A small electric dipole 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  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? [𝜀0 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 √ 2𝜎𝑝0 𝜖0𝐼 at r = 2𝑅. (D) The dipole will undergo small oscillations with an angular frequency of √ 𝜎𝑝0 100𝜖0𝐼 at r = 10R.

 To solve this problem, we need to analyze the behavior of a small electric dipole 

${\mathbf{p}}_0$ placed near a uniformly charged spherical shell. The dipole is free to rotate about its center and is initially oriented at a small angle $\theta$. We will determine the conditions under which the dipole undergoes small oscillations and calculate the angular frequency of these oscillations.

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Step 1: Electric Field Due to the Spherical Shell

A uniformly charged spherical shell of radius $R$ with surface charge density $\sigma$ produces an electric field. Outside the shell ($r>R$), the electric field is equivalent to that of a point charge at the center of the shell:

$$E=\frac{\sigma R^2}{{ϵ}_0{r}^2}$$

where:

• $\sigma$ is the surface charge density,

• $R$ is the radius of the shell,

• $r$ is the distance from the center of the shell to the dipole,

• ${ϵ}_0$ is the permittivity of free space.

Inside the shell ($r<R$), the electric field is zero due to the symmetry of the spherical shell.

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Step 2: Torque on the Dipole

The dipole ${\mathbf{p}}_0$ experiences a torque $\mathbf{\tau }$ in the presence of the electric field $\mathbf{E}$. The torque is given by:

$$\mathbf{\tau }={\mathbf{p}}_0\times \mathbf{E}$$

For small angles $\theta$, the magnitude of the torque is:

$$\tau =p_{0}E\sin \theta \approx p_{0}E\theta$$

Substituting the electric field $E$ for $r>R$:

$$\tau =p_0\left(\frac{\sigma R^2}{{ϵ}_0{r}^2}\right)\theta$$

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Step 3: Equation of Motion

The torque causes the dipole to undergo rotational motion. The equation of motion for small oscillations is:

$$I\frac{d^2\theta }{d{t}^2}=-\tau$$

Substituting the expression for $\tau$:

$$I\frac{d^2\theta }{d{t}^2}=-p_0\left(\frac{\sigma R^2}{{ϵ}_0{r}^2}\right)\theta$$

This is a simple harmonic oscillator equation of the form:

$$\frac{d^2\theta }{d{t}^2}+{\omega }^2\theta =0$$

where the angular frequency $\omega$ is:

$$\omega =\sqrt{\frac{p_0\sigma R^2}{{ϵ}_{0}I{r}^2}}$$

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Step 4: Analyzing the Statements

Now, let's analyze the given statements:

(A) The dipole will undergo small oscillations at any finite value of $r$.

• This is incorrect because the dipole will only oscillate if $r>R$. For $r<R$, the electric field inside the shell is zero, and there is no torque to cause oscillations.

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

• This is correct because for $r>R$, the electric field is non-zero, and the dipole will experience a torque, leading to small oscillations.

(C) The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{2\sigma p_0}{{ϵ}_{0}I}}$ at $r=2R$.

• Substituting $r=2R$ into the expression for $\omega$:

  $$\omega =\sqrt{\frac{p_0\sigma R^2}{{ϵ}_{0}I(2R{)}^2}}=\sqrt{\frac{p_0\sigma }{4{ϵ}_{0}I}}$$

  This does not match the given frequency. Hence, this statement is incorrect.

(D) The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{\sigma p_0}{100{ϵ}_{0}I}}$ at $r=10R$.

• Substituting $r=10R$ into the expression for $\omega$:

  $$\omega =\sqrt{\frac{p_0\sigma R^2}{{ϵ}_{0}I(10R{)}^2}}=\sqrt{\frac{p_0\sigma }{100{ϵ}_{0}I}}$$

  This matches the given frequency. Hence, this statement is correct.

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Final Answer:

The correct statements are:

$$\boxed{{\displaystyle \text{(B) and (D)}}}$$