3.12 Cells in Series and in Parallel

NCERT Class 12 physics book for blind and visually impaired students.

Like resistors, cells can be combined together in an electric circuit. And like resistors, one can, for calculating currents and voltages in a circuit, replace a combination of cells by an equivalent cell.

Cells in Series

FIGURE 3.20 Two cells of EMF’s ε1 and ε2 in the series. r1, r2 are their internal resistances. For connections across A and C, the combination can be considered as one cell of EMF εeq and an internal resistance rEq

Consider first two cells in series (Fig. 3.20), where one terminal of the two cells is joined together leaving the other terminal in either cell free. ε1, ε2 are the EMFs 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. 3.20. Then V (A) − V (B) is the potential difference between the positive and negative terminals of the first cell. We have already calculated it in Equation (3.57) and hence,

V AB is equivalent to V(A) − V(B) = ε1 − I r1 .. .. (3.60)

Similarly,


VBC is equivalent to V(B) − V(C) = ε2 − I r2 .. .. (3.61)

Hence, the potential difference between the terminals A and C of the combination is


V ac = V(A) − V(C) = [V(A) −− V(B)] + [V(B) − V(C)]


= (ε1 + ε2) − I(r1 + r2) .. .. (3.62)

If we wish to replace the combination by a single cell between A and C of EMF εeq and internal resistance r eq, we would have


V AC = εq − I req .. .. (3.63)


Comparing the last two equations, we get


εeq = ε1 + ε2 .. .. (3.64)


and req = r1 + r2 .. .. (3.65)

In Fig.3.20, we had connected the negative electrode of the first to the positive electrode of the second. If instead we connect the two negatives, Equation (3.61) would change to VBC = − ε2 – I r2 and we will get


εeq = ε1 − ε2 where (ε1 > ε2) .. .. (3.66)

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 problem

This is so, when the current leaves each cell from the positive electrode. If in the combination, the current leaves any cell from the negative electrode, the EMF of the cell enters the expression for εeq with a negative sign, as in Equation (3.66).

Cells in Parallel

Next, consider a parallel combination of the cells (Fig. 3.21).

FIGURE 3.21 Two cells in parallel. For connections across A and C, the combination can be replaced by one cell of EMF εeq and internal resistances req whose values are given in Eqs. (3.73) and (3.74).

I1 and I2 are the currents leaving the positive electrodes of the cells. At the point B1, I1 and I2 flow in whereas the current I flows out. Since as much charge flows in as out, we have

I = I1 + I2 .. .. (3.67)

Let V(B1) and V(B2) be the potentials at B1 and B2, respectively. Then, considering the first cell, the potential difference across its terminals is V(B1) − V(B2). Hence, from Equation (3.57)

V ≡ V(B1) − V(B2) = ε1 − I1r1 .. .. (3.68)

Points B1 and B2 are connected exactly similarly to the second cell. Hence considering the second cell, we also have


V ≡ V(B1) − V(B2) = ε2 − I2r2 .. .. (3.69)

Combining the last three equations


I = I1 + I2


= (ε1+ V)/r1 + (ε2 + V)/r2 = (ε1/r1 + ε2/r2) − V(1/r1 + 1/r2) .. .. (3.70)

Hence, V is given by,


V = (ε1r2 + ε2r1)/(r1 + r2) − I × r1r2/(r1 + r2) .. .. (3.71)

If we want to replace the combination by a single cell, between B1 and B2, of EMF εeq and internal resistance req, we would have


V = εeq – I req .. .. (3.72)

The last two equations should be the same and hence


εeq = (ε1 r2 + ε2 r1) /( r1 + r2) .. .. (3.73)


req= r1 r2/(r1 + r2) .. .. (3.74)

We can put these equations in a simpler way,

1/req = 1/r1 + 1/r2 .. .. (3.75)


εeq/req = ε1/r1 + ε2/r2 .. .. (3.76)

In Fig. (3.21), we had joined the positive terminals together and similarly the two negative ones, so that the currents I1, I2 flow out of positive terminals. If the negative terminal of the second is connected to positive terminal of the first, Equations. (3.75) and (3.76) would still be valid with ε2 → −ε2

Equations (3.75) and (3.76) can be extended easily. If there are n cells of EMFs ε1, .. .. εn and of internal resistances r1, .. .. rn respectively, connected in parallel, the combination is equivalent to a single cell of EMF εeq and internal resistance req, such that


1/req = 1/r1 + ...+ 1/rn .. .. (3.77)


εeq/req = ε1/r1 + ... + εn/rn .. .. (3.78)

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[Gustav Robert Kirchhoff (1824 - 1887) German physicist, professor at Heidelberg and at Berlin. Mainly known for his development of spectroscopy, he also made many important contributions to mathe¬matical physics, among them, his first and second rules for circuits.]


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