Question 5

For $r < R$, the electric field inside a uniformly charged spherical shell is zero, so the dipole experiences no torque and will not oscillate. Thus, small oscillations only occur for $r > R$.

The electric field at a distance $r$ outside the shell is given by $E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} = \frac{4\pi R^2 \sigma}{4\pi\epsilon_0 r^2} = \frac{\sigma R^2}{\epsilon_0 r^2}$.

The torque on the dipole is:

$\tau = -p_0 E \sin\theta \approx -p_0 E \theta$.

The angular frequency of oscillation is $\omega = \sqrt{\frac{p_0 E}{I}} = \sqrt{\frac{p_0 \sigma R^2}{\epsilon_0 I r^2}}$.

Substituting $r = 10R$, we get $\omega = \sqrt{\frac{\sigma p_0}{100 \epsilon_0 I}}$, making option (D) correct and (B) correct.