The escape speed of an electron launched from the surface of a glass sphere of diameter $1 \ cm$ that has been charged to $10 \ nC$ is $x \times 10^7 \ m/s$. The value of $x$ is:

$A$ 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 $300 \ V$.

$A$ parallel plate capacitor is charged and then disconnected from the source of steady $E.M.F.$ The plates are then drawn apart farther. Again it is connected to the same source. Then:

The material filled between the plates of a parallel plate capacitor has resistivity $200 \, \Omega \, m$. The capacitance of the capacitor is $2 \, pF$. If a potential difference of $40 \, V$ is applied across the plates,the leakage current flowing through the capacitor is (given the relative permittivity of the material is $50$):

$A$ capacitor of capacity $C$ is connected to a battery of potential $V$ in parallel. The distance between its plates is reduced to half at once,while the battery remains connected. Then,the energy supplied by the battery to charge the capacitor to potential $V$ again is:

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.