**Electromotive Force (emf):**

*The property of a cell by virtue of which the charge moves in the circuit in a particular direction is known as electromotive force (e.m.f). The electromotive force is not a force but just the historical name*.

The emf of a cell can be defined as the work done by the cell to move unit positive charge (+1C) throughout the complete electrical loop. It can also be defined as the potential difference between the two terminals of the cell in open circuit (i.e. when no current flows). It is denoted by ‘E’. Its unit is volt.

Mathematically,

emf, E = $\frac{Work\text{ }done\text{ }(energy)}{charge}$

## Terminal potential difference:

Terminal potential difference of a cell is defined as the amount of work done by the cell to move a unit positive charge (+1C) through external/load resistance. It can also be defined as the potential difference between the two terminals of the cell in closed circuit (i.e. when current flows). It is denoted by ‘V’. It’s unit is volt.

Mathematically,

Terminal potential difference, V = $\frac{Work\text{ }done\text{ }(energy)}{charge}$

## Internal resistance:

The resistance offered by electrolyte between the electrodes of a cell is known as internal resistance. It is denoted by ‘r’. Its unit is Ohm ($\Omega $).

**Factors on which the internal resistance of a cell depends**

Factors on which the internal resistance of a cell depends are:

(i) Nature of electrolyte:

Internal resistance (r) is inversely proportional to conductivity ($\sigma $)

(ii) Distance between the electrodes: “internal resistance, r” is directly proportional to distance

(iii) Area of electrodes:

Internal resistance (r) is inversely proportional to area of electrodes (A)

(iv) Concentration:

Internal resistance (r) is inversely proportional to concentration of electrolyte

(v) Temperature:

Internal resistance (r) is inversely proportional to temperature of electrolyte.

**Relation between emf (E), terminal p.d. (V) and internal resistance (r):**

*Fig: circuit of emf, terminal p.d. and internal resistance*

Let us consider a cell of e.m.f. (E) and internal resistance (r) is connected with external resistance (R). Let (I) be the current flowing through the circuit and terminal potential difference is (V). Since internal resistance always in series combination with external resistance, so total resistance of circuit becomes R+r.

Current, I = $\frac{Total\text{ }e.m.f.}{Total\text{ }resistance}$

or, I = $\frac{E}{R+r}$

or, E = I (R + r)

or, E = IR + Ir

or, E = V + Ir (where V = IR)

which is the relation between e.m.f., terminal p.d and internal resistance for discharging circuit.

In case of of charging circuit, current flows in reverse direction. So, the above relation becomes

E = V–Ir

## Cell:

It is a device which can convert chemical energy to electrical energy.

The combination of cells is known as battery.

## Grouping of cells:

The cells can be grouped in three ways; (i) Series combination of cells, (ii) Parallel combination of cells and (iii) Mixed combination of cells.

### (i) Series combination:

Let us consider ‘n’ identical cells each of emf ‘E’ and internal resistance ‘r’ are connected in series across an external resistance ‘R’.

Then, total e.m.f. = nE

Total internal resistance = nr

Total resistance of circuit, R_{eq} = R + nr

Now current, I = $\frac{Total\text{ }e.m.f.}{Total\text{ }resistance}$

I = $\frac{nE}{R+nr}$

This is the required expression for current in the series combination of cells.

**Special cases:**

(i) If R >>>> nr

In this case, value of nr can be neglected as compared to R

I = $\frac{nE}{R+nr}$

$\therefore $ I = $\frac{nE}{R}$

It is the condition of maximum current.

(ii) If nr >>>>R

In this case, value of R can be neglected as compared to nr.

I = $\frac{nE}{R+nr}$

Or, I = $\frac{nE}{nr}$

Or, I = $\frac{E}{r}$

It is the condition of current due to one cell only.

### (ii) Parallel combination:

Let us consider ‘n’ identical cells each of emf ‘E’ and internal resistance ‘r’ are connected in parallel across an external resistance ‘R’.

Then, total e.m.f. = e.m.f. due to one cell = E

and total internal resistance = r’

Or, $\frac{1}{r’}$ = $\frac{1}{r}$ + $\frac{1}{r}$ + $\frac{1}{r}$……….upto m-cells

or, $\frac{1}{r’}$ = $\frac{m}{r}$

or, r′ = $\frac{r}{m}$

$\therefore $Total resistance of circuit, R_{eq} = R + $\frac{r}{m}$

Now,

Current, I = $\frac{Total\text{ }e.m.f.}{Total\text{ }resistance}$

I = $\frac{E}{R+r/m}$

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

This is the required expression for current in the parallel combination of cells.

**Special cases:**

(i) If mR >>>>r,

In this case, value of ‘r’ can be neglected as compared to mR

We have, I = $\frac{mE}{mR+r}$

Or, I = $\frac{mE}{mR}$

$\therefore $ I = $\frac{E}{R}$

This is the condition of current due to one cell only.

(ii) If r >>>> mR

In this case value of mR can be neglected as compared to r.

We have I = $\frac{mE}{mR+r}$

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

This is the condition of maximum current.

### (iii) Mixed combination of cells:

Let us consider a number of identical cells each of emf ‘E’ and internal resistance ‘r’ are connected in mixed combination in m-rows and n-columns. The combination is then connected across an external resistance ‘R’.

Then, total e.m.f. = e.m.f. due to cells in each row = nE

Internal resistance in each row = r + r + r …..upto ‘n’ cells = nr

Total internal resistance = r’

$\frac{1}{r’}$ = $\frac{1}{nr}$ + $\frac{1}{nr}$ + $\frac{1}{nr}$……….m-cells

or, $\frac{1}{r’}$ = $\frac{m}{nr}$

or, r′ = $\frac{nr}{m}$

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

Now,

Current, I = $\frac{Total\text{ }e.m.f.}{Total\text{ }resistance}$

I = $\frac{E}{R+nr/m}$

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

This is the required expression for current in the mixed combination of cells with m-rows and n-columns.

**Condition for maximum current:**

For maximum current,

mR + nr = Minimum

Or, ($\sqrt{mR}$ –$\sqrt{nr}$ )^{2} = 2 $\sqrt{mR}$×$\sqrt{nr}$ = minimum

Or, ($\sqrt{mR}$ –$\sqrt{nr}$ )^{2} = minimum = 0

Or, $\sqrt{mR}$ = $\sqrt{nr}$

Or, mR = nr

Or, R = $\frac{nr}{m}$

Hence for maximum current, the external resistance R should be equal to the total internal resistance of combination of cells.

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