$A$ 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

$A$ battery of $emf$ $E$ is being charged from a charger such that the positive terminal of the battery is connected to terminal $A$ of the charger and the negative terminal of the battery is connected to terminal $B$ of the charger. The internal resistance of the battery is $r$.

The thermo emf of a hypothetical thermocouple varies with the temperature $\theta$ of the hot junction as $E = a\theta + b\theta^2$ in volts,where the ratio $a/b$ is $700^{\circ}C$. If the cold junction is kept at $0^{\circ}C$,then the neutral temperature is:

The charge flowing through a resistance $R$ varies with time as $Q = 2t - 8t^2$. The total heat produced in the resistance is (for $0 \leq t \leq \frac{1}{8} \ s$)

$A$ silver voltmeter and a copper voltmeter are connected in parallel to a $12\ V$ battery. In $30\ minutes$,$1\ g$ of silver and $1.8\ g$ of copper are deposited. The power supplied by the battery is .... $(Z_{Cu} = 6.6 \times 10^{-4}\ g/C$ and $Z_{Ag} = 11.2 \times 10^{-4}\ g/C)$ (in $J/s$)

At time $t = 0$,terminal $A$ in the circuit shown in the figure is connected to $B$ by a key and alternating current $I(t) = I_0 \cos(\omega t)$,with $I_0 = 1 \text{ A}$ and $\omega = 500 \text{ rad s}^{-1}$ starts flowing in it with the initial direction shown in the figure.
At $t = \frac{7\pi}{6\omega}$,the key is switched from $B$ to $D$. Now onwards only $A$ and $D$ are connected. $A$ total charge $Q$ flows from the battery to charge the capacitor fully. If $C = 20 \mu\text{F}$,$R = 10 \Omega$ and the battery is ideal with emf of $50 \text{ V}$,identify the correct statement$(s)$.
$(A)$ Magnitude of the maximum charge on the capacitor before $t = \frac{7\pi}{6\omega}$ is $1 \times 10^{-3} \text{ C}$.
$(B)$ The current in the left part of the circuit just before $t = \frac{7\pi}{6\omega}$ is clockwise.
$(C)$ Immediately after $A$ is connected to $D$,the current in $R$ is $10 \text{ A}$.
$(D)$ $Q = 2 \times 10^{-3} \text{ C}$.