Motion of an electron in a uniform magnetic field An electron enters a homogeneous magnetic field with its velocity perpendicular to the field direction. What path does it follow (neglecting electric fields and radiation)?

Introduction / Context:
Charged particles in magnetic fields experience the Lorentz force, which bends their trajectories. Understanding the resulting motion is fundamental in cyclotrons, mass spectrometers, and magnetron orbits, and in plasma confinement.

Given Data / Assumptions:

• Uniform magnetic field B.
• Initial velocity v is exactly perpendicular to B.
• No electric field and no collisions; non-relativistic speeds.

Concept / Approach:

The Lorentz force is F = q (v × B). With v ⟂ B, the force magnitude is q v B and is always perpendicular to the instantaneous velocity, doing no work (speed constant). A perpendicular, constant-magnitude force produces uniform circular motion. The radius r satisfies r = m v / (|q| B), and angular frequency (cyclotron frequency) is ω_c = |q| B / m, independent of speed and radius in this simple model.

Step-by-Step Solution:

Verification / Alternative check:

If v had a component along B, the motion would be a helix; with purely perpendicular v, the pitch is zero and the path is a circle.

Why Other Options Are Wrong:

(a) ignores the magnetic force; (b) requires v ∥ B; (c) arbitrary angle without cause; (e) would need changing B or energy loss.

Common Pitfalls:

Confusing force direction (always perpendicular) with velocity direction, or assuming magnetic fields can change kinetic energy; they do not in this ideal case.

Final Answer:

A circular path in a plane perpendicular to the field