JEE Advanced · 2024 · 26 May 2024 · Paper 2 · Electrostatics

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 $r > R$ .✓ Correct
(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 $\frac{σ p _{0}}{100 \in _{0} I}$ at $r = 10 R$ .✓ Correct

Answer: (B), (D)

Solution

Inside the spherical shell (

r < R

), the electric field is zero, so the dipole experiences no torque and will not oscillate.

For

r > R

, the electric field at distance

r

from the center is:

E = \frac{σ R ^{2}}{ε _{0} r ^{2}}

The restoring torque on the dipole is

τ = - p_{0} E sin θ \approx - p_{0} E θ

for small

θ

.

The angular frequency of oscillation is:

ω = \frac{p _{0} E}{I} = \frac{p _{0} σ R ^{2}}{I ε _{0} r ^{2}}

r = 2 R

ω = \frac{p _{0} σ}{4 I ε _{0}}

(not matching option C)

At

r = 10 R

ω = \frac{p _{0} σ}{100 I ε _{0}}

(matches option D)

Therefore, options (B) and (D) are correct.

JEE Advanced · 2024 · 26 May 2024 · Paper 2 · One or more correct options