Question

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? \[ \epsilon_0 \text{ is the permittivity of free space.} \]
\includegraphics[width=0.4\linewidth]{ph5.png}

• 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 \( \sqrt{\frac{2 \sigma p_0}{\epsilon_0 I}} \) at \( r = 2R \).
• D. The dipole will undergo small oscillations with an angular frequency of \( \sqrt{\frac{\sigma p_0}{100 \epsilon_0 I}} \) at \( r = 10R \).

Hint:

For oscillations of a dipole, calculate the torque and angular acceleration using the electric field and moment of inertia.

Answer 1

The Correct Option is B

For \( r>R \), the electric field outside the shell is: \[ E_0 = \frac{\sigma \cdot 4 \pi R^2}{4 \pi \epsilon_0 r^2}. \] The torque on the dipole: \[ \tau = \mathbf{p} \times \mathbf{E} = p_0 E \sin\theta. \] Using the moment of inertia: \[ I \alpha = p_0 E \sin\theta \quad \Rightarrow \quad \alpha \approx \frac{p_0 E}{I}. \] Substituting: \[ \alpha = \frac{p_0 \cdot \sigma \cdot 4 \pi R^2}{4 \pi \epsilon_0 I r^2} \cdot \theta. \] The angular frequency is: \[ \omega = \sqrt{\frac{p_0 R^2}{\epsilon_0 I r^3}}. \] For \( r = 2R \), \( \omega \) does not match the given options. For \( r = 10R \), we get: \[ \omega = \sqrt{\frac{\sigma p_0}{100 \epsilon_0 I}}. \] Thus, Options (2) and (4) are correct.