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Question #43787

Difficulty: 8/10 | 🟢 Free

The Question

An electron of mass $m$ and charge $-e$ moves in a circular orbit of radius $r_0$ with speed $v_0$ around a stationary nucleus of charge $+Ze$ in the $xy$ -plane. Its initial angular momentum is in the $+\hat{k}$ direction. A uniform magnetic field $\vec{B} = B(t)\hat{k}$ , confined to a coaxial cylindrical region of radius $R$ ( $R < r_0$ ), is slowly increased from $0$ to $B_0$ . The magnetic field outside this cylindrical region is always zero. Assuming the electron settles into a new circular orbit of equilibrium radius $r_{eq}$ , what is the first-order change in the radius $\Delta r = r_{eq} - r_0$ ?

Options

A.
$\frac{e R^2 B_0}{m v_0}$
B.
$\frac{e R^2 B_0}{2 m v_0}$
C.
$-\frac{e R^2 B_0}{m v_0}$
D.
$0$

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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

Coulombic Central Force|Induced Electric Fields|Maxwell's Correction to Ampere's Law|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the distinction between local and global time-varying magnetic fields and their effect on orbital dynamics.

A common trap is blindly applying Larmor's theorem (where orbital radius remains constant, Option D) without realizing it requires a local magnetic field at the electron's position to provide a compensating radial Lorentz force.

Here, the magnetic field is confined to $R < r_0$ .

The time-varying flux induces a non-conservative azimuthal electric field that exerts a torque, increasing the electron's angular momentum.

Since the local magnetic field at the electron's orbit is zero, there is no additional radial Lorentz force.

The increased angular momentum against the unchanged conservative Coulombic central force strictly forces the equilibrium radius to expand.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Coulombic Central Force
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free