Small Oscillations of an Electric Dipole Near a Uniformly Charged Spherical Shell

To determine the correct statements regarding the small oscillations of the electric dipole, we analyze the electric field due to the uniformly charged spherical shell and the torque acting on the dipole.

1. Electric Field of the Spherical Shell

A spherical shell of radius $R$ with uniform surface charge density $\sigma$ has a total charge given by: $Q = 4\pi R^2 \sigma$

Using Gauss's Law, the electric field $\vec{E}$ at a distance $r$ from the center of the shell is:

• Inside the shell ($r < R$): $\vec{E} = 0$

• Outside the shell ($r > R$): $\vec{E}(r) = \frac{Q}{4\pi \varepsilon_0 r^2} \hat{r} = \frac{4\pi R^2 \sigma}{4\pi \varepsilon_0 r^2} \hat{r} = \frac{\sigma R^2}{\varepsilon_0 r^2} \hat{r}$ where $\hat{r}$ is the unit radial vector directed outward from the center of the shell.

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2. Dynamics of the Electric Dipole

When a small electric dipole of dipole moment $\vec{p}_0$ is kept at a distance $r$ and oriented at a small angle $\theta$ with respect to the radial line, it experiences a torque given by: $\vec{\tau} = \vec{p}_0 \times \vec{E}$

The magnitude of the restoring torque tending to align the dipole back with the electric field line is: $\tau = -p_0 E \sin\theta$

For small angular displacements ($\theta \ll 1$), we use the approximation $\sin\theta \approx \theta$: $\tau \approx -p_0 E \theta$

Using Newton's second law for rotation about the center of the dipole: $I \frac{d^2\theta}{dt^2} = -p_0 E \theta \implies \frac{d^2\theta}{dt^2} + \left( \frac{p_0 E}{I} \right) \theta = 0$

This is the standard equation of Simple Harmonic Motion (SHM): $\frac{d^2\theta}{dt^2} + \omega^2 \theta = 0$

where the angular frequency $\omega$ is given by: $\omega = \sqrt{\frac{p_0 E}{I}}$

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3. Evaluation of Options

• For $r < R$: Since $E = 0$, the torque on the dipole is zero ($\tau = 0$). Hence, there is no restoring force, and the dipole will not oscillate.

  • Option (A) states that the dipole will undergo small oscillations at any finite value of $r$, which is incorrect because it fails for $r < R$.

• For $r > R$: Since $E = \frac{\sigma R^2}{\varepsilon_0 r^2} > 0$, a non-zero restoring torque acts on the dipole, causing it to undergo small oscillations for any finite value of $r > R$.

  • Option (B) is correct.

• At $r = 2R$: Substituting $r = 2R$ into the expression for electric field: $E = \frac{\sigma R^2}{\varepsilon_0 (2R)^2} = \frac{\sigma}{4\varepsilon_0}$

  The angular frequency $\omega$ becomes: $\omega = \sqrt{\frac{p_0}{I} \cdot \frac{\sigma}{4\varepsilon_0}} = \sqrt{\frac{\sigma p_0}{4\varepsilon_0 I}} = \frac{1}{2} \sqrt{\frac{\sigma p_0}{\varepsilon_0 I}}$

  • Option (C) gives an angular frequency of $\sqrt{\frac{2\sigma p_0}{\varepsilon_0 I}}$, which is incorrect.

• At $r = 10R$: Substituting $r = 10R$ into the expression for electric field: $E = \frac{\sigma R^2}{\varepsilon_0 (10R)^2} = \frac{\sigma}{100\varepsilon_0}$

  The angular frequency $\omega$ becomes: $\omega = \sqrt{\frac{p_0}{I} \cdot \frac{\sigma}{100\varepsilon_0}} = \sqrt{\frac{\sigma p_0}{100\varepsilon_0 I}}$

  • Option (D) is correct.

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Conclusion

The correct statements are (B) and (D).