Plain-English Meaning
Put a charged particle in a magnetic field and it goes in circles. The rate at which it orbits is the Larmor frequency. A heavier particle circles more slowly. A stronger magnetic field or larger charge makes it spin faster. This is why MRI machines use very strong magnets to make hydrogen nuclei precess.
Keep charge, field, potential, and current distinct. That single habit fixes a large fraction of electromagnetism errors.
Deeper Explanation
ω_L = qB/m (rad/s). The Lorentz force provides centripetal force: qvB = mv²/r → r = mv/(qB) (cyclotron radius). Frequency f = qB/(2πm) (independent of speed for non-relativistic particles). In MRI, the Larmor precession frequency of ¹H is f = (42.58 MHz/T) × B₀.
Worked Example
Problem: A proton (q = 1.6×10⁻¹⁹ C, m = 1.67×10⁻²⁷ kg) enters a 1.5 T field. Find the Larmor frequency.
- ω_L = qB/m
- ω_L = 1.6×10⁻¹⁹ × 1.5 / 1.67×10⁻²⁷
- = 2.4×10⁻¹⁹ / 1.67×10⁻²⁷ = 1.44×10⁸ rad/s
- f = ω/(2π) = 1.44×10⁸/6.28 ≈ 22.9 MHz
Result: f ≈ 22.9 MHz (this is why MRI typically operates at ~64 MHz for ¹H at 1.5 T)
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At A Glance
Category: Electromagnetism
Levels covered: High School, College, Masters, PhD
Best use: Start with the formula meaning, then move to the worked example and quiz so the equation turns into a tool instead of a memorised line.