A small electric dipole , having a moment of inertia about its center, is kept at a distance from the center of a spherical shell of radius . The surface charge density is uniformly distributed on the spherical shell. The dipole is initially oriented a

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 $σ$ 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 [2024]

[ $ε_{0}$ is the permittivity of free space.]

If released from rest, then which of the following statement(s) is(are) correct? [2024]

[ $ε_{0}$ is the permittivity of free space.]

Smart Achievers · Accepted Answer

(2, 4)

The electric field inside the sphere is zero, so the dipole will oscillate when $r > R$ .

Hence option (2) is correct and option (1) is incorrect.

For $R$ , $θ$

$ω = \sqrt{\frac{P E}{I}} = \sqrt{\frac{P_{0} σ R^{2}}{I ε_{0} r^{2}}}$

When $r = 2 R;$ $ω = \sqrt{\frac{P_{0} σ R^{2}}{I ε_{0} {(2 R)}^{2}}}$

or, $ω = \sqrt{\frac{P_{0} σ}{4 I ε_{0}}}$

Therefore option (3) is incorrect.

When $r = 10 R;$ $ω = \sqrt{\frac{P_{0} σ R^{2}}{I ε_{0} {(10 R)}^{2}}}$

$∴$ $ω = \sqrt{\frac{P_{0} σ}{100 I ε_{0}}}$

Hence option (4) is correct.

Solution

1 The dipole will undergo small oscillations at any finite value of $r$ .

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

3 The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{2 σ p_{0}}{ε_{0} I}}$ at $r = 2 R$ .

4 The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{σ p_{0}}{100 ε_{0} I}}$ at $r = 10 R$ .

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Solution

1 The dipole will undergo small oscillations at any finite value of r.

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

3 The dipole will undergo small oscillations with an angular frequency of 2σp0ε0I at r=2R.

4 The dipole will undergo small oscillations with an angular frequency of σp0100 ε0I at r=10R.

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1 The dipole will undergo small oscillations at any finite value of $r$ .

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

3 The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{2 σ p_{0}}{ε_{0} I}}$ at $r = 2 R$ .

4 The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{σ p_{0}}{100 ε_{0} I}}$ at $r = 10 R$ .