A battery cannot be made directly like a single cell; it's constructed by connecting multiple cells in series or parallel, depending on the desired output. An ordinary cell has an e.m.f. of 1.5v and can deliver up to 1/8 ampere of current continuously. If the desired output needs to increase, then by connecting the cells in series, the voltage can be added, and by connecting the cells in parallel, the current is added.

19.1 : Cells in Series

If the negative terminal of the first cell is linked to the positive terminal of the second, and so on, the cells are said to be connected in series.

When the voltage is required more than the single cell e.m.f., then by connecting the number of cells in series, the voltage can be added to meet the requirement.

Consider a resistor R that is connected in series with n cells with an internal resistance of r and e.m.f. E as shown in figure

Total battery e.m.f. = n E

Internal resistance of battery = n r

Total resistance of the circuit = R + n r

Current, I = n$\frac{\mathrm{E}}{\mathrm{R+nr}}$

In some cases the R is might be less than or greater than the internal resistance .They are

Case 1:

If R >> n r. then n r can be neglected as compared to R

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

= n * current due to one cell

Case 2:

If R << n r, then R can be neglected as compared to R

I = n $\frac{\mathrm{E}}{\mathrm{nr}}$

=$\frac{\mathrm{E}}{\mathrm{r}}$ = current due to one cell

Condition for maximum current:

In order to get maximum current in series combination, the external resistance (R) should be very high as compared to the internal resistance of the battery.

19.2 Cells in parallel

The cells are said to be parallel if the positive terminal and negative terminal of n cells are connected to a common point.

When the current is required more than a single cell current, then by connecting the number of cells in parallel, the current can be added to meet the requirement.

Consider a resistor R connected in series with n cells with an internal resistance of r and e.m.f. E as shown in figure.

Since all cells are connected in parallel, so the internal resistance is also in parallel, then the total internal resistance will be

$\frac{1}{{\mathrm{r}}_{\mathrm{eq}}}$=$\frac{1}{{\mathrm{r}}_1}$+$\frac{1}{{\mathrm{r}}_2}$+$\frac{1}{{\mathrm{r}}_3}$+......+m terms

= 1+1+1+ ………..m terms/r

=$\frac{\mathrm{m}}{\mathrm{r}}$

r_eq=$\frac{\mathrm{m}}{\mathrm{r}}$

Total circuit resistance = R + r_eq

= R + r/m

Current I =$\frac{\mathrm{E}}{\mathrm{R}}$+($\frac{\mathrm{r}}{\mathrm{m}}$)

I=$\frac{\mathrm{mE}}{\mathrm{mR+r}}$

In some cases the R is might be less than or greater than the internal resistance .They are,

Case i

If R << r , then mR may be neglected as compared to r

I=$\frac{\mathrm{mE}}{\mathrm{r}}$

= m * current due to one cell

Case ii

If R>> r, then r may be neglected as compared to m R.

I=$\frac{\mathrm{mE}}{\mathrm{mR}}$

= E /R = current due to one cell

Condition for maximum current:

In order get maximum current in parallel combination the external resistance (R) should be very low as compared to the internal resistance of each cell

19.3 Cells in parallel and series combination

The cells are said to be a series-parallel combination if the cells are connected in parallel and all parallel cells are connected in series, as shown in figure.

To get more current, the cells are connected in parallel, and to get more voltage along with current, all the parallel cells are connected in series, so the battery can generate more voltage and current than the single cell.

Consider external resistance R connected to the number of cells in series connected in parallel as shown in figure

Internal resistance of one series of the cells = n r

There are many parallel series, each having an internal resistance of n r If the number of n r is connected in parallel, then the total internal resistance of the battery is

$\frac{1}{{\mathrm{rt}}_{}}$=$\frac{1}{{\mathrm{nr}}_{}}$+$\frac{1}{{\mathrm{nr}}_{}}$+......+m terms

= 1+1+1+…..m terms/n r

=$\frac{\mathrm{m}}{\mathrm{nr}}$

rt=$\frac{\mathrm{nr}}{\mathrm{m}}$

Total circuit resistance = R +$\frac{\mathrm{nr}}{\mathrm{m}}$

Total battery e.m.f. = e.m.f. due to series of cells = nE

Current, I = $\frac{\mathrm{nE}}{\mathrm{R+(nr/m)}}$

=$\frac{\mathrm{mnE}}{\mathrm{mR+nr}}$

I=$\frac{\mathrm{mnE}}{\mathrm{mR+nr}}$

Condition for maximum current:

I=$\frac{\mathrm{mnE}}{\mathrm{mR+nr}}$

I=$\frac{\mathrm{mnE}}{{(\sqrt{mR}-\sqrt{nr})}^2+2\sqrt{\mathrm{mR}}\sqrt{\mathrm{nr}}}$

As m n E are constant then current will be maximum when denominator is minimum then,

$\frac{{(\sqrt{\mathrm{mR}}-\sqrt{\mathrm{nr}})}^2}{}$=0

Or

m R = n r

R = nr/m

The external resistance = total internal resistance of the battery

In order to get maximum current in the battery, the external resistance is equal to the total internal resistance of the battery.