Question

How cells are combined in parallel? Derive the expression for current flowing in the external circuit. When is this combination useful.

Answer 1

Parallel combination of cells: In this combination the positive plates of all the cells are connected together at a point and the their negative plates are connected at another point and a suitable resistance is connected between these two points.
Expression for the current in the external circuit: Let m cells be connected in parallel combination as shown in Fig. The internal resistance of each cell is r and the e.m.f. of each cell is E. The resistance of the external circuit is R.
If the equivalent internal resistance of the combination is X then
$\frac{1}{X}=\frac{1}{r}+\frac{1}{r}+\frac{1}{r}+........$upon m terms
or $\frac{1}{X}=\frac{m}{r}$
or $X=\frac{r}{m}$
Total resistance of the circuit $=\frac{r}{m}+R$
Since the cells are connected in parallel thus the total e.m.f. of the circuit will be equal to the e.m.f. E of a single cell.
Thus, current in the circuit, $I=\frac{Totale.m.f.}{Totalresistance}=\frac{E}{\frac{r}{m}+R}$
or $I=\frac{mE}{r+mR}$
This is the required expression.
Advantageous combination- If $r>>mR$, then mR is considered negligible.
$I=\frac{mE}{r}$
or $I=m\times$current obtained from one cell
Thus, this combination is advantageous when the internal resistance (r) of the cells is high, because the current obtained from a single cell becomes m times. In brief, the parallel combination of cells advantageous only when the internal resistance of cell is very large as compared to the external circuit.