Cells in Series and Parallel Derivation

Table of Contents

Cells in Series and Parallel:

Cells in Series:

Cells in Series - Cells in Series and Parallel Derivation

The cells are said to be connected in series, if the positive terminal of one cell is connected to the negative terminal of the second, and so on as shown in the figure. The external resistance ‘R’ is connected to the free terminals of the first and last cells.

Let ‘n’ identical cells are connected in series, each of e.m.f. ‘E’ and internal resistance ‘r’. Let ‘R’ be the resistance of the external resistor. Here,

The total internal resistance of all the cells = nr
The total resistance in the circuit = R +nr
Total (effective) EMF of all the cells = nE
Current in the resistance R = Total EMF/Total Resistance
⇒ I = nE/R+nr ————-(I)

Special Cases:

(a) If R >> nr i.e. nr can be neglected From (I),

I = nE/R

Thus the current in the external resistance is n times the current due to a single cell.

(b) If R << nr i.e. R can be neglected. From (I),

I = nE/nr
⇒ I = E/r

Thus, in this case, the current in the external resistance is the same as due to a single cell.

Thus we can conclude that the maximum current can be drawn from the series combination of cells if the external resistance R is very high as compared to the internal resistance of the cells.

Cells in Parallel:

Cells in Parallel - Cells in Series and Parallel Derivation

The cells are said to be connected in parallel, if the positive terminals of all the cells are connected to one point A and their negative terminals are connected to another point B. The external resistance is connected between A and B.

Let ‘m’ identical cells are connected in parallel, each of e.m.f. ‘E’ and internal resistance ‘r’. Let ‘R’ be the resistance of an external resistor. Here, the total internal resistance of all the cells is given by-

1/rp = 1/r + 1/r + 1/r ——— up to m terms
⇒ 1/rp = m/r
⇒ rp = r/m
Total resistance of the circuit = R + r/m
(Because r/m and R are in series)
Total (effective) E.M.F. of all the cells = E

Current in resistor R = Total E.M.F/Total Resistance

⇒ I = E/R+r/m = E/mR+r/m = mE/mR+r
⇒ I = mE/mR+r ———–(II)

Special Cases:

(a) If R<<r, i.e. mR can be neglected. From (II)

I = mE/r

Thus current in external resistance is m times the current due to a single cell.

(b) If R>>r, i.e. r can be neglected from (II)

I = mE/mR
⇒ I = E/R

Thus the current in external resistance is the same as due to a single cell. Hence, we conclude that the maximum current can be drawn from the parallel combination of cells if the external resistance is very small as compared to the internal resistance of the cells.


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