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 $y$?

An electric dipole of dipole moment $\vec{P}$ is placed in a uniform electric field $\vec{E}$. Which of the following statements are correct?
Statement $I$: The torque on the dipole is $\vec{\tau} = \vec{P} \times \vec{E}$
Statement $II$: The potential energy of the dipole is $U = -\vec{P} \cdot \vec{E}$
Statement $III$: The net force on the dipole is non-zero

An electric dipole is placed on the $x$-axis as shown in the figure. Consider a spherical closed surface around it. Then,

An electric dipole consisting of two opposite charges of $2 \times 10^{-6} \, C$ each,separated by a distance of $3 \, cm$,is placed in an electric field of $2 \times 10^5 \, N/C$. The maximum torque on the dipole will be:

$A$ point electric dipole placed at the origin has a potential given by $V(r, \theta) = \frac{p \cos \theta}{4 \pi \varepsilon_0 r^2}$,where $\theta$ is the angle made by the position vector with the direction of the dipole. Then,

$A$ point particle of mass $M$ is attached to one end of a massless non-conducting rod of length $L$. Another point particle of the same mass is attached to the other end of the rod. The two particles carry charges $+q$ and $-q$ respectively. This arrangement is placed in a region of uniform electric field $E$. When the rod makes a small angle $(< 5^o)$ with the direction of the field,what is the minimum time required for the rod to become parallel to the field?