Parallel Combinations of Cells

A parallel combination of cells is a way of connecting batteries where all the positive terminals are linked together, and all the negative terminals are connected as well. This setup keeps the overall voltage the same as just one cell, but it increases how much current the system can supply. It's especially useful when you need your power source to last longer without raising the voltage. Another advantage is that if one cell stops working, the others can still keep the device running. However, it's important to use cells with the same voltage to prevent damage or uneven performance.

1.0Definition of Parallel Combination of Cells

• When all the positive terminals of the cells are connected to a common point and all the negative terminals to another common point, the cells are said to be connected in parallel.
• The total current supplied to the external circuit is the sum of the currents supplied by individual cells.
• If one cell fails (open circuit), others can still supply power — greater reliability in some cases.
• Since voltage remains the same, the load is shared among all cells, reducing the burden on any single cell.

2.0Expression For Parallel Combination of Cells

• Two cells of emfs ${\epsilon }_1$ and ${\epsilon }_2$ and internal resistances $r_1$ and $r_2$  are connected in parallel between two points.
• Currents $I_1$ and $I_2$ from the positive terminals of the two cells flow towards the junction B1 and current I flows out, $I=I_1+I_2$
• As the two cells are connected in parallel between the same two points $B_1$ and $B_2$, the potential difference V across both cells must be the same.
• The potential difference between the terminals of first cell is

$\begin{array}{rl}V& =V_{B_1}-V_2={\epsilon }_1-I_1{r}_1\\ I_1& =\frac{{\epsilon }_1-V}{r_1}\end{array}$

• The potential difference between the terminals of second cell is

$\begin{array}{rl}V& =V_{B_1}-V_2={\epsilon }_2-I_2{r}_2\\ I_2& =\frac{{\epsilon }_2-V}{r_2}\end{array}$

Hence $I=I_1+I_2=\frac{{\epsilon }_1-V}{r_1}+\frac{{\epsilon }_2-V}{r_2}$

$\begin{array}{rl}& I=\left(\frac{{\epsilon }_1}{r_1}+\frac{{\epsilon }_2}{r_2}\right)-V\left(\frac{1}{r_1}+\frac{1}{r_2}\right)\\ & V\left(\frac{r_1+r_2}{r_1{r}_2}\right)=\frac{{\epsilon }_1{r}_2+{\epsilon }_2{r}_1}{r_1{r}_2}-I\\ & V=\frac{{\epsilon }_1{r}_2+{\epsilon }_2{r}_1}{r_1{r}_2}-I\frac{r_1{r}_2}{r_1+r_2}\end{array}$

When cells support each other

$\begin{array}{r}{\epsilon }_{eq}=\frac{{\epsilon }_1{r}_2+{\epsilon }_2{r}_1}{r_1{r}_2}\\ r_{eq}=\frac{r_1{r}_2}{r_1+r_2}\end{array}$

When cells are connected with opposite polarity

$\begin{array}{rl}{E}_1>E_2& \\ {\epsilon }_{eq}=\frac{{\epsilon }_1{r}_2-{\epsilon }_2{r}_1}{r_1{r}_2}& \\ r_{eq}& =\frac{r_1{r}_2}{r_1+r_2}\end{array}$

Note: If n identical cells are connected in parallel and then :-

$\begin{array}{rl}{E}_{eq}& =n\frac{\frac{E}{r}}{\frac{n}{r}}\\ E_{eq}& =E\\ r_{eq}& =\frac{r}{n}\end{array}$

Current in the circuit $i=\frac{E}{R+\frac{r}{n}}=\frac{nE}{nR+r}$

If $r\ll nR\Rightarrow i=\frac{E}{R}=$ Current from any one cell when connected with the external resistance.

If $r\gg nR\Rightarrow i=\frac{nE}{R}=n\times$ Current from any cell when short-circuited