Cells in Parallel

In a PARALLEL combination of cells, all the POSITIVE terminals of the cells are joined together at one common node, and all the NEGATIVE terminals are joined together at another common node (Fig. 11.13b). Here the total current supplied to the external circuit divides among the individual cells (branches) -- $I_1$ from the first cell, $I_2$ from the second, and so on -- rather than the voltage dividing, as it does in series.

Consider two cells connected this way, with their shared terminals at potentials $V_{B1}$ and $V_{B2}$. For the FIRST cell (emf $\varepsilon_1$, internal resistance $r_1$, delivering current $I_1$), the potential difference across its own terminals is

and, because points $B_1$ and $B_2$ connect to the SECOND cell in exactly the same way, that cell gives the analogous pair

Since the total current is $I=I_1+I_2$, substituting the two branch currents from Eq. (11.46) and its analogue for cell 2, and collecting terms, leads (after some algebra) to

If this two-cell combination is now replaced by a SINGLE equivalent cell of emf $\varepsilon_{eq}$ and internal resistance $r_{eq}$ connected between the same points $B_1$ and $B_2$,

comparing Eq. (11.49) and Eq. (11.50) term by term gives the equivalent internal resistance and equivalent emf of the parallel pair:

These generalise for $n$ cells (emfs $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_n$; internal resistances $r_1,r_2,\dots,r_n$) connected in parallel to

When substituting emf values into these formulas, each $\varepsilon_i$ must be given its correct ALGEBRAIC sign according to the actual polarity with which that particular cell has been connected into the combination. …