A small electric dipole p _0 , having a moment of inertia I about...

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. A small electric dipole p _0 , having a moment of inertia I about its center, is kept at a distance r from the center

A small electric dipole p _0 , having a moment of inertia I about its center, is kept at a distance r from the center

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

\varepsilon_0

is the permittivity of free space.]

Options

AThe dipole will undergo small oscillations at any finite value of $r$ .
BThe dipole will undergo small oscillations at any finite value of $r>R$ .
CThe dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{2 \sigma p_0}{\epsilon_0 I}}$ at $r=2 R$ .
DThe dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{\sigma p_0}{100 \epsilon_0 I}}$ at $r=10 R$ .

Correct answer

B. The dipole will undergo small oscillations at any finite value of $r>R$ .

Step-by-step 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)$

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