When two identical batteries of internal resistance $1 \Omega$ each are connected in series across a resistor $R$,the rate of heat produced in $R$ is $J_1$. When the same batteries are connected in parallel across $R$,the rate is $J_2$. If $J_1 = 2.25 J_2$,then the value of $R$ in $\Omega$ is

$A$ battery of $24$ cells,each of emf $1.5\, V$ and internal resistance $2\, \Omega$,is to be connected to send the maximum current through a $12\, \Omega$ resistor. The correct arrangement of cells will be:

When current supplied by a cell to a circuit is $0.3 \,A$,its terminal potential difference is $0.9 \,V$. When the current supplied becomes $0.25 \,A$,its terminal potential difference becomes $1.0 \,V$. The internal resistance of the cell is ............ $\Omega$.

Two non-ideal identical batteries are connected in parallel. Consider the following statements:
$(i)$ The equivalent e.m.f. is smaller than either of the two e.m.f.s.
$(ii)$ The equivalent internal resistance is smaller than either of the two internal resistances.

A circuit consists of three batteries of emf $E_{1} = 1 \text{ V}$, $E_{2} = 2 \text{ V}$, and $E_{3} = 3 \text{ V}$ and internal resistances $1 \Omega$, $2 \Omega$, and $1 \Omega$ respectively, which are connected in parallel as shown in the figure. The potential difference between points $P$ and $Q$ is: (in $\text{ V}$)

What is a series connection of cells and what is a parallel connection of cells? Explain.