Like resistors, cells can be combined together in electric circuits to get more current or voltages.
Cells in Series :
Cells are connected in series when they are joined end to end so that the same quantity of electricity must flow through each cell.
Consider two cells in series (Fig.), where one terminal of the two cells is joined together leaving the other terminal in either cell free. ε1, ε2 are the emf’s of the two cells and r1, r2 their internal resistances, respectively.
Let V (A), V (B), V (C) be the potentials at points A, B and C shown in Fig. Then V (A) – V (B) is the potential difference between the positive and negative terminals of the first cell.
VAB = V (A) – V (B) = ε1 - I r1
Similarly, VBC = V (B) – V (C) = ε2 - I r2
Hence, the potential difference between the terminals A and C of the combination is
VAC = V (A) – V (C) = V (A) – V (B) + V (B) – V (C) = (ε1 + ε2) - I(r1 + r2)
If we replace the combination by a single cell between A and C of emf εeq and internal resistance req, we get
VAC = εeq – I req
Comparing the last two equations, we get,
εeq = ε1 + ε2 and req = r1 + r2
If we connected the negative electrode of the first to the positive electrode of the second. If instead we connect the two negatives, Eq. would change to VBC = – ε2 –I r2 and we will get,
εeq = ε1 – ε2 (ε1 > ε2)
The rule for series combination clearly can be extended to any number of cells:
(i) The equivalent emf of a series combination of n cells is just the sum of their individual emf’s, and
(ii) The equivalent internal resistance of a series combination of n cells is just the sum of their internal resistances. Thus in series combination of cells
(1) The emf of the battery is the sum of the individual emfs.
(2) The current in each cell is the same and is identical with the current in the entire arrangement.
(3) The total internal resistance of the battery is the sum of the individual internal resistances.
