The electric field in a region is given by $\overrightarrow{E}=(\left(2x\right)\hat{\mathrm{i}}+\left(4y\right)\hat{\mathrm{j}})\times {10}^6 \mathrm{N} {\mathrm{C}}^{-1}$, where $x$ and $y$ are in $\text{meters}$. A $1 \mathrm{\mu C}$ charged particle is moved from the origin to the co-ordinates $(2 \mathrm{m}, 4 \mathrm{m})$, first along the $y$-axis and then along the $x$-axis. The work done by the electric field is,

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:

The switch $S$ shown in figure is kept closed for a long time and then opened at $t=0,$ then the current in the middle $20 \mathrm{\Omega }$ resistor at $t=0.25 \mathrm{ms}$ is