The net electric field at the given point $O$ due to all the four identical short electric dipoles of dipole moment $p$ each,as shown in the figure,is $\left( k = \frac{1}{4\pi \varepsilon_0} \right)$.

An electric dipole of moment $\overrightarrow{p}$ is placed normal to the lines of force of electric intensity $\overrightarrow{E}$. The work done in deflecting it through an angle of $180^\circ$ is:

Two small spherical balls of mass $10 \ g$ each with charges $-2 \ \mu C$ and $2 \ \mu C$,are attached to two ends of a very light rigid rod of length $20 \ cm$. The arrangement is now placed near an infinite nonconducting charge sheet with a uniform charge density of $100 \ \mu C / m^2$ such that the length of the rod makes an angle of $30^{\circ}$ with the electric field generated by the charge sheet. The net torque acting on the rod is (Take $\varepsilon_0 = 8.85 \times 10^{-12} \ C^2 / Nm^2$) (in $Nm$)

Two charges $+q$ and $-q$ are kept at a certain distance apart. Then, at any point on the right bisector of the line joining the two charges:

$A$ and $B$ are two points on the axis and the perpendicular bisector respectively of an electric dipole. $A$ and $B$ are far away from the dipole and at equal distance from it. The electric fields at $A$ and $B$ are $\vec{E}_A$ and $\vec{E}_B$.

An electric dipole having each charge of magnitude $2 \mu C$ is placed in an electric field of intensity $8 \times 10^{4} \ N/C$. If the maximum torque acting on the dipole is $4 \times 10^{-3} \ N \cdot m$,the length of the dipole is: (in $mm$)