An electric dipole of length $2 \ cm$ is placed with its axis making an angle of $60^{\circ}$ to a uniform electric field of $10^{5} \ N/C$. If it experiences a torque of $8 \sqrt{3} \ Nm$,calculate the magnitude of the charge on the dipole. (Given: $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$)

An electric dipole is kept in a non-uniform electric field. It experiences

The electric field due to a short electric dipole at a distance $r$ on the axial line from its mid-point is $x$ times the electric field at a distance $2r$ on the equatorial line from the mid-point of the dipole. Then,the value of $x$ is

The electric field intensity at a point on the axis of an electric dipole in air is $4 \text{ NC}^{-1}$. Then the electric field intensity at a point on the equatorial line which is at a distance equal to twice the distance on the axial line and if the dipole is in a medium of dielectric constant $4$ is

$A$ small electric dipole $\vec{p}_0$,having a moment of inertia $I$ about its center,is kept at a distance $r$ from the center of a spherical shell of radius $R$. The surface charge density $\sigma$ is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle $\theta$ as shown in the figure. While staying at a distance $r$,the dipole is free to rotate about its center. If released from rest,then which of the following statement$(s)$ is (are) correct? [$\varepsilon_0$ is the permittivity of free space.]
$(A)$ The dipole will undergo small oscillations at any finite value of $r$.
$(B)$ The dipole will undergo small oscillations at any finite value of $r > R$.
$(C)$ The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{\sigma p_0}{4 \varepsilon_0 I}}$ at $r = 2R$.
$(D)$ The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{\sigma p_0}{100 \varepsilon_0 I}}$ at $r = 10R$.

If $E_{a}$ and $E_{q}$ represent the electric field intensity due to a short dipole at a point on its axial line and on the equatorial line at the same distance $r$ from the centre of the dipole,then