Parallel and Mixed Grouping of Cells - JEE Important Topic

In an electric circuit, we have seen several ways of grouping the resistors and other electrical components. The most common ways are series and parallel. Likewise, we can group the cells in the circuits. In a series circuit, all the cells are connected side by side through the wires and the same value of current flows through all the components; the potential difference may or may not differ across all the components.

In a parallel circuit, the cells are connected in a parallel format and the potential difference is the same across all the components and the current may or may not be the same through all the components. In this article, we will see the cells connected in series, parallel, and mixed form.

Resistance in Series Formula

When all the resistances are connected side by side in a circuit, then that system is called the series system. In this case, the same value of current flows through all the resistances.

Consider that we have ‘n’ resistances of resistance ‘R’ connected in series in the circuit. The battery has an emf of ‘E’ and internal resistance ‘r’. The total resistance will become Rt = nR + r. The current will be $I=\frac{E}{nR+r}$.

Electric Cell

The electric cell diagram describes the whole structure of an electric cell. A typical electric cell diagram consists of a cylindrical cell having a cathode, anode, metal cap, and electrolytes. The metal cap acts as a positive terminal and the base acts as a negative terminal. In such cells, the redox reaction occurs and the flow of electrons is generated. Such flow of electrons produces an electric current.

Electric Cell

Combination of Two or More Cells

There are three types of combinations of cells:

• Series Combination: In a series combination, all the cells are connected side by side. The current through all the cells is the same. When two or more cells are connected in series, the combination is called ‘Battery’.

Series Combination

• Parallel Combination: In a parallel combination, all the cells are connected in a parallel manner. The voltage across all the cells may or may not be the same, depending upon the nature of all the cells.

Parallel Combination

• Mixed Combination: In a mixed combination, some cells are in series and other cells are in parallel.

Mixed Combination

Formula for Cells in Parallel

In a parallel combination, the cells are connected in parallel form. Consider ‘n’ cells of emf E1, E2, E3,…, En and internal resistance r1, r2, r3,…, rn connected in parallel across a resistor of resistance ‘R’. The net current from all the cells will add up at the junction and form the equivalent current.

$ {{I}_{eq}}={{I}_{1}}+{{I}_{2}}+{{I}_{3}}+...+{{I}_{n}} $ 

$\frac{{{E}_{eq}}}{{{R}_{eq}}}=\frac{{{E}_{1}}}{{{r}_{1}}}+\frac{{{E}_{2}}}{{{r}_{2}}}+\frac{{{E}_{3}}}{{{r}_{3}}}+...+\frac{{{E}_{n}}}{{{r}_{n}}} $ 

$\frac{1}{{{R}_{eq}}}=\frac{1}{{{r}_{1}}}+\frac{1}{{{r}_{2}}}+\frac{1}{{{r}_{3}}}+...+\frac{1}{{{r}_{n}}}$

Formula for Cells in Series

In a series combination, the cells are connected in series form or side by side. Consider ‘n’ cells of emf E1, E2, E3,…, En and internal resistance r1, r2, r3,…, rn connected in parallel across a resistor of resistance ‘R’. 

The value of current passing through all the cells will be the same. When two or more cells are connected in series, the emf of all the cells add up and we get a net emf. Also, since all the internal resistances are connected in series, the resistances will add up and we will get a net resistance.

 $ {{E}_{eq}}={{E}_{1}}+{{E}_{2}}+{{E}_{3}}+...+{{E}_{n}} $ 

 $ {{R}_{eq}}={{r}_{1}}+{{r}_{2}}+{{r}_{3}}+...+{{r}_{n}} $ 

Formula for Cells in Mixed Combination

In a mixed combination, the cells are connected both in series and parallel format. Consider ‘m’ rows in a parallel combination such that each row consists of ‘n’ cells of emf ‘E’ and internal resistance ‘r’. Consider that the whole combination is across an external resistance ‘R’. Here, ‘mn’ is a constant. 

In this case, the current is given by $I=\frac{mnE}{mR+nr}$.

Also, the maximum current is drawn from the battery when external resistance matches the net internal resistance. The maximum current is given by ${{I}_{\max }}=\frac{mE}{2r}$.

Conclusion

The cells can be combined in various ways according to the need and necessity. Like, the cells connected in parallel have the advantage that if one cell gets damaged, it will not harm the working of other cells. While the cells connected in series give greater resultant voltage than individual cells and such a combination does get overheated. Such combinations are used in batteries, households, power plants, transformers, and other electrical sites.