N cells in series — maximum current

An application of optimisation to physics: one seeks the configuration of a set of cells that maximises the current delivered to a load.

Example — $N$ cells in series: maximum current

$N$ identical cells (voltage $E$, internal resistance $r$) are connected in $n$ parallel rows, each with $N/n$ cells in series, to the load $R$. The current is: $i = \frac{N\cdot n\cdot E}{N\cdot R + n^2\cdot r}.$ For which $n$ is the current maximum?

Solution

One differentiates with respect to $n$ (treating $n$ as a continuous variable) with the quotient rule and sets $i'(n)=0$. Putting the numerator constant $NE$ and the denominator $D(n)=NR+n^2r$: $i'(n) = NE\cdot\frac{D(n) - n\cdot D'(n)}{D(n)^2} = NE\cdot\frac{(NR+n^2r) - n\cdot 2nr}{D(n)^2} = NE\cdot\frac{NR - n^2 r}{D(n)^2}.$ $i'(n)=0 \iff NR = n^2 r \iff n = \sqrt{NR/r}$: the current is maximum when the overall internal resistance ($n^2 r/N$ … ) equals the load, that is when the parallel rows number $\boxed{n = \sqrt{NR/r}}$.

Links

Topics: Derivatives
Concepts: Objective function · Maximum and minimum
Methods: Maximum minimum via derivative
Skills: Differentiating · Modelling