An electric dipole consisting of two opposite charges of $2 \times 10^{-6} \, C$ each,separated by a distance of $3 \, cm$,is placed in an electric field of $2 \times 10^5 \, N/C$. The maximum torque on the dipole will be:

$A$ rigid dipole undergoes a simple harmonic motion about its centre in the presence of an electric field $\vec{E}_1 = E_0\hat{i}$. If another electric field $\vec{E}_2 = 2E_0(\hat{j} + \hat{k})$ is introduced to the system, what will be the percentage change in the frequency of the oscillation (approximate) (in $\%$)?

Electric charges $q, q, -2q$ are placed at the corners of an equilateral triangle $ABC$ of side $l$. The magnitude of the electric dipole moment of the system is

An electric dipole of dipole moment $\vec{P}$ is placed parallel to a uniform electric field of intensity $\vec{E}$. On rotating it through $180^{\circ}$,the amount of work done 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 $r = 10R$.

An electric dipole is situated in an electric field as shown in the figure. The dipole and electric field are both in the plane of the paper. The dipole is rotated about an axis perpendicular to the paper at point $A$ in an anti-clockwise direction. If the angle of rotation is measured with respect to the direction of the electric field,then the torque for different values of the angle of rotation $\theta$ is correctly represented by which graph among $a, b, c, d$ given in the figure?