In the given circuit,with steady current,the potential drop across the capacitor must be

The two ends of a uniform conductor are joined to a cell of $e.m.f.$ $E$ and some internal resistance. Starting from the midpoint $P$ of the conductor,we move in the direction of current and return to $P$. The potential $V$ at every point on the path is plotted against the distance covered $(x)$. Which of the following graphs best represents the resulting curve?

The temperature coefficient of resistance of tungsten is $4.5 \times 10^{-3} \, ^{\circ}C^{-1}$ and that of germanium is $-5 \times 10^{-2} \, ^{\circ}C^{-1}$. $A$ tungsten wire of resistance $100 \, \Omega$ is connected in series with a germanium wire of resistance $R$. The value of $R$ for which the resistance of the combination does not change with temperature is .......... $\Omega$.

In a copper voltameter,the mass of copper deposited in $6 \, \text{min}$ is $m \, \text{g}$. If the current-time graph is as shown in the figure,the electrochemical equivalent of copper will be:

Given below are two statements:
Statement $I$: $A$ uniform wire of resistance $80\,\Omega$ is cut into four equal parts. These parts are now connected in parallel. The equivalent resistance of the combination will be $5\,\Omega$.
Statement $II$: Two resistances $2\,R$ and $3\,R$ are connected in parallel in an electric circuit. The value of thermal energy developed in $3\,R$ and $2\,R$ will be in the ratio $3:2$.
In the light of the above statements,choose the most appropriate answer from the options given below.

$A$ wire of length $L$ and $3$ identical cells of negligible internal resistances are connected in series. Due to current,the temperature of the wire is raised by $\Delta T$ in a time $t$. $A$ number $N$ of similar cells is now connected in series with a wire of the same material and cross-section but of length $2L$. The temperature of the wire is raised by the same amount $\Delta T$ in the same time $t$. The value of $N$ is