$A$ radioactive substance in the form of a metallic sphere of diameter $10^{-3} \ m$ emits particles at a constant rate of $6.25 \times 10^{10}$ particles per second. If the sphere is electrically isolated,how long will it take for its potential to increase by $1.0 \ V$? Assume that $80\%$ of the emitted particles escape from the surface. The time taken is $... \mu s$.

Two neutral conducting spheres of diameters $8 \ cm$ and $2 \ cm$ separated with a distance of $15 \ cm$ between their centres are joined by a thin conducting wire. $A$ charge of $100 \ nC$ is given to one of the spheres and the system is allowed to reach electrostatic equilibrium. The electric potential at a point on the line joining the centres of the two spheres where the net electric field becomes zero is . . . . . . $V$. (Neglect the charge acquired by the wire and $\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \ Nm^2 C^{-2}$)

$A$ fully charged capacitor of capacity $C$ is discharged through a small resistance wire embedded in a solid block of specific heat $s$ and mass $m$ under thermally isolated conditions. If the temperature of the block is raised by $\Delta T$,the initial potential difference across the capacitor is

Two capacitors $C_1 = 2\,\mu F$ and $C_2 = 6\,\mu F$ in series are connected in parallel to a third capacitor $C_3 = 4\,\mu F$. This arrangement is then connected to a battery of $e.m.f. = 2\,V$,as shown in the figure. How much energy is lost by the battery in charging the capacitors?

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 $\mu C$.

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):