EMF, Internal Resistance and Potential Dividers

EMF and internal resistance

A power source (e.g. a battery) does work to move charge around a circuit. Its electromotive force (EMF, ε) is the energy given to each coulomb of charge:

ε = energy ÷ charge   (volts)

Real sources have internal resistance (r) — resistance within the source itself. Some energy is transferred (as heat) inside the source, so the voltage available to the external circuit (the terminal potential difference) is less than the EMF:

ε = I(R + r) = V + Ir

where V = terminal p.d. across the external resistance R, and Ir = the "lost volts" inside the source.

Finding EMF and internal resistance

Measure terminal p.d. V for different currents I and plot V against I:

V = ε − Ir

• y-intercept = ε (EMF).
• gradient = −r (internal resistance).

Kirchhoff's laws

• First law (charge conservation): the total current into a junction = total current out of it.
• Second law (energy conservation): around any closed loop, the sum of EMFs = the sum of the p.d.s across the components.

Potential dividers

A potential divider uses two (or more) resistors in series to produce a chosen fraction of the supply voltage:

V_out = V_in × R₂ ÷ (R₁ + R₂)

• Replacing one resistor with a thermistor or LDR makes a sensor circuit whose output voltage changes with temperature or light — used to trigger alarms/switches.
• A potentiometer (variable) gives a continuously adjustable output.

Worked example

A battery of EMF 6.0 V and internal resistance 0.5 Ω drives a current of 2.0 A. Find the terminal p.d.

• V = ε − Ir = 6.0 − (2.0 × 0.5) = 6.0 − 1.0 = 5.0 V. ✓

Common mistakes

• Forgetting the lost volts (Ir) — terminal p.d. is less than EMF when current flows.
• Reading r as the gradient (it's the negative gradient of a V–I graph).
• Getting the potential-divider ratio the wrong way round (V_out is across R₂).

Exam tips

• Use ε = V + Ir and interpret a V–I graph (intercept = ε, gradient = −r).
• Apply Kirchhoff's laws to solve multi-loop circuits.
• Use the potential-divider equation for sensor circuits (thermistor/LDR).

Key facts to remember

• EMF ε = I(R + r) = V + Ir; terminal p.d. V = ε − Ir (lost volts); from a V–I graph, ε = intercept, r = −gradient.
• Kirchhoff: current conserved at junctions; EMFs = Σ p.d.s around a loop.
• Potential divider: V_out = V_in × R₂/(R₁+R₂); with a thermistor/LDR it makes a sensor.