical printed voltages can power the same circuit differently if their internal resistances differ, which is precisely why internal resistance, not EMF, is often the more practically important number for anything that draws significant current.

The full derivation of this, along with how internal resistance is measured in practice and how it interacts with circuits in series and parallel, is worked through in the Current Electricity notes.