4.3ElectricityCore

EMF and Internal Resistance

Every real battery quietly resists its own current — which is why the voltage you actually measure across its terminals is always a little less than its labelled value once it’s driving a circuit. That gap is entirely explained by the battery’s own internal resistance.

35 min Video by vt.physics Module 4.3 Electrical circuits

EMF and Internal Resistance — A Level PhysicsWatch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

Define e.m.f. as the total energy transferred per unit charge, including energy dissipated internally
Explain internal resistance and lost volts
Use $ε = I (R + r)$
Describe the graphical method for finding e.m.f. and internal resistance

EMF and internal resistance

The $e.m.f.$ , $ε$ , of a source is the total energy transferred to each unit of charge — including energy dissipated inside the source itself, due to its own $internal resistance$ , $r$ . The energy "used up" driving current through this internal resistance is the $lost volts$ ; what remains is the $terminal potential difference$ available to the external circuit.

ε = I (R + r) = I R + I r

ε = terminal p.d. (IR) + lost volts (Ir).

1Lost volts:

I r = 1.5 \times 0.30 = 0.45

2Terminal p.d.:

V = ε - I r = 4.5 - 0.45

AnswerLost volts = 0.45 V; terminal p.d. = 4.05 V.

Tip — E.m.f. equals terminal p.d. only when no current flows — that’s the only moment there are no lost volts at all.

Finding e.m.f. and internal resistance graphically

Rearranging in terms of terminal p.d. gives $V = ε - I r$ — a straight-line equation. Plotting terminal p.d. against current (varying the external resistance) gives a line with $y$ -intercept $ε$ and gradient $- r$ .

V = ε - I r

Terminal p.d. against current: intercept = ε, gradient = −r.

1Comparing

V = ε - I r

with the graph: intercept

= ε = 6.0

2Gradient

= - r = - 0.50 Ω

, so

r = 0.50 Ω

Answerε = 6.0 V, r = 0.50 Ω

Equation recap

ε = I (R + r)

E.m.f. in terms of external and internal resistance.

V = ε - I r

Terminal p.d., used for the graphical ε/r method.

Common mistakes to avoid

Treating e.m.f. and terminal p.d. as always equal.

They are only equal at zero current; whenever current flows, terminal p.d. = ε − Ir, which is less than ε.

Forgetting internal resistance contributes to the total circuit resistance when finding current.

Use ε = I(R+r) — internal resistance adds to the external resistance, exactly like another series resistor.

Reading internal resistance directly (with its sign) off the gradient of a V-I graph.

The gradient of V against I is −r, so internal resistance r is the positive magnitude of that (negative) gradient.

Key takeaways

E.m.f. is the total energy per unit charge a source provides, including internal losses.
Internal resistance causes "lost volts" Ir, reducing terminal p.d. below ε.
ε = I(R+r); plotting V against I gives intercept = ε, gradient = −r.

Test yourself

Ready to lock in EMF and Internal Resistance? Pick a mode and earn XP & Dobloons.

EMF and Internal Resistance

35 min Video by vt.physics Module 4.3 Electrical circuits