An electric dipole is formed by two equal and opposite charges $q$ with separation $d$. The charges have the same mass $m$. It is kept in a uniform electric field $E$. If it is slightly rotated from its equilibrium orientation,then its angular frequency $\omega$ is

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

In the situation shown in the diagram,if $q \ll |Q|$ and $r \gg a$,find the net force on the free charge $-q$ and the net torque on it about $O$ at the instant shown. ($p = 2aQ$ is the dipole moment).

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

An electric dipole with dipole moment $\vec{p} = \frac{p_0}{\sqrt{2}}(\hat{i}+\hat{j})$ is held fixed at the origin $O$ in the presence of a uniform electric field $\vec{E} = E_0 \hat{i}$. If the potential is constant on a circle of radius $R$ centered at the origin as shown in the figure,then the correct statement$(s)$ is/are:
($\varepsilon_0$ is the permittivity of free space,$R \gg$ dipole size)
$(1)$ $R = \left(\frac{p_0}{4 \pi \varepsilon_0 E_0}\right)^{1/3}$
$(2)$ The magnitude of the total electric field on any two points of the circle will be the same.
$(3)$ The total electric field at point $A$ is $\vec{E}_A = \sqrt{2} E_0(\hat{i}+\hat{j})$
$(4)$ The total electric field at point $B$ is $\vec{E}_B = 0$

$A$ point dipole $\vec{p} = -p_0\hat{x}$ is kept at the origin. The potential and electric field due to this dipole on the $y$-axis at a distance $d$ are,respectively: (Take $V = 0$ at infinity)