A charge $q$ is placed at the centre of the line joining two equal charges $Q$. The system of the three charges will be in equilibrium, if $q$ is equal to

Two identical conducting spheres, having charges of opposite sign, attract each other with a force of $0.108$ $N$ when separated by $0.5$ $m$. The spheres are connected by a conducting wire, which is then removed, and thereafter, they repel each other with a force of $ 0.036$ $N$. The initial charges on the spheres are

Two equal charges of magnitude $Q$ each are placed at a distance $d$ apart. Their electrostatic energy is $E$. A third charge $-Q / 2$ is brought midway between these two charges. The electrostatic energy of the system is now

Two charges placed in air repel each other by a force of ${10^{ – 4}}\,N$. When oil is introduced between the charges, the force becomes $2.5 \times {10^{ – 5}}\,N$. The dielectric constant of oil is

Two charges $-\mathrm{q}$ each are fixed separated by distance $2\mathrm{d}$. A third charge $\mathrm{d}$ of mass $m$ placed at the midpoint is displaced slightly by $x (x \,<\,<\, d)$ perpendicular to the line joining the two fixed charged as shown in figure. Show that $\mathrm{q}$ will perform simple harmonic oscillation of time period.  $T =\left[\frac{8 \pi^{3} \epsilon_{0} m d^{3}}{q^{2}}\right]^{1 / 2}$