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 \(\Omega\)
no. of rows = \(n\)
external resistance, \(R\) = 0.4 \(\Omega\)
Let, \(m\) be the number of columns.
Now,
\[\begin{align*} mn&=12\\ m&=\frac{12}{n}\\ \end{align*}\] Now,
\[\begin{align*} I&=\frac{nm\mathcal{E}}{nR+mr}\\ &=\frac{n\times {\frac{12}{n}}\times 2}{n \times 0.4 + {\frac{12}{n}}\times 0.5}\\ &=\frac{24n}{0.4 n^2+6}\\ &=\frac{240n}{4n^2+60}\\ \therefore I&=\frac{60 n}{n^2+15}\\ \end{align*}\] Hence, the current flowing through the resistance is \(I=\frac{60 n}{n^2+15}\).
Here is the attached description of the numerical problem: