A small electric dipole vector p0, having a moment of inertia I about its center, is kept at a distance r from the center

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?

[\(\varepsilon_0\) is the permittivity of free space.]

(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 \(\sqrt{\frac{2\sigma p_0}{\varepsilon_0I}}\) at \(r = 2R\)

(D) The dipole will undergo small oscillations with an angular frequency of \(\sqrt{\frac{\sigma p_0}{100\varepsilon_0I}}\) at \(r =10R\)

Answer 1

Correct options are (B) and (D)

\(\tau=|\vec{p} \times \vec{E}| \)

\(I \alpha=p_0 E \sin \theta\)

\(\alpha=\frac{p \cdot \theta}{I}\left(\frac{1}{4 \pi \varepsilon_0} \frac{\sigma 4 \pi R^2}{r^2}\right)\)

\(\alpha=\left(\frac{p_0 \sigma R^2}{I \varepsilon_0 r^2}\right) \cdot \theta\)

\(\therefore \omega=\sqrt{\frac{p_0 \sigma R^2}{I \varepsilon_0 r^2}}\)

For r = 2R

\(\omega=\sqrt{\frac{p_0 \sigma}{4 / \varepsilon_0}} \quad ( \mathrm{C \ is\ incorrect)}\)

Also, for \(r=10 R\)

\(\omega=\sqrt{\frac{p_0 \sigma}{4 l(100)}} \quad ( \mathrm{D\ is \ correct)}\)

It will oscillate for any finite value of \(r>R\). (B is correct)