A small electric dipole $\vec{p}_0$, having a moment of inertia $I$ about its center, is kept at a distance…
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.]
The dipole will undergo small oscillations at any finite value of $r$.
The dipole will undergo small oscillations at any finite value of $r>R$.
The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{2 \sigma p_0}{\epsilon_0 I}}$ at $r=2 R$.
The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{\sigma p_0}{100 \epsilon_0 I}}$ at $r=10 R$.
Solution
\(\begin{aligned}
& \tau=|\vec{p} \times \vec{E}| \\
& I \alpha=p_0 E \sin \theta \\
& \alpha=\frac{p \cdot \theta}{I}\left(\frac{1}{4 \pi \varepsilon_0} \frac{\sigma 4 \pi R^2}{r^2}\right) \\
& \alpha=\left(\frac{p 0 \sigma R^2}{I \varepsilon_0 r^2}\right) \cdot \theta \\
& \therefore \omega=\sqrt{\frac{p 0 \sigma R^2}{I \varepsilon_0 r^2}}
\end{aligned}\)
For \(r=2 R\)
\(\omega=\sqrt{\frac{p_0 \sigma}{4 / \varepsilon_0}} \quad(\mathrm{C}\) is incorrect \()\)
Also, for \(r=10 R\)
\(\omega=\sqrt{\frac{p_0 \sigma}{4 l(100)}}\) (D is correct)
It will oscillate for any finite value of \(r>R\). (B is correct)
If released from rest, then which of the following statement(s) is(are) correct?
[ $\varepsilon_0$ is the permittivity of free space.]