Question

A small electric dipole \(\bar{P_0}\), 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 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?

If released from rest, then which of the following statement(s) is(are) correct?

• A. The dipole will undergo small oscillations at any finite value of r > 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 \(\sqrt\frac{2\sigma p_0}{\epsilon_0I} \ at\ r=2R\)
• D. The dipole will undergo small oscillations with an angular frequency of \(\sqrt\frac{6 p_0}{100\epsilon_0I} \ at\ r=10R\)

Answer 1

The Correct Option is B, D

Electric Field and Torque on the Dipole 

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 is:

\[ \tau = \mathbf{p} \times \mathbf{E} = p_0 E \sin\theta \]

At \( r = 2R \):

\[ \frac{2\sigma p_0}{\epsilon_0 I} \]

At \( r = 25 \):

\[ \frac{\sigma p_0}{100 \epsilon_0 I} \]

Using the Moment of Inertia:

\[ I \alpha = p_0 E \sin\theta \quad \Rightarrow \quad \alpha \approx \frac{p_0 E}{I} \]

Substituting:

The angular frequency is:

\[ \alpha = \frac{p_0 \cdot \sigma \cdot 4\pi R^2}{4\pi \epsilon_0 I r^2} \cdot \theta \] \[ \omega = \frac{p_0 R^2}{\epsilon_0 I r^3} \]

For \( r = 2R \), \( \omega \) does not match the given options. For \( r = 10R \), we get:

\[ \omega = \dots \]

Conclusion:

Thus, Options (2) and (4) are correct.