If there are $3$ parallelly connected cells of emf $\varepsilon_1 = 1.2 \ V, \varepsilon_2 = 1.4 \ V$ and $\varepsilon_3 = 1.5 \ V$ and of internal resistances $r_1 = 0.1 \ \Omega, r_2 = 0.2 \ \Omega$ and $r_3 = 0.3 \ \Omega$,then find $\frac{\varepsilon_{eq}}{r_{eq}} = $ . . . . . . $V \Omega^{-1}$.

Write the equation for the equivalent electromotive force (emf) of a series connection of cells.

$100$ cells, each of $e.m.f.$ $5\, V$ and internal resistance $1\, \Omega$, are to be arranged to produce maximum current in a $25\, \Omega$ external resistance. Each row must contain an equal number of cells. The number of rows should be:

$A$ battery consists of $n$ rows of cells connected in parallel,with each row containing $m$ cells in series. The maximum current is obtained through an external resistance of $3 \, \Omega$. If the total number of cells is $24$ and the internal resistance of each cell is $0.5 \, \Omega$,find the values of $m$ and $n$.

When a current of $2 \ A$ flows through a resistor of $2 \ \Omega$,the terminal voltage across the cell is $E/2$. The internal resistance of the cell is ........ $\Omega$.

Two cells of emf $1 \ V$ and $2 \ V$ and internal resistance $2 \ \Omega$ and $1 \ \Omega$,respectively,are connected in series with an external resistance of $6 \ \Omega$. The total current in the circuit is $I_1$. Now the same two cells in parallel configuration are connected to the same external resistance. In this case,the total current drawn is $I_2$. The value of $\left(\frac{I_1}{I_2}\right)$ is $\frac{x}{3}$. The value of $x$ is . . . . . . .