Cells in series :

As shown in fig. suppose two cells of emf ɛ₁ and ɛ₂ and internal resistances r₁ and r₂ are connected in series between points A and C. let I be the current flowing through the series combination.

Let V_A,V_B,V_C the potentials at points A, B and C respectively. The potential difference across the terminal of the two cells will be

V_AB = V_A – V_B = ɛ₁ – Ir₁ and

V_BC = V_B - V_C = ɛ₂ – Ir₂

Thus the potential difference between the terminals A and C of the series combination is

V_AC = V_A – V_C = (V_A - V_B) + (V_B - V_C) = (ɛ₁ – Ir₁) + (ɛ₂ – Ir₂)

V_AC = ɛ₁ + ɛ₂ – I(r₁ + r₂)

If we wish to replace the series combination by a single cell of emf ɛ _equ and internal resistance r_equ then

VAC = ɛ_equ – I r_equ

Comparing the last two equation we get

ɛ_equ = ɛ= + ɛ₂

and

r_equ = r₁ + r₂

for a series combinations of n cells,

ɛ_equ = ɛ₁ + ɛ₂+……………………………+ ɛ_n

r_equ = r₁ + r₂+……………………………+ r_n

condition for maximum current from a series combination of cells :

 total emf of n cells in series = sum of emfs of all cells = nξ

total internal resistance of n cells in series

= r + r + r ………… + r = nr

Total resistance in the circuit =  R + nr

The current in the circuit is

Special cases :

1. if R >> nr, then    

2. If R << nr, then   

Thus, when external resistance is much higher than the net internal resistance, the cells should be connected n series to get maximum current.

Thus, when external resistance is much higher than the net internal resistance, the cells should be connected n series to get maximum current.

Cell in parallel :

When the positive terminals of all cells are connected to one point and all their negative terminals to another points, the cell are said to be connected in parallel. In parallel combinations the potential differences across the cells remains same. The potential difference between the terminals of the first cell is

V = V_B1 – V_B2 = ξ₁ – I₁r₁

The potential difference between the terminals of second cell is

V = V_B1 – V_B2 = ξ₂ – I₁r₂

If we wish to replace the parallel combination by a single cell of emf ξ_equ and internal resistance r_equ then,

V = ξ_equ – r_equ

Comparing the last two equations, we get

We can express the above results in a simpler way as follows :

Condition for maximum current :

As shown in fig, suppose m cells each of internal resistance r is connected in parallel is given by

Total resistance is

Therefore the current in the circuit is given by

Special cases :

Thus, when external resistance is much smaller than the net internal resistance, the cells should be connected in parallel to get maximum current.

Mixed grouping of cells:

 In this combination, a certain number of identical cells are joined in series and all such rows are then connected in parallel with each other.

Total number of cells = mn

Net emf of each row of n cells in series and also total emf of parallel combination = n ξ

Total internal resistance is given by

Total resistance of the circuit

The current through the external resistance R,

Clearly, the current I will be maximum if the denominator (mR + nr) is minimum. Now,

 As the perfect square cannot be negative, so mR + nr will be minimum if

mR = nr

Or

External resistance = total internal resistance of the cells.