Before You Begin

Question #42497

Difficulty: 8/10 | 🟢 Free

The Question

A rigid uniform thin spherical shell of mass $m$ , radius $R$ , and total charge $Q$ (uniformly distributed over its surface) is pivoted at its center such that it can rotate freely but cannot translate. It is situated in a uniform, constant magnetic field $\vec{B} = B_0 \hat{i}$ . At $t=0^-$ , the shell has an instantaneous angular velocity $\vec{\omega}_0 = \omega_0 \hat{k}$ . At $t=0$ , an impulsive force $\vec{F} = J \delta(t) \hat{j}$ is applied to the shell at the coordinates $(0, 0, R)$ , where $\delta(t)$ is the Dirac delta function. Determine the angular velocity vector $\vec{\omega}$ of the shell immediately after the impulse at $t=0^+$ .

Options

A.
$-\frac{3 J}{2 m R} \hat{i} + \omega_0 \hat{k}$
B.
$-\left( \frac{3 J}{2 m R} + \frac{Q B_0}{2 m} \right) \hat{i} + \omega_0 \hat{k}$
C.
$\frac{3 J}{2 m R} \hat{i} + \omega_0 \hat{k}$
D.
$-\frac{3 J}{2 m R} \hat{i} + \frac{Q B_0}{3 m} \hat{j} + \omega_0 \hat{k}$

Check Your Answer

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

Collision Impulses|Magnetic Dipole Moment Orientation|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between finite and impulsive torques in the context of the Angular Impulse-Momentum Theorem.

A common trap in JEE Advanced is to introduce a magnetic field and a rotating charged body, tempting students to calculate a magnetic torque ( $\vec{\mu} \times \vec{B}$ ) or apply Larmor's theorem.

However, because the magnetic field is constant and the resulting magnetic torque is finite, its integral over an infinitesimal collision time ( $\Delta t \to 0$ ) is strictly zero.

The magnetic field does not impart any impulsive angular momentum.

Students who fail to recognize this will incorrectly add a magnetic term (such as the Larmor frequency) to the final angular velocity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Collision Impulses
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free