Cells in Parallel

When you connect two or more cells side-by-side so that all their positive terminals meet at one node and all their negative terminals meet at another, the cells are said to be in parallel. Unlike cells in series, parallel cells do not stack their voltages — the terminal voltage of the combination stays the same as that of a single cell. What changes instead is the circuit's ability to supply current. Think of it like several taps feeding water into the same pipe: each tap adds to the total flow rate, even though the pressure (voltage) remains unchanged. The practical advantage of a parallel arrangement is increased current capacity and reduced internal resistance. Each cell contributes a share of the total current demanded by the external circuit, so no single cell is drained as quickly. This is exactly why laptop batteries and electric vehicles use large banks of cells wired in parallel (as well as in series): you get the endurance of many cells without sacrificing the operating voltage. The cells must have the same EMF; otherwise, circulating currents flow between them and waste energy as heat. Understanding parallel cells requires you to apply Kirchhoff's laws carefully. The equivalent EMF of identical parallel cells equals the EMF of one cell, while the equivalent internal resistance is found using the parallel-resistance formula. Together, these rules predict how the combination behaves as a source in any circuit you attach to it.

Key takeaways
  • Identify the configuration
  • Equivalent EMF
  • Equivalent internal resistance
  • Current sharing
  • Terminal voltage

Identify the configuration

Cells are in parallel when every positive terminal connects to a common positive rail and every negative terminal connects to a common negative rail. Current can travel through any of the cells to reach the external circuit. Label each cell with its EMF (ε) and internal resistance (r).

Equivalent EMF

For n identical cells each with EMF ε, the equivalent EMF of the parallel combination is simply ε — the same as one cell. Intuitively, each branch sits between the same two nodes; nodes cannot simultaneously be at two different potential differences. If the cells have different EMFs, a weighted average applies (derived from Kirchhoff's current law), but at A-level you will almost always deal with identical cells.

Equivalent internal resistance

Each cell's internal resistance r is in parallel with the others. Using the standard parallel-resistance formula for n identical cells: 1/r_eq = 1/r + 1/r + … (n times) = n/r, so r_eq = r/n. This is the key result: connecting n identical cells in parallel reduces the effective internal resistance by a factor of n, which increases the short-circuit current the combination can deliver and reduces wasted power inside the source.

Current sharing

When the combination drives a current I through an external resistance R, each of the n identical cells supplies a current I/n. No single cell is overloaded. The total current is I = ε / (r_eq + R) = ε / (r/n + R). As n → ∞, r_eq → 0 and the source approaches an ideal voltage source — one with no internal voltage drop regardless of current drawn.

Terminal voltage

The terminal voltage across the combination is V = ε − I·r_eq. Because r_eq = r/n is smaller than r for a single cell, the voltage drop across the internal resistance is lower for the same external current, keeping V closer to ε. This is why rechargeable battery packs maintain a steadier voltage under heavy load compared with a single cell of the same chemistry.

The Key Equation

εeq=ε,req=rn,I=εrn+R\varepsilon_{\text{eq}} = \varepsilon, \quad r_{\text{eq}} = \frac{r}{n}, \quad I = \frac{\varepsilon}{\dfrac{r}{n} + R}
  • \varepsilon_{\text{eq}} — Equivalent EMF (V): The effective electromotive force of the parallel combination — equal to the EMF of a single cell when all cells are identical.
  • \varepsilon — EMF of one cell (V): The electromotive force of each individual cell in the parallel bank.
  • r — Internal resistance (single cell) (Ω): The resistance inside each individual cell due to its electrochemical components.
  • n — Number of cells (dimensionless): The total number of identical cells connected in parallel.
  • r_{\text{eq}} — Equivalent internal resistance (Ω): The effective internal resistance of the whole parallel combination, equal to r/n.
  • R — External resistance (Ω): The load resistance connected across the terminals of the parallel cell bank.
  • I — Total current (A): The current flowing through the external circuit, shared equally among the n cells.

Worked Example

Problem. Three identical cells, each with EMF 1.5 V and internal resistance 0.6 Ω, are connected in parallel and then connected to an external resistor of 2.0 Ω. Calculate (a) the equivalent internal resistance of the combination, (b) the total current through the external resistor, and (c) the current supplied by each cell.

req=0.2 Ω,I≈0.682 A,Icell≈0.227 Ar_{\text{eq}} = 0.2\,\Omega,\quad I \approx 0.682\,\text{A},\quad I_{\text{cell}} \approx 0.227\,\text{A}

Answer. Equivalent internal resistance 0.2 Ω; total current ≈ 0.682 A; current per cell ≈ 0.227 A.

Figure
Diagram
Three Identical Cells in Parallel+−long = +rCell 1ε, rrCell 2ε, rrCell 3ε, rRII/3I/3I/3ε_eq = ε | r_eq = r/n | Each cell carries I/n
Origins · 1 minGeorg Simon Ohm and how this was discovered
GS
Georg Simon Ohm
1789–1854 · German

Ohm formulated the relationship between voltage, current, and resistance (V = IR), which underpins every calculation involving cells in circuits — including the analysis of parallel cell combinations. His work on internal resistance of galvanic cells directly explained why real sources behave differently from ideal ones, and his mathematical framework made it possible to derive the equivalent resistance of any combination of cells and resistors. Ohm's original 1827 paper was so poorly received by the Prussian education ministry that he was forced to resign his teaching post. It took nearly two decades and recognition from the Royal Society of London before his genius was widely acknowledged.

  • Cells in Series — The direct counterpart to parallel cells — series cells add EMFs but also stack internal resistances, giving a complementary picture of how cell arrangements affect circuit behaviour.
  • Internal Resistance — The concept of internal resistance is central to understanding why parallel combinations reduce r_eq — you need to know what internal resistance is before you can analyse how connecting cells in parallel changes it.
  • Resistors in Parallel — The formula r_eq = r/n for parallel cells is derived directly from the parallel-resistor rule — understanding resistors in parallel makes the derivation transparent.
  • Kirchhoff's Current Law — KCL (the sum of currents at a node is zero) is the fundamental law that explains how current splits equally among identical parallel cells and how unequal cells would exchange circulating currents.