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Question #42932

Difficulty: 8/10 | 🟢 Free

The Question

A thin non-conducting spherical shell of mass $M$ , radius $R$ , and total charge $Q$ uniformly distributed over its surface is initially at rest in a vacuum. An external impulsive mechanical torque $\vec{\tau}_{ext}$ is applied to the shell for a very short duration $\Delta t$ , delivering a total angular impulse $\int_0^{\Delta t} \vec{\tau}_{ext} dt = J \hat{k}$ . Assuming no other external mechanical forces act on the system and ignoring radiation losses, what is the final angular velocity $\vec{\omega} = \omega \hat{k}$ of the shell immediately after the impulse? (Here $\mu_0$ is the permeability of free space).

Options

A.
$\frac{J}{\frac{2}{3} M R^2}$
B.
$\frac{J}{\frac{2}{3} M R^2 + \frac{\mu_0 Q^2 R}{18 \pi}}$
C.
$\frac{J}{\frac{2}{3} M R^2 + \frac{\mu_0 Q^2 R}{36 \pi}}$
D.
$\frac{J}{\frac{2}{5} M R^2 + \frac{\mu_0 Q^2 R}{18 \pi}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Non-conducting Charged Sphere|Angular Momentum conservation|Frame of Reference Dependency

Why This Question Is Tricky

Common Mistakes

This question tests a profound trap in classical electrodynamics and mechanics: the failure to account for the 'electromagnetic inertia' or the angular momentum stored in the self-field of an accelerating charge distribution.

Students typically apply the standard angular impulse-momentum theorem ( $J = I_{mech}\omega$ ) treating the shell purely as a mechanical rigid body.

However, in a non-inertial (accelerating) state, the time-varying magnetic field generated by the rotating charge induces a non-conservative electric field via Faraday's Law.

This induced field exerts a back-torque on the shell itself, opposing the angular acceleration.

Equivalently, the applied impulse must supply not only the mechanical angular momentum of the mass but also the field angular momentum ( $\vec{L}_{field} = \epsilon_0 \int \vec{r} \times (\vec{E} \times \vec{B}) dV$ ).

Ignoring this leads to the incorrect naive answer, punishing rote application of $L=I\omega$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Non-conducting Charged Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free