n Cells of Unequal EMF in Series

Consider $n$ cells of EMFs $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n$ and internal resistances $r_1, r_2, \ldots, r_n$, all connected in series so as to aid one another (each cell's positive terminal joined to the next cell's negative terminal, all driving current the same way around the loop). Since the same current $I$ flows through every cell in the chain, applying Kirchhoff's loop rule (Section 3.11) around the complete circuit, with an external resistance $R$, gives

so that the combination behaves exactly as a single equivalent cell with

-- i.e., EMFs and internal resistances of series-connected cells simply add algebraically, REGARDLESS of whether the individual cells are identical or not, exactly as the equal-cell case of Section 3.10 already showed for the special case $\varepsilon_1=\varepsilon_2=\cdots=\varepsilon$ and $r_1=r_2=\cdots=r$ (where the sums collapse to $n\varepsilon$ and $nr$).

If one cell in the chain happens to be connected with REVERSED polarity relative to the others (so it opposes rather than aids the net current), its own EMF must be entered into the sum with a NEGATIVE sign (it is being charged by the rest of the chain rather than discharging), while its internal resistance still adds normally with a positive sign, since internal resistance always opposes current regardless of the cell's polarity: …