Two concentric rings, one of radius $a$ and the other of radius $b$, have the charges $+q$, and $-{\left(2/5\right)}^{-3/2}q$, respectively, as shown in the figure. Find the ratio $\frac{b}{a}$ if a charged particle placed on the axis at $z=a$ is in equilibrium

Four charged particles each having charge $Q=1 \mathrm{C}$ are fixed at the corners of the base ( $A\text{, }B\text{,} C$ and $D$) of a square pyramid with slant length $a\left(AP=BP=PD=PC=a=\sqrt{2} \mathrm{m}\right)$, a charge $-Q$ is fixed at point $P$. A dipole with dipole moment $p=1 \mathrm{C} \mathrm{m}$ is placed at the centre of the base and perpendicular to its plane as shown in figure. Force on the dipole due to the charge particles is $\frac{x}{4\pi {\epsilon }_0}\mathrm{N}$. Find the value of $x$?

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.