Cells in Mixed Groupings:
If the cells are connected as shown in the figure, they are said to be connected in mixed grouping. Let there be n cells in series in one row and m rows of each cells in parallel. Suppose all the cells are identical. Let e.m.f. of each cell be E and internal resistance r.
In each row, there are n cells in series.
Total internal resistance in one row = nr.
There are m rows in parallel. Therefore, the total internal resistance of all the cells is given by
|1/rp = 1/nr + 1/nr + 1/nr ——– up to m terms|
⇒ 1/rp = m/nr
⇒ rp = nr/m
Total resistance in the circuit = R + nr/m
Total resistance in the circuit = mR+nr/m
Total (effective) emf of all the cells = nE
The current in external resistance is given by-
I = Total EMF/Total resistance = nE/(mR+nR)/m
⇒ I = mnE/mR+nr
The current I will be maximum, if (mR+nr) is minimum. It can be shown mathematically that (mR+nr) is minimum.
If mR = nr
⇒ R = nr/m
i.e. external resistance = Total internal resistance of all the cells
Thus, we get maximum current in a mixed grouping, if the value of external resistance is equal to the total internal resistance of all the cells.
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