An electric dipole of dipole moment $p$ is aligned parallel to a uniform electric field $E$. The energy required to rotate the dipole by $90^{\circ}$ is $\left[\begin{array}{ll}\sin 0^{\circ}=0, & \sin 90^{\circ}=1 \\ \cos 0^{\circ}=1, & \cos 90^{\circ}=0\end{array}\right]$

Two opposite and equal charges of $4 \times 10^{-8} \ C$ are placed at a distance of $2 \times 10^{-2} \ cm$ to form a dipole. If the dipole is placed in an external electric field of $4 \times 10^8 \ N/C$,calculate the maximum torque and the work done to rotate it by $180^{\circ}$.

An electric dipole of moment $\overrightarrow{p}$ is placed normal to the lines of force of electric intensity $\overrightarrow{E}$. The work done in deflecting it through an angle of $180^\circ$ is:

At the midpoint of the axis of an electric dipole,which of the following is true?

$A$ 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 $r = 10R$.

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. . . . . .