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Question #42343

Difficulty: 8/10 | 🟢 Free

The Question

An electron (charge $-e$ , mass $m$ ) is in a circular orbit of radius $R_0$ in the $xy$ -plane under the influence of a spatially varying, time-dependent magnetic field $\mathbf{B}(r, t) = B(t) \left( 1 - \alpha r^2 \right) \hat{k}$ , where $r$ is the distance from the origin. The field strength parameter $B(t)$ is slowly increased over time. It is observed that the orbital radius of the electron remains strictly constant at $R_0$ during this entire process. What is the value of the constant $\alpha$ ?

Options

A.
$\frac{1}{2 R_0^2}$
B.
$\frac{2}{3 R_0^2}$
C.
$\frac{1}{R_0^2}$
D.
$\frac{3}{4 R_0^2}$

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

Vertical Circular Motion|Magnetic Moment of Orbiting Electron|Moving Charge in Non-Uniform B|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the advanced application of Faraday's law and the conservation of canonical angular momentum in a non-uniform magnetic field (the Wideröe Betatron condition).

A common trap is assuming the adiabatic invariance of the magnetic moment ( $\mu \propto v^2/B$ ) holds universally, which would incorrectly imply the radius must change as $B$ increases.

Students must recognize that the induced tangential electric field provides a specific torque, and for the radius to remain constant, the magnetic field at the orbit must be exactly half the average magnetic field enclosed by the orbit.

Quick Facts About This Question

Question Information

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