Twelve cells each of emf 2 V and of internal resistance 0.5 Ω are arranged in a battery of n rows and an external resistance 0.4 Ω is connected to the poles of the battery. Estimate the current flowing through the resistance in terms of n.

Twelve cells each of emf 2 V and of internal resistance 0.5 Ω are arranged in a battery of n rows and an external resistance 0.4 Ω is connected to the poles of the battery. Estimate the current flowing through the resistance in terms of n.
Solution:
no. of cells, $N$ = 12
emf of each cells, $\mathcal{E}$ = 2 V
internal resistance of each cell, $r$ = 0.5 $\mathrm{\Omega }$
no. of rows = $n$
external resistance, $R$ = 0.4 $\mathrm{\Omega }$
Let, $m$ be the number of columns.
Now,
$$\begin{array}{rl}mn& =12\\ m& =\frac{12}{n}\end{array}$$ Now,
$$\begin{array}{rl}I& =\frac{nm\mathcal{E}}{nR+mr}\\ & =\frac{n\times \frac{12}{n}\times 2}{n\times 0.4+\frac{12}{n}\times 0.5}\\ & =\frac{24n}{0.4n^2+6}\\ & =\frac{240n}{4n^2+60}\\ \therefore I& =\frac{60n}{n^2+15}\end{array}$$ Hence, the current flowing through the resistance is $I=\frac{60n}{n^2+15}$.
Here is the attached description of the numerical problem: