E.m.f. and Internal Resistance

Internal resistance (r) — the resistance of the source itself, arising from the materials the current must pass through inside the cell. Every real source has some.

E.m.f. (ε) — the total energy transferred to electrical energy per unit charge by the source.

Terminal potential difference (V) — the p.d. actually available across the terminals, which is what a voltmeter across the cell reads.

ε = I(R + r)

V = ε − Ir

Lost volts (Ir) — the energy per unit charge dissipated inside the cell. It is not lost from the universe, but it is unavailable to the external circuit, heating the cell instead.

Example: a 9.0 V cell of internal resistance 1.5 Ω connected to a 6.0 Ω resistor. The current is 9.0 ÷ (6.0 + 1.5) = 1.2 A, so the terminal p.d. is 9.0 − (1.2 × 1.5) = 7.2 V. The remaining 1.8 V is dissipated inside the cell.

Why terminal p.d. falls as current rises — the lost volts Ir grow with the current, so V = ε − Ir shrinks. Drawing a large current from a cell with significant internal resistance makes the terminal p.d. sag noticeably, which is why car headlights dim when the starter motor turns.

Measuring ε and r graphically — vary the external resistance, recording V and I at each setting, and plot V against I. Comparing V = ε − Ir with y = mx + c shows the graph is a straight line with:

y-intercept = ε, the e.m.f., which is the terminal p.d. when no current flows

gradient = −r, so the internal resistance is the magnitude of the gradient

Why the intercept gives ε — at zero current there are no lost volts, so the terminal p.d. equals the e.m.f. This is also why a high-resistance voltmeter connected across a cell on its own reads very nearly the e.m.f.: it draws almost no current.