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.]

$$\tau=\vert{}\vec{p} \times \vec{E}\vert{}$$

$$I \alpha=p_0 E \sin \theta$$

$$\alpha=\frac{p \cdot \theta}{I}\left(\frac{1}{4 \pi \varepsilon_0} \frac{\sigma 4 \pi R^2}{r^2}\right)$$

$$\alpha=\left(\frac{p_0 \sigma R^2}{I \varepsilon_0 r^2}\right) \cdot \theta$$

$$\therefore \omega=\sqrt{\frac{p_0 \sigma R^2}{I \varepsilon_0 r^2}}$$

For $$r = 2R$$:

$$\omega=\sqrt{\frac{p_0 \sigma}{4 I \varepsilon_0}} \quad ( \mathrm{C \ is \ incorrect})$$

Also, for $$r=10 R$$:

$$\omega=\sqrt{\frac{p_0 \sigma}{I \varepsilon_0 (100)}} \quad ( \mathrm{D \ is \ correct})$$

It will oscillate for any finite value of $$r>R$$. (B is correct)