JEE Advanced 2024 · Paper 2 · Physics Q5

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PhysicsMultiple correct+4 / -2Question 22 of 51

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.

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

[ $\varepsilon_0$ is the permittivity of free space.]

A spherical shell of radius R (circle, radius arrow drawn from its centre). A dashed vertical line rises from the centre; at distance r above the centre (r marked with a vertical double arrow) sits a small dipole p₀, drawn as an arrow tilted at a small angle θ from the dashed radial line.

The dipole will undergo small oscillations at any finite value of

r

The dipole will undergo small oscillations at any finite value of

r > R

The dipole will undergo small oscillations with an angular frequency of

\sqrt{\dfrac{2\sigma p_0}{\varepsilon_0 I}}

r = 2R

The dipole will undergo small oscillations with an angular frequency of

\sqrt{\dfrac{\sigma p_0}{100\varepsilon_0 I}}

r = 10R

Answer and worked solution

Official answer: B, D

Field of the shell.

Inside ( $r<R$ ) the field is zero.
Outside it is radial: $E=\dfrac{\sigma R^2}{\varepsilon_0r^2}$ .

Small oscillations. A dipole tilted by a small $\theta$ from the radial field feels a restoring torque $p_0E\sin\theta\approx p_0E\theta$ . So
$\omega=\sqrt{\frac{p_0E}{I}}=\sqrt{\frac{\sigma p_0R^2}{\varepsilon_0Ir^2}}.$

(A) For $r<R$ there is no field and no torque, so no oscillation. ✗
(B) For any $r>R$ there is a radial field, so it oscillates. ✓
(C) At $r=2R$ , $\omega=\sqrt{\frac{\sigma p_0}{4\varepsilon_0I}}$ , not $\sqrt{\frac{2\sigma p_0}{\varepsilon_0I}}$ . ✗
(D) At $r=10R$ , $\omega=\sqrt{\frac{\sigma p_0}{100\varepsilon_0I}}$ . ✓

Topics:Electric dipole Electric flux and Gauss's law Simple harmonic motion

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JEE Advanced 2024 · Paper 2, Physics question 5