Result

Terminal voltage

Δ V = ε - I r

Statement

What you actually measure at the battery terminals when current I flows.

Symbols and units

$ε$	emf of the source	V
$r$	internal resistance	Ω
$I$	current drawn	A
$Δ V$	terminal voltage	V

When it applies

Steady current, source modelled as an ideal emf plus a series resistance.

Don't use when

Models the source as an ideal EMF plus a fixed series resistance — a real battery's internal resistance can vary with current, temperature, and charge state.

Derivation

Go once round the loop, adding every potential change:

ε - I r - I R = 0

The terminal voltage is what is left after the internal drop, and it is what appears across the load:

Δ V = ε - I r = I R

Solving the loop equation for the current:

I = \frac{ε}{R + r}

Multiplying the loop equation by I shows where the energy goes:

I ε = I^{2} r + I^{2} R

Worked examples

Starter motor load

ε = 12 V, r = 0.05 Ω, I = 100 A:

Δ V = 12 - (100) (0.05) = 7 V

The cause of headlights dimming while cranking, and of a battery that reads 12 V yet fails to start.

Measuring internal resistance

12.6 V open circuit, 12.1 V under a 5 A load:

r = \frac{12.6 - 12.1}{5} = 0.10 Ω

Watch out for

Maximum power reaches the load when R = r, but efficiency is then only 50 %. Power transfer and efficiency are different design goals.

Connects to

Junction and loop rules Power Combining resistors