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Question #42977

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 of charge $+e$ . A uniform magnetic field $\vec{B}$ , perpendicular to the plane of the orbit, is suddenly switched on. The direction of $\vec{B}$ is parallel to the initial orbital angular momentum of the electron. Assuming the duration of the switch-on process is extremely short such that the orbital radius remains approximately $r_0$ during this time, what is the minimum final magnitude of the magnetic field $B$ required for the electron to just escape the Coulombic binding of the nucleus (i.e., its total mechanical energy becomes zero)?

Options

A.
$\frac{2m v_0}{e r_0} (\sqrt{2} - 1)$
B.
$\frac{m v_0}{e r_0}$
C.
$\frac{m v_0}{e r_0} (\sqrt{2} - 1)$
D.
$\frac{2m v_0}{e r_0}$

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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

Faraday’s Law Vector Form|Magnetic Moment of Orbiting Electron|Coulombic Central Force|Turning Point Identification

Why This Question Is Tricky

Common Mistakes

This question tests a critical misconception regarding the work done by magnetic fields.

While a static magnetic field does no work on a moving charge, a time-varying magnetic field induces a non-conservative electric field (Faraday's Law) that does perform work, thereby changing the particle's kinetic energy.

Students often fall into the trap of assuming kinetic energy is conserved and attempt to solve for escape by merely balancing the new Lorentz force with the Coulomb force.

Another common trap is incorrectly applying the potential energy of a static magnetic dipole ( $\Delta E = -\vec{\mu} \cdot \vec{B}$ ) without accounting for the fact that the orbital magnetic moment itself changes during the magnetic pulse.

The correct approach requires integrating the induced tangential impulse to find the new velocity and applying the standard zero-total-energy escape condition.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Faraday’s Law Vector Form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free