Cells in Series

For n identical cells in series: ε_eq = nε, r_eq = nr. Series combination increases EMF and internal resistance equally — beneficial when external resistance is much larger than r.

Derivation

$n$ cells with EMFs $\mathcal{E}_1, \mathcal{E}_2, \ldots$ and internal resistances $r_1, r_2, \ldots$ connected end-to-end (anode of one to cathode of next).

Total EMF (KVL — EMFs add when cells are aiding):

$$\mathcal{E}_{eq} = \mathcal{E}_1 + \mathcal{E}_2 + \cdots + \mathcal{E}_n$$

Total internal resistance (series):

$$r_{eq} = r_1 + r_2 + \cdots + r_n$$ $$\boxed{\mathcal{E}_{eq} = \sum \mathcal{E}_i, \quad r_{eq} = \sum r_i}$$

For $n$ identical cells

$$\mathcal{E}_{eq} = n\mathcal{E}, \quad r_{eq} = nr$$

Current through external resistance $R$:

$$I = \frac{n\mathcal{E}}{R + nr}$$

When to use series

Series combination is beneficial when $R \gg nr$ — the internal resistance overhead is a small fraction of total resistance, and current is approximately $n\mathcal{E}/R$.

Opposing cell

If one cell is reversed: $\mathcal{E}_{eq} = \mathcal{E}_1 - \mathcal{E}_2$, $r_{eq} = r_1 + r_2$. Internal resistance still adds.