A small electric dipole 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 ?[ε0 is the permittivity of free space.]

Engineering

Physics

Electric Dipole

Electric Flux and Gauss Law and its Applications

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 ?

[ε₀ is the permittivity of free space.]

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

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

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

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

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The electric field inside sphere is zero. So dipole will oscillate when r > R.

For r > R $E = \frac{{σR}^{2}}{ε_{0} r^{2}}$

$ω = \sqrt{\frac{PE}{I}} = \sqrt{\frac{P_{0} {σR}^{2}}{{Iε}_{0} r^{2}}}$

when r = 2R

$ω = \sqrt{\frac{P_{0} σ}{4 {Iε}_{0}}}$

when r = 10 R

$ω = \sqrt{\frac{P_{0} σ}{100 {Iε}_{0}}}$