Question

If $24$ cells each of internal resistance $I$ $\Omega$ are to be connected in series and parallel combination to get the maximum power output across a load of $10$ $\Omega$. If $x$ represents the number of rows and $y$ represents the number of cells in each row. Which of the following relation between $x$ and $y$ is correct?

• A. $y-x=23$
• B. $y-x=10$
• C. $y-x=23$
• D. $y-x=5$

Answer 1

The correct option is B $y-x=10$


There are $x$ rows and each row has $y$ cells. The total number of cells is $x\times y=24$. Here $I=\frac{ye}{R+\left(\frac{yr}{x}\right)}$
Maximum power is generated when external load resistance is equal to internal resistance
$\therefore$ for maximum power $\frac{yr}{x}=R$ or $y=10x$ or $10x^2=24$ or $x=\sqrt{2.4}=1.55$

$\left(I\right)$. If $x=1$, $y=24$, $I=\frac{24e}{34}$ , where $e$ is the emf of the single cell, then
$P_1={\left(\frac{24e}{34}\right)}^2\times 10=4.98e^2$
$\left(I\right)$. If $x=2$, $y=12$, $I=\frac{12e}{16}$ , where $e$ is the emf of the single cell, then
$P_1={\left(\frac{12e}{16}\right)}^2\times 10=5.625e^2$
So we have two rows $(x=2)$ each containing $12$ cells $(y=12)$ in series
$\therefore$ $y-x=10$