Grouping of Cells

Grouping of Cells

To get high voltage or current, a number of cells are combined in a group. The combination of cells thus obtained is called a battery. Cells can be grouped into series, parallel and mixed grouping.

Cells in Series

Consider n identical cells, each of emf E and internal resistance r, connected in series across an external resistance R as shown in figure.

\begin{align*} \text {Total emf of the battery} &= n E \\ \text {Internal resistance of the battery} &= r+ r+\dots + r = nr \\\text {Total circuit resistance} &= R + nr \\ \text {The current on the circuit, I} &= \frac {\text {total emf}}{\text {total resistance }}\\ &= \frac {nE}{R + nr} \\\end{align*}

Special Cases

1. If R >> nr, then nr can be neglected as compared to R.
   $$ \therefore I = n\frac ER = n \times text {current due to one cell.}$$
2. If R<< nr, then R can be neglected as compared to nr.
   $$ \therefore I = \frac {nE}{nr} = \frac Er = \text {current due to one cell.} $$

Hence, the maximum current is obtained in a series combination of cells when the external resistance R is very high as compared to the total internal resistance of the battery, nr.

Cells in Parallel

Consider m identical cells each of emf E and internal resistance r connected in parallel across an external resistance R as shown in figure.

\begin{align*} \text {Emf of the battery between A and B} &= E \\\text {Total internal resistance of the battery is given by} \\ \frac {1}{r_T} &= \frac 1r + \frac 1r + \dots + \frac 1r \\ &= \frac nr \\\text {or,} \: r_T &= \frac rn \\ \text {Total circuit resistance} &= R + \frac rm \\ &= \frac {mR + r}{m} \\ \text {Total current in the circuit, I} &= \frac {\text {total emf}}{\text {total resistance}} \\ &= \frac {E}{(mR + r)/m} \\ &= \frac {mE} {mR + r} \\ \end{align*}

Special Cases

1. If R << r, then mR can be neglected as compared to r.
   $$ \therefore I = m\frac Er = m \times \text {current due to one cell.}$$
2. If r<< R, then r can be neglected as compared to mR.
   $$ \therefore I = \frac {mE}{mR} = \frac ER = \text {current due to one cell.} $$

Hence, the maximum current is obtained in a parallel combination of cells, when the external resistance R is very low.

Mixed Grouping of Cells

Consider N identical cells each of emf E and internal resistance r, connected in a group with n cell in series and m such rows are connected in parallel across an external resistance R.

\begin{align*} \text {Total number of cells N} &= mn \\ \text {Internal resistance of n cells connected in series} &= nr \\ \text {Total internal resistance of N cells} &= \frac {nr}{m} \\ \text {Total emf of N cells} \\ &= \text {emf of n cells in a row} = nE \\ \text {Total resistance of the circuit} &= R + \frac {nr}{m} \\ \text {Current in the circuit, I} &= \frac {mnE}{mR + nr} \\ \end{align*}

\begin{align*} \text {Current I will be maximum if} \: mR + nr \: \text {is minimum.} \\ \text {Above equation can be written as } \\ I &= \frac {mnE}{(\sqrt {mR} - \sqrt {nr})^2 + 2\sqrt {mnrR}} \\ \text {For I to be maximum} \: \sqrt {mR} - \sqrt {nr} = 0 \\ \text {or,} \: mR &= nr \\ \text {or,} \: R &= \frac {nr}{m} \\ \end{align*}

Hence in order to get maximum current in mixed grouping of cells, the external resistance R should be equal to the total internal resistance of the battery in mixed grouping.

Efficiency of a Cell

In general, the efficiency η of any system is defined by

$$ \eta = \frac {\text {output power}}{\text {input power}} $$

Consider a cell of emf E and internal resistance r delivering power to a resistance R. if I be the current in circuit, then power developed \( = I^2 (r + R) = EI \)

\begin{align*} \text {The power wastage in the internal resistance of cell} &= I^2r \\ \text {Output power} &= I^2R \\ \therefore \eta &= \frac {I^2R}{I^2R + I^2r} \\&= \frac {R}{R + r} \\ &= \frac {1}{1 + \left (\frac rR \right )} \\ \end{align*}

For higher efficiency, external resistance must be of greater value.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, et al. Principles of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.