$A$ dipole with dipole moment $p$ is rotated by an angle $\theta$ in an electric field $E$ from the direction of the electric field. The work done on the dipole is ......

$A$ 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,

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. . . . . .

Two identical electric point dipoles have dipole moments $\overrightarrow{p}_{1} = p\hat{i}$ and $\overrightarrow{p}_{2} = -p\hat{i}$ and are held on the $x$-axis at a distance $'a'$ from each other. When released,they move along the $x$-axis with the direction of their dipole moments remaining unchanged. If the mass of each dipole is $'m'$,their speed when they are infinitely far apart is

The dipole moment of a system of charge $+q$ distributed uniformly on an arc of radius $R$ subtending an angle $\pi /2$ at its centre, where another charge $-q$ is placed at the centre, is:

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 $\vec{E}_B = 0$