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Question #43041

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $q > 0$ is initially at rest at a radial distance $r_0$ from the z-axis. At $t=0$ , a time-varying, spatially non-uniform magnetic field $\vec{B} = \beta r t \hat{k}$ is switched on, where $\beta$ is a positive constant and $r$ is the distance from the z-axis. What is the magnitude of the angular velocity of the particle about the z-axis at a later time $t$ , in terms of its instantaneous radial distance $r$ ?

Options

A.
$\frac{q \beta r t}{3m}$
B.
$\frac{q \beta r t}{2m}$
C.
$\frac{q \beta r_0^3 t}{3m r^2}$
D.
$\frac{q \beta (r^3 - r_0^3) t}{3m r^2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Moving Charge in Non-Uniform B|Motional EMF for Arbitrary Shapes|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of Faraday's Law and the Lorentz force in a non-uniform, time-varying magnetic field.

A common trap is for students to blindly apply the standard formula for an induced electric field in a uniform magnetic region ( $E = \frac{r}{2}\frac{dB}{dt}$ ), which incorrectly yields Option B.

Another trap is neglecting the magnetic force component in the azimuthal direction ( $-q\dot{r}B$ ) caused by radial motion, which is essential to form the exact time derivative for angular momentum.

By forcing the derivation from first principles, it distinguishes students who rely on rote formulas from those who understand the fundamental Maxwell-Lorentz framework and the conservation of canonical angular momentum.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Moving Charge in Non-Uniform B
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free