Two identical small equally charged conducting balls are suspended from long threads secured at one point. The charges and masses of the balls are such that they are in equilibrium when the distance between them is $a$ (the length of the thread $L\gg a$). One of the balls is then discharged. Again for the certain value of distance $b\left(b\ll l\right)$ between the balls, the equilibrium is restored. Find the value of $\left(\frac{a^3}{b^3}\right)$.

A particle having charge that of an electron and mass $1.6\times {10}^{-30} \mathrm{k}\mathrm{g}$ is projected with an initial speed $u$ at an angle ${45}^{\mathrm{o}}$ to the horizontal from the lower plate of a parallel plate capacitor as shown in the figure. The plates are sufficiently long and have separation $2 \mathrm{c}\mathrm{m}$. The maximum value of velocity of particle not to hit the upper plate is $\sqrt{W}\times {10}^6 \mathrm{m} {\mathrm{s}}^{-1}$. Take electric field between the plates as ${10}^3 \mathrm{V} {\mathrm{m}}^{-1}$ directed upward. Find the $W$.

In the figure shown, find the ratio of the linear charge densities ${\lambda }_1$ (on semi – infinite straight wire) and ${\lambda }_2$ (on semi – circular part) that is ${\lambda }_1/{\lambda }_2$ so that the field at $O$ is along $y$ direction.

A thin semi-circular ring of radius $r$ has a positive charge $q$ distributed uniformly over it. The net field $E$ at the centre $O$ is