Cells in Series and Parallel: Class 12 Physics Notes, Definition, Working Principle, Formula & Real-Life Applications

A cell or an electrochemical cell is a single-unit device that generates electrical energy through chemical reactions. There are two main types of electrochemical cells: including Galvanic/Voltaic or Electrolytic cells. An electrochemical cell consists of an anode and cathode (two electrodes), electrolyte solutions containing ions, a salt bridge/porous barrier and an external circuit that connects the electrodes in galvanic cells. 

Please note:

Three questions are asked from Current Electricity chapter which includes a topic on Combination of cells in series and parallel. It is important to have a thorough understanding of the chapter from JEE exam point of view. 

 

Equivalent EMF 

${\mathrm{E}}_{\mathrm{e}\mathrm{q}}={\mathrm{E}}_1+{\mathrm{E}}_2+\dots \dots .+{\mathrm{E}}_{\mathrm{n}}\mathrm{}\text{[write EMF's with polarity]}$

Equivalent internal resistance $r_{\text{eq}}=r_1+r_2+r_3+r_4+\dots +r_n$
If $n$  cells each of emf $E$ , arranged in series and if $r$  is internal resistance of each cell, then total emf $=\mathrm{n}\mathrm{E}$  so current in the circuit

$\mathrm{I}=\frac{\mathrm{n}\mathrm{E}}{\mathrm{R}+\mathrm{n}\mathrm{r}}$

If $\mathrm{n}\mathrm{r}\ll \mathrm{R}$  then $\mathrm{I}=\frac{\mathrm{n}\mathrm{E}}{\mathrm{R}}⟶\mathrm{}$  Series combination is advantageous.
If $\mathrm{n}\mathrm{r}\gg \mathrm{R}$  then $\mathrm{I}=\frac{\mathrm{E}}{\mathrm{r}}\mathrm{}⟶\mathrm{}$  Series combination is not advantageous.
Note : If polarity of $m$  cells is reversed, then equivalent emf $=(n-2m)E$  while the equivalent resistance is still $nr+R$ , so current in $R$ will be $\mathrm{i}=\frac{(\mathrm{n}-2\mathrm{m})\mathrm{E}}{\mathrm{n}\mathrm{r}+\mathrm{R}}$

When you put electrochemical cells in series, the following happens:

1. EMFs Add Up

Each cell has its own open-circuit voltage (EMF) 𝐸. When cell are put in series combination, the total EMF is the sum:

E total = E1 + E2 +⋯+En 

So three 1.5 V cells in series will result in 4.5 V at the terminals.

2. Internal Resistances Will Add Up

Every cell has a little internal resistance 𝑟. In series, these resistance will add up. 

𝑟total = 𝑟1 + 𝑟2 + ⋯+ 𝑟𝑛

That extra resistance slightly limits how much current you can draw.

3. When You Connect a Load

Once you attach a load 𝑅L, Kirchhoff’s loop law gives

$E_{\mathrm{total}} = I ( R{L}_{} + r_{\mathrm{total}} ) \Rightarrow I = \frac{E_{\mathrm{total}}}{ R{L}_{} + r_{\mathrm{total}} }$

The same current 𝐼 flows through every cell

The following industries use the combination of cell in series:

Industry	Use Case
Consumer Electronics	Digital cameras and portable gaming devices use multiple cells in series
Automotive Industry	Car batteries (~12V) consist of six 2V cells that are connected in series Electric vehicles use hundreds of cells in series to create high-voltage battery packs (~400-800V)
Renewable Energy	Cells are connected in series in solar panels to increase the voltage output

 

${\mathrm{E}}_{\mathrm{e}\mathrm{q}}=\frac{{\epsilon }_1/{\mathrm{r}}_1+{\epsilon }_2/{\mathrm{r}}_2+\dots .+\frac{{\epsilon }_{\mathrm{n}}}{r_{\mathrm{n}}}}{1/{\mathrm{r}}_1+1/{\mathrm{r}}_2+\dots ..+1/{\mathrm{r}}_{\mathrm{n}}}\mathrm{}$  [Use emf's with polarity]

$\frac{1}{r_{eq}}=\frac{1}{r_1}+\frac{1}{r_2}+\dots .+\frac{1}{r_n}$
If $m$  cells each of emf $E$  and internal resistance $r$  be connected in parallel and if this combination is connected to an external resistance then equivalent emf of the circuit $=\mathrm{E}$ .

Internal resistance of the circuit $=\frac{r}{m}$ .
and $I=\frac{E}{R+\frac{r}{m}}=\frac{mE}{mR+r}$ .

If $mR\ll r;I=\frac{mE}{r}⟶$ then, parallel combination of cells is advantageous.
If $mR\gg r;I=\frac{E}{R}⟶$ then, parallel combination of cells is not advantageous.

The following points explain the applications of cell in parallel:

Industry	Use Case
Consumer Electronics	Smartphones and laptops use parallel cell configurations to increase battery capacity and support higher current demands without increasing voltage
Automotive Industry	Electric vehicle battery packs combine parallel cell groups to increase current output and range while maintaining voltage requirements
Renewable Energy	Home battery storage systems connect cells in parallel to increase energy storage capacity and handle peak loads
Industrial Applications	Data centers employ parallel battery arrangements in UPS systems to handle high current loads
Marine/RV	Boats and recreational vehicles use parallel battery banks to extend runtime for onboard systems without changing system voltage
Telecommunications	Cell towers and network equipment use parallel battery systems for extended backup power during outages
Medical Equipment	Medical carts and portable equipment use parallel battery configurations for longer runtime between charges

$\mathrm{m}\mathrm{n}=$  number of identical cells.
$\mathrm{n}=$  number of rows
$\mathrm{m}=$  number of cells in each row.
The combination of cells is equivalent to single cell of $\mathrm{e}\mathrm{m}\mathrm{f}=\mathrm{m}\mathrm{E}$
and internal resistance $=\frac{\mathrm{m}\mathrm{r}}{\mathrm{n}}$

Current $I=\frac{mE}{R+\frac{mr}{n}}$
For maximum current $n\mathrm{R}=\mathrm{m}\mathrm{r}$
or $R=\frac{\mathrm{m}\mathrm{r}}{\mathrm{n}}=$  internal resistance of the equivalent battery.

$I_{\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{nE}{2r}=\frac{mE}{2R}.$