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Question #43651

Difficulty: 8/10 | 🟢 Free

The Question

An electron of mass $m$ and charge $-e$ is moving in a circular orbit of radius $r_0$ with speed $v_0$ around a stationary nucleus. A uniform magnetic field of magnitude $B$ is slowly established perpendicular to the plane of the orbit. The direction of the magnetic field is such that the magnetic Lorentz force on the electron points towards the nucleus. Assuming the magnetic field is weak ( $eB \ll m v_0/r_0$ ) and neglecting any radiative losses, what is the change in the kinetic energy of the electron?

Options

A.
$\frac{1}{2} e B v_0 r_0$
B.
$e B v_0 r_0$
C.
$0$
D.
$\frac{1}{3} e B v_0 r_0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Radius and Velocity Scaling|Motional EMF in Rotating Rods|Lorentz Force Vector Direction

Why This Question Is Tricky

Common Mistakes

This question tests the adiabatic invariant of canonical angular momentum and the application of Faraday's Law in a changing magnetic field.

A common trap is assuming that because magnetic forces do no work, the kinetic energy of the electron remains constant (Option C).

Students fail to account for the non-conservative induced electric field that accelerates the electron during the ramp-up phase.

Another lethal trap is blindly applying the conservation of mechanical angular momentum ( $mvr = \text{constant}$ ), which leads to an incorrect change in orbital radius and yields Option B.

The correct approach requires recognizing that the canonical angular momentum is conserved, which, combined with the radial force balance, proves that the orbital radius remains constant to first order (Larmor's Theorem) while the kinetic energy increases.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Radius and Velocity Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free