$A$ parallel plate capacitor of plate area $A$ and plate separation $d$ is charged to a potential $V$ and then the battery is disconnected. $A$ slab of dielectric constant $k$ is then inserted between the plates of the capacitor so as to fill the space between the plates. If $Q$,$E$,and $W$ denote respectively the magnitude of charge on each plate,the electric field between the plates (after the slab is inserted),and the work done on the system in the process of inserting the slab,then state the incorrect relation from the following:

In the adjoining figure,three capacitors $C_1$,$C_2$,and $C_3$ are connected to a battery of potential $V$. The correct condition is (Symbols have their usual meanings):

Two capacitors of equal capacitance $(C_1 = C_2)$ are shown in the figure. Initially,while the switch $S$ is open,one of the capacitors is uncharged and the other carries charge $Q_0$. The energy stored in the charged capacitor is $U_0$. After the switch is closed,the capacitors $C_1$ and $C_2$ carry charges $Q_1$ and $Q_2$,respectively; the voltages across the capacitors are $V_1$ and $V_2$; and the energies stored in the capacitors are $U_1$ and $U_2$. Which of the following statements is $INCORRECT$?

$A$ thin metallic partition of negligible thickness is inserted between two shaded metallic plates as shown. The remaining ends are then packed with insulating plates to form a container-like structure. Two taps shown are opened at $t = 0$ and finally closed at $t = 5 \ s$. Find the capacitance of the system between $A$ and $B$ after closing the taps. (Assume the liquid to be non-conducting). Volumetric flow rates and dielectric constants of the liquid are given.

For the circuit shown in the figure,the charge on the $4 \mu F$ capacitor is $....... \mu C$.

$A$ positive point charge of $10^{-8} \ C$ is kept at a distance of $20 \ cm$ from the center of a neutral conducting sphere of radius $10 \ cm$. The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of $10 \ cm$ further away from the center of the sphere along the radial direction. Taking $\frac{1}{4 \pi \epsilon_0} = 9 \times 10^9 \ N \cdot m^2 / C^2$ (where $\epsilon_0$ is the permittivity of free space),which of the following statements is/are correct:
$(A)$ Before the grounding,the electrostatic potential of the sphere is $450 \ V$.
$(B)$ Charge flowing from the sphere to the ground because of grounding is $5 \times 10^{-9} \ C$.
$(C)$ After the grounding is removed,the charge on the sphere is $-5 \times 10^{-9} \ C$.
$(D)$ The final electrostatic potential of the sphere is $300 \ V$.