Write the expression for the torque experienced by an electric dipole placed in a uniform electric field.

An electric dipole in a uniform electric field experiences (when it is placed at an angle $\theta$ with the field):

$A$ given charge is situated at a certain distance from an electric dipole in the axial position and experiences a force $F$. If the distance of the charge is doubled,the force acting on the charge will be:

Two electric dipoles of dipole moments $1.2 \times 10^{-30} \, C \cdot m$ and $2.4 \times 10^{-30} \, C \cdot m$ are placed in two different uniform electric fields of strengths $5 \times 10^{4} \, N \cdot C^{-1}$ and $15 \times 10^{4} \, N \cdot C^{-1}$ respectively. The ratio of maximum torque experienced by the electric dipoles will be $\frac{1}{x}$. The value of $x$ is $.....$

An electric dipole having dipole moment $P = q \times 2\ell$ is placed in a uniform electric field $E$. The dipole moment is along the direction of the field. The force acting on it and its potential energy are respectively

$A$ 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 $r = 10R$.