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 $\left( k = \frac{1}{4\pi \varepsilon_0} \right)$.

If the electric field at a distance $x$ on the axis of an electric dipole is equal to the electric field at a distance $y$ on its equatorial line,what is the ratio $x : y$?

One end of a spring of negligible unstretched length and spring constant $k$ is fixed at the origin $(0,0)$. $A$ point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $\overrightarrow{p}$ pointing towards the charge $q$ is fixed at the origin,the spring gets stretched to a length $\ell$ and attains a new equilibrium position. If the point mass is now displaced slightly by $\Delta \ell \ll \ell$ from its equilibrium position and released,it is found to oscillate at frequency $\frac{1}{\delta} \sqrt{\frac{k}{m}}$. The value of $\delta$ is. . . . . .

An electric dipole of moment $p$ is placed in an electric field of intensity $E$. The dipole acquires a position such that the axis of the dipole makes an angle $\theta$ with the direction of the field. Assuming that the potential energy of the dipole is zero when $\theta = 90^o$,the torque and the potential energy of the dipole will respectively be

Two point charges $-q$ and $+q$ are placed at a distance of $L$,as shown in the figure. The magnitude of electric field intensity at a distance $R$ $(R \gg L)$ varies as:

Three identical dipoles are arranged as shown below. What will be the net electric field at $P$ $\left( {k = \frac{1}{{4\pi {\varepsilon _0}}}} \right)$?