JEE Advanced 2024 — Paper 2 — PHY-II solution

JEE Advanced2024Paper 2PHY-II

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 $σ$ 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?

[ $ε_{0}$ 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 $\frac{2 σ p _{0}}{ε _{0} I}$ at $r = 2 R$ .
D
The dipole will undergo small oscillations with an angular frequency of $\frac{σ p _{0}}{100 ε _{0} I}$ at $r = 10 R$ .

Solution

The electric field outside the uniformly charged spherical shell at distance $r$ from its centre is that of a point charge $Q = σ \cdot 4 π R^{2}$ placed at the centre:

$E = \frac{1}{4 π ε _{0}} \cdot \frac{σ \cdot 4 π R ^{2}}{r ^{2}} = \frac{σ R ^{2}}{ε _{0} r ^{2}} .$

Inside the shell ( $r < R$ ) the field is zero, so no restoring torque exists and the dipole cannot oscillate. Hence oscillation requires $r > R$ .

The restoring torque on the dipole tilted by small angle $θ$ is $τ = - p_{0} E sin θ \approx - p_{0} E θ .$

Equation of motion: $I \ddot{θ} = - p_{0} E θ \Rightarrow ω = \frac{p _{0} E}{I} = \frac{p _{0} σ R ^{2}}{ε _{0} I r ^{2}} .$

For $r = 2 R$ : $ω = \frac{p _{0} σ}{4 ε _{0} I}$ , which does not match option (C).

For $r = 10 R$ : $ω = \frac{p _{0} σ}{100 ε _{0} I}$ , matching option (D).

Hence the correct options are (B) and (D).

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