The linear charge density on a dielectric ring of radius $R$ varies with $\mathrm{\theta }$ as $\lambda ={\lambda }_0\mathrm{cos\theta }/2,$ where ${\lambda }_0$ is constant. Find the potential at the center $O$ of the ring (in volt).

An electric dipole having dipole moment $4\times {10}^{-9} \mathrm{C} \mathrm{m}$ is placed at origin along $x$-axis. If the electric field that should be applied along $x$-axis so that we get an equipotential spherical surface of radius $3 \mathrm{m}$ centred at origin is $E \mathrm{V} {\mathrm{m}}^{-1}$, what is the value of $3E$?

(Take $k=9\times {10}^9 \mathrm{N} {\mathrm{m}}^2 {\mathrm{C}}^{-2}$)

Two smooth spherical non-conducting shells, each of radius $R$ and having uniformly distributed charges $Q$ and $-Q$ on their surfaces are released on a smooth non-conducting surface when the distance between their centres is $5R$. The mass of $A$ is $m$ and that of $B$ is $2m$. The speed of $A$ just before $A$ and $B$ collide is (neglect gravitational interaction, take $k=\frac{1}{4\pi {\epsilon }_0}$),

$AB$ and $CD$ are two large, non-parallel metallic plates connected to a battery as shown. If ${\sigma }_{\mathrm{A}}$ and ${\sigma }_{\mathrm{B}}$ be the charge densities at points $A$ and $B$, respectively, then,

A uniformly charged and infinitely long line having a linear charge density $\lambda$ is placed at a normal distance $\mathrm{y}$ from a point $\mathrm{O}$ . Consider a sphere of radius $\mathrm{R}$ with $\mathrm{O}$ as centre and $R>y$ . Electric flux through the surface of the sphere is: