If the capacitance of each capacitor in the tetrahedron shown is $C$, then the effective capacitance of the network across any two junctions is ......$C$.

When a soap bubble is charged,its size:

Consider a simple $RC$ circuit as shown in Figure $1$.
Process $1$: In the circuit,the switch $S$ is closed at $t=0$ and the capacitor is fully charged to voltage $V_0$ (i.e.,charging continues for time $T \gg RC$). In the process,some dissipation $(E_D)$ occurs across the resistance $R$. The amount of energy finally stored in the fully charged capacitor is $E_C$.
Process $2$: In a different process,the voltage is first set to $V_0/3$ and maintained for a charging time $T \gg RC$. Then the voltage is raised to $2V_0/3$ without discharging the capacitor and again maintained for time $T \gg RC$. The process is repeated one more time by raising the voltage to $V_0$ and the capacitor is charged to the same final voltage $V_0$.
These two processes are depicted in Figure $2$.
$(1)$ In Process $1$,the energy stored in the capacitor $E_C$ and heat dissipated across resistance $E_D$ are related by:
$[A]$ $E_C = E_D$
$[B]$ $E_C = E_D \ln 2$
$[C]$ $E_C = \frac{1}{2} E_D$
$[D]$ $E_C = 2 E_D$
$(2)$ In Process $2$,the total energy dissipated across the resistance $E_D$ is:
$[A]$ $E_D = \frac{1}{2} CV_0^2$
$[B]$ $E_D = 3 \left( \frac{1}{2} CV_0^2 \right)$
$[C]$ $E_D = \frac{1}{3} \left( \frac{1}{2} CV_0^2 \right)$
$[D]$ $E_D = 3 CV_0^2$
Select the correct pair of answers for $(1)$ and $(2)$.

The plates $S$ and $T$ of an uncharged parallel plate capacitor are connected across a battery. The battery is then disconnected and the charged plates are now connected in a system as shown in the figure. The system shown is in equilibrium. All the strings are insulating and massless. The magnitude of charge on one of the capacitor plates is: [Area of plates = $A$]

The charge on each capacitor when a voltage source of $15 \ V$ is connected in the circuit as shown,is (in $\mu C$)

Two parallel plate capacitors with capacitances $C$ and $2C$ are connected in parallel. They are charged to a potential difference $V$. Now,if the battery is disconnected and a dielectric of dielectric constant $K$ is inserted between the plates of the capacitor with capacitance $C$,find the potential difference across each capacitor.