Cells, emf and internal resistance

A cell pushes charge round a circuit. Its emf ε is the voltage across its terminals when no current flows, the most it can give.

A real cell has internal resistance r: the chemicals inside it also resist the current. So the current in a circuit with external resistance R is:

I = ε / (R + r)

Some of the emf is used up inside the cell. The voltage across the terminals, the terminal voltage, is:

• Giving current (discharging): V = ε – Ir, less than ε.
• Being charged (current forced in backwards): V = ε + Ir, more than ε.

Example: ε = 6 V, r = 1 Ω, R = 5 Ω. I = 6 / 6 = 1 A. V = 6 – 1 × 1 = 5 V, which is also IR = 1 × 5.

Short circuit (R = 0): the largest possible current, ε/r, here 6 A. All the energy turns into heat inside the cell, which can damage it.

Combining cells:

• Series: emfs add, and internal resistances add. n equal cells: nε and nr.
• Parallel (equal cells): the emf stays ε, but the internal resistance becomes r/n. This gives more current with less heating inside.

A car battery has a very small r, so it can deliver the large current needed to start the engine.