Two Cells of Unequal EMF in Parallel

Consider two cells of EMFs $\varepsilon_1, \varepsilon_2$ and internal resistances $r_1, r_2$, connected in parallel between two common nodes $A$ and $B$ (like terminals joined together), with an external resistance $R$ connected across $A$ and $B$. Let $I_1$ be the current supplied by cell 1, $I_2$ the current supplied by cell 2, and $I$ the current through the external resistance $R$; by Kirchhoff's junction rule at node $A$,

Let $V$ be the potential difference across the common terminals $A$ and $B$ (also the potential difference across $R$, so $V = IR$). Applying Kirchhoff's loop rule separately to each cell's own branch gives

Substituting both into the junction-rule equation $I = I_1+I_2 = V/R$ and solving for $V$ leads, after simplification, to the pair behaving as a SINGLE equivalent cell of EMF $\varepsilon_{\text{eq}}$ and internal resistance $r_{\text{eq}}$ given by

so that $V = \varepsilon_{\text{eq}} - I r_{\text{eq}}$, exactly the same form as for a single cell (Section 3.9), with $I = I_1+I_2$ playing the role of the total current drawn from this equivalent cell. Notice that the equivalent internal resistance follows the ordinary parallel-resistor rule, but the equivalent EMF is a resistance-WEIGHTED average of the two individual EMFs, not a simple average -- the cell with the SMALLER internal resistance contributes more strongly to $\varepsilon_{\text{eq}}$, since it is more effective at maintaining its own EMF against the common terminal voltage. …