The force on a charge situated on the axis of a dipole is $F$. If the charge is shifted to double the distance,the new force will be

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. . . . . .

The figure shows an electric field in which an electric dipole $\overrightarrow{P}$ is placed. Which of the following statements is correct?

For an electric dipole with charge $q = 2 \times 10^{-6} \ C$ and separation distance $d = 0.01 \ m$,if the external electric field is $E = 5 \times 10^{5} \ N/C$,calculate the maximum torque acting on the dipole.

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 $mm$)

Electrical field intensity due to an electric dipole on its axis at distance $x$ $(x \gg a)$ and on the equatorial line at distance $y$ $(y \gg a)$ are same. What is the ratio of $x$ and $y$?