Question

To get maximum current through a resistance of $2.5\Omega$, one can use $m$ rows of cells, each row having $n$ cells. The internal resistance of each cell is $0.5\Omega$. What are the values of $n$ and $m$ if the total number of cells is $45$?

• A. $m=3,n=15$
• B. $m=5,n=9$
• C. $m=9,n=5$
• D. $m=15,n=3$

Answer 1

The correct option is A $m=3,n=15$

Let each cell have emf $E$ and internal resistance $r.$

Since, each row have $n$ cells, so equivalent emf and internal resistance of each row will be

$E_{row}=nE$ and $r_{row}=nr$

Now $m$ row have parallel combination, So, net equivalent emf and internal resistance will be

$E_{eq}=\frac{\sum \frac{nE}{nr}}{\sum \frac{1}{nr}}$

$\therefore E_{eq}=\frac{\frac{mnE}{nr}}{\frac{m}{nr}}=nE$

And, $r_{eq}=\frac{nr}{m}$

So, current flow through the circuit will be

$i=\frac{nE}{R+\frac{nr}{m}}=\frac{mnE}{mR+nr}$

Given:
$R=2.5\Omega ;r=0.5\Omega$

For maximum current,

$mR=nr$

$\Rightarrow 2.5m=0.5n$

$5m=n..........\left(1\right)$

Given that,
$mn=45$

putting, $n=5m$, we get

$m\times 5m=45$

$\Rightarrow m=\pm 3$, since value can't be negative,

$\therefore m=3$ and $n=15$

Hence, the correct answer is option $\left(a\right)$.