Question

(MN) identical cells, of e.m.f. E and internal resistance r, connected in N parallel rows where each row contains M cells in series. This combination of cells is connected across an external resistance R. What should be arrangment of the cells for getting maximum current through R if the total number of cells is fixed?

(1) Number of cells in a row = RN/r and number of rows = Nr/R

(2) Number of cells in a row = √RN/r and number of rows = √Nr/R

(3) Number of cells in a row = √MR/r and number of rows = √MR/R

(4) Number of cells in a row = √Nr/R and number of rows = √nR/r

Answer 1

(2) Number of cells in a row = √RN/r and number of rows = √Nr/R

The simplified version of the given circuit is shown in diagram (2) we have,

If the total number of cell is fixed, than m is fixed. Then we need to find the value of the m and n so that the current is maximum. We can write

As mn is fixed. I is maximum when denominator is minimum

If N is the total number of cells then we can get mn = N and m/n = R/r

Hence for the maximum current. Number of cell in a row = \(\sqrt{\frac{RN}{r}}\)

and number of rows = \(\sqrt{\frac{NR}{R}}\)