EMF, internal resistance and combinations of cells
A cell converts chemical energy into electrical energy. Its emf (E) is the potential difference between its terminals when no current is drawn; it equals the work done by the cell per unit charge moved through the whole circuit. Unit: volt.
Every real cell has an internal resistance r (the electrolyte resists the flow of ions). When the cell drives a current I through an external resistance R:
- I = E / (R + r)
- Terminal voltage V = E – Ir = IR, which is less than E while the cell is discharging.
- While a cell is being charged, V = E + Ir.
Worked example: E = 2 V, r = 0.5 Ω, R = 3.5 Ω. I = 2/4 = 0.5 A; V = 2 – 0.5 × 0.5 = 1.75 V.
Cells in series (n identical cells): total emf nE, total internal resistance nr, so I = nE / (R + nr). Series is useful when R is large compared with r.
Cells in parallel (n identical cells): emf stays E, internal resistance r/n, so I = E / (R + r/n). Parallel is useful when R is small compared with r.
For two different cells in series (helping each other), E_eq = E₁ + E₂ and r_eq = r₁ + r₂. For two cells in parallel, E_eq = (E₁r₂ + E₂r₁)/(r₁ + r₂) and r_eq = r₁r₂/(r₁ + r₂).
Key points
- Terminal voltage V = E – Ir (discharging); V = E only when I = 0.
- Short-circuit current = E/r (maximum current).
- Series: emf adds, internal resistance adds. Identical cells in parallel: emf same, r divides.
Exam tips
- Many questions give E, r and R and ask for I or V; write I = E/(R + r) first.
- If a cell is reversed in a series chain, subtract its emf but still add its internal resistance.
Common mistakes
- Treating terminal voltage as always equal to emf.
- Dividing emf by n for parallel identical cells (emf stays the same).
- Forgetting that a reversed cell still adds internal resistance.