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Cells in Parallel

For n identical cells in parallel: EMF stays the same, internal resistance reduces to r/n. Parallel combination is beneficial when external resistance is much smaller than r.
Class 11Class 12
Derivation

For identical cells

nn identical cells (each EMF E\mathcal{E}, internal resistance rr) with all positive terminals joined and all negative terminals joined.

By symmetry, each cell carries current I/nI/n. Terminal voltage:

V=E−In⋅rV = \mathcal{E} - \frac{I}{n}\cdot r

Equivalent single source must satisfy V=Eeq−IreqV = \mathcal{E}_{eq} - Ir_{eq}:

Eeq=E,req=rn\mathcal{E}_{eq} = \mathcal{E}, \quad r_{eq} = \frac{r}{n} Eeq=E,req=rn\boxed{\mathcal{E}_{eq} = \mathcal{E}, \quad r_{eq} = \frac{r}{n}}

General case (non-identical cells)

Using KCL and the condition of equal terminal voltages across the parallel combination:

Eeq=∑Ei/ri∑1/ri,1req=∑1ri\mathcal{E}_{eq} = \frac{\sum \mathcal{E}_i/r_i}{\sum 1/r_i}, \quad \frac{1}{r_{eq}} = \sum \frac{1}{r_i}

When to use parallel

Current through external RR:

I=ER+r/nI = \frac{\mathcal{E}}{R + r/n}

When R≪r/nR \ll r/n is not satisfied — i.e., when R≪rR \ll r — parallel combination is beneficial: req=r/n≪Rr_{eq} = r/n \ll R, so most of E\mathcal{E} drives the external circuit. Useful for high-current, low-voltage applications.

Note
Parallel combination does not increase the available voltage — it reduces the effective internal resistance, allowing higher current delivery. Series increases voltage; parallel increases current capacity.