A small electric dipole $\stackrel{\to <!--\; \to \; -->}{p_0}$, 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 σ is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle 𝜃 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 ?

[𝜀₀ is the permittivity of free space.]

Correct Statements:
1. The dipole will undergo small oscillations at any finite value of r > R.
2. The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{\sigma p_0}{100{\epsilon }_{0}I}}$ at r = 10R.

Step-by-Step Explanation:

1. Electric Field due to the Spherical Shell:
The spherical shell has radius $R$ and uniform surface charge density $\sigma$. The total charge on the shell is given by:

$Q=\sigma ·4\pi R^2$

Using Gauss's Law, the electric field $E$ at a distance $r$ from the center of the shell is:
- For $r<R$ (inside the shell): The electric field is zero ($E=0$).
- For $r>R$ (outside the shell): The electric field points radially outwards and has the magnitude:

$E=\frac{Q}{4\pi {\epsilon }_0{r}^2}=\frac{\sigma ·4\pi R^2}{4\pi {\epsilon }_0{r}^2}=\frac{\sigma R^2}{{\mathrm{��}}_0{r}^2}$

2. Restoring Torque on the Dipole:
As shown in the figure, the dipole ${\overrightarrow{p}}_0$ is placed at a distance $r$ along the radial line from the center, making a small angle $\theta$ with the radial line (which represents the direction of the electric field $\overrightarrow{E}$).
The torque acting on the dipole is:

$\overrightarrow{\tau }={\overrightarrow{p}}_0\times \overrightarrow{E}$

The magnitude of the restoring torque is:

$\tau =-p_{0}Esin\theta$

For small angular displacement $\theta$, we approximate $sin\theta \approx \theta$:

$\tau \approx -p_{0}E\theta$

3. Equation of Motion and Angular Frequency:
The rotational equation of motion about the center of the dipole is:

$I\frac{d^2\theta }{d{t}^2}=-p_{0}E\theta$

Rearranging the terms:

$\frac{d^2\theta }{d{t}^2}+\left(\frac{p_{0}E}{I}\right)\theta =0$

This matches the standard equation for simple harmonic motion ($\alpha +{\omega }^2\theta =0$). Thus, the dipole undergoes small oscillations for any finite value of $r>R$ where the field is non-zero.
The angular frequency of oscillation is:

$\omega =\sqrt{\frac{p_{0}E}{I}}=\sqrt{\frac{p_0\sigma R^2}{{\epsilon }_{0}I{r}^2}}$

4. Calculation at r = 10R:
Substituting $r=10R$ into the expression for angular frequency:

$\omega =\sqrt{\frac{p_0\sigma R^2}{{\epsilon }_{0}I{\left(10R\right)}^2}}=\sqrt{\frac{\sigma p_0}{100{\epsilon }_{0}I}}$