Internal Resistance by Potentiometer Explained for School

Every real cell has internal resistance, written r: the materials inside the cell oppose the current just as an external resistor would. Because of r, the voltage at the terminals drops whenever the cell delivers current, which is why a torch dims as its batteries age even though they still show a voltage on a meter.

Internal resistance cannot be read directly with a voltmeter, but the potentiometer exposes it with two null readings. First the cell's EMF is balanced on open circuit, then the cell is loaded with a known resistor and balanced again at a shorter length; the shrinkage of the balancing length reveals r exactly.

Key Concepts

• Internal resistance (r): the cell's own resistance, in ohms, arising from its electrolyte and electrodes.
• EMF (E): the open-circuit voltage, balanced at length l₁ with no load connected.
• Terminal voltage (V): the voltage under load, balanced at the shorter length l₂ when a known resistor R is connected across the cell.
• Voltage drop inside the cell: E − V = Ir, the portion of the EMF consumed by the internal resistance.
• Load resistor (R): a known external resistance shunted across the cell during the second reading.

Formulas

Internal resistance: r = R (l₁ − l₂) / l₂. Derivation sketch: on open circuit E = k l₁; with R connected, the terminal voltage is V = k l₂, and also V = E R / (R + r) from the voltage divider formed by r and R. Eliminating E, V and k gives r = R (l₁ − l₂) / l₂. Assumptions: the cell's EMF and internal resistance stay constant between the two readings, the wire's potential gradient is unchanged, and the connecting leads have negligible resistance.

Worked Example

A cell is balanced on open circuit at l₁ = 300 cm. A 5.0 Ω resistor is then connected across its terminals and the new balance is found at l₂ = 250 cm.

1. Write the formula: r = R (l₁ − l₂) / l₂.
2. Find the shrinkage: l₁ − l₂ = 300 − 250 = 50 cm.
3. Substitute: r = 5.0 × 50 / 250.
4. Compute: 50 / 250 = 0.20, so r = 5.0 × 0.20 = 1.0 Ω.
5. Interpretation: with a 1.0 Ω internal resistance, drawing 1 A would drop 1 V inside the cell itself, noticeably dimming any load.

Common Misconceptions

• Internal resistance is invisible from outside. You cannot see it or measure it with an ohmmeter across the terminals; it only shows up as missing voltage under load.
• A cell showing full voltage can still be dead. An aged cell may keep its EMF while its r grows huge, so it collapses the moment a real current is demanded.
• r is not constant forever. It rises as the cell discharges, dries out or cools, which is why the two potentiometer readings must be taken promptly one after the other.

Real-World Applications

• Battery-health testers estimate r to grade used cells and predict remaining life.
• Choosing batteries for high-current devices (starters, power tools): low r is essential.
• Inverter and UPS designers account for r to predict voltage sag under sudden loads.
• Electric-vehicle battery packs are monitored for rising internal resistance as an ageing signal.

Internal resistance is the hidden tax every cell levies on its own EMF, and the potentiometer measures it with two quiet null readings. The fractional shrinkage of the balancing length, multiplied by the known load, is the cell's internal resistance in ohms.