Show that for a combination of capacitors in series or parallel,the total energy stored is the sum of the energies stored in individual capacitors.

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 $(2)$.

$A$ capacitor $A$ has a capacitance of $15\ \mu F$ when filled with a dielectric of constant $K = 15$. Another capacitor $B$ is air-filled and has a capacitance of $1\ \mu F$. Both are charged to $100\ V$. After charging, the dielectric is removed from capacitor $A$, and the two capacitors are connected in parallel. The common potential difference across them will be ... $V$.

$A$ capacitor $C_1 = 1\,\mu F$ can withstand a maximum voltage of $V_1 = 6\,kV$,and a capacitor $C_2 = 3\,\mu F$ can withstand a maximum voltage of $V_2 = 4\,kV$. If these two capacitors are connected in series,what is the maximum voltage (in $kV$) that can be applied to the combination?

In the given circuit,a charge of $+80 \ \mu C$ is given to the upper plate of the $4 \ \mu F$ capacitor. Find the charge on the upper plate of the $3 \ \mu F$ capacitor in the steady state in $\mu C$.

$A$ metal ball of radius $R$ is placed concentrically inside a hollow metal sphere of inner radius $2R$ and outer radius $3R$. The ball is given a charge $+2Q$ and the hollow sphere is given a total charge $-Q$. The electrostatic potential energy of this system is: