EMF & Internal Resistance — Quick Summary

Mr Toogood's Physics · Electricity basics

AQA 3.5.1.6

ε = I(R+r)

Core equation

ε = IR + Ir

Expanded

V = IR

Terminal p.d.

"Lost volts"

Internal resistance

No cell is 100% efficient — some chemical energy is lost as heat inside it. This is modelled as an internal resistance, r, in series with the cell (not a real physical resistor, just a useful analogy).

Diagram showing emf equals terminal p.d. plus lost volts

emf splits between the terminal p.d. and the "lost volts" across r.

emf = terminal p.d. + lost volts

ε = emf (V) I = current (A)
R = external (circuit) resistance (Ω)
r = internal resistance (Ω) — typically 100–900 mΩ

Measuring emf: use a high-resistance voltmeter/multimeter across the open-circuit terminals — negligible current flows, so there are no lost volts and the reading equals ε exactly.

Terminal p.d. vs. current graph

Graph of terminal p.d. against current showing emf as the y-intercept and internal resistance as the gradient

y-intercept = ε; magnitude of gradient = r.

Plotting terminal p.d. (V) against current (I) gives a straight line: V = ε − Ir. The y-intercept is ε (I=0, no lost volts) and the gradient's magnitude is r.

Maximum power theorem

Graph of power dissipated in external resistor against its resistance, peaking when R equals r

Power in the external resistor peaks when R = r.

Power dissipated in the external resistor R is greatest when R = r. Below this, R has a smaller p.d. than r; above it, power falls again since P=V²/R decreases as R grows once p.d. is roughly constant.

Exam essentials

Rearrangements

V = ε − Ir (terminal p.d.)
r = (ε−V)/I
I = ε/(R+r)

Reading the graphs

V–I graph: intercept = ε, |gradient| = r.
P–R graph: peak at R = r (max power theorem).
Negligible r → terminal p.d. ≈ ε regardless of I.

Common slips

Watch mΩ ↔ Ω conversions — r is usually small.
Don't forget the square root when finding I from P=I²R.
"Describe" questions need graph values quoted, not just shape.