On this page:
Introduction Motion in Electric Fields Linear Acceleration Deflection in Electric Fields Motion in Magnetic Fields Circular Motion in Magnetic Fields Helical Motion Velocity Selector Mass Spectrometer Cyclotron: Particle Accelerator Comparing Motions Hall Effect Applications in Technology Why This Matters
In the previous lesson, we learned that electric and magnetic fields exert forces on charged particles. Now we explore what happens when particles actually move through these fields.
This isn't just theoretical physics. When you watch an old cathode ray tube TV, you're seeing electrons steered by electric and magnetic fields to paint images on the screen. When doctors use an MRI machine, they're manipulating the motion of hydrogen nuclei in your body using powerful magnetic fields. When physicists discover new particles at CERN's Large Hadron Collider, they're accelerating protons to near light speed using electric fields and bending their paths with magnets.
Understanding how charged particles move in electromagnetic fields is the foundation for particle accelerators, mass spectrometers, particle detectors, electron microscopes, and countless other technologies.
Let's explore how we can control particle motion using fields.
F = qE = ma
a = qE/m
For a particle with charge q and mass m in a uniform electric field, the acceleration depends on the charge-to-mass ratio.
qΔV = ΔKE = ½mv² - ½mu²
Starting from rest: v = √(2qΔV/m)
Example: An electron (m = 9.11 × 10⁻³¹ kg) is accelerated from rest through 1000 V. Find final speed.
v = √(2qΔV/m) = √[(2)(1.60 × 10⁻¹⁹)(1000)/(9.11 × 10⁻³¹)] = 1.87 × 10⁷ m/s (about 6% of light speed)
When a particle enters perpendicular to the field:
The path is parabolic (like projectile motion).
Example: Electron enters horizontally at 2.0 × 10⁷ m/s between plates with E = 5.0 × 10⁴ N/C. Plates are 0.10 m long. Find vertical deflection.
t = L/v₀ = 5.0 × 10⁻⁹ s
a = qE/m = (1.60 × 10⁻¹⁹)(5.0 × 10⁴)/(9.11 × 10⁻³¹) = 8.78 × 10¹⁵ m/s²
y = ½at² = ½(8.78 × 10¹⁵)(5.0 × 10⁻⁹)² = 0.110 m = 11.0 cm
Magnetic forces are always perpendicular to velocity:
qvB = mv²/r → r = mv/(qB) = p/(qB)
T = 2πm/(qB) (period)
f = qB/(2πm) (cyclotron frequency)
Example: A proton (m = 1.67 × 10⁻²⁷ kg) moves at 3.0 × 10⁶ m/s perpendicular to a 0.50 T magnetic field.
r = mv/(qB) = (1.67 × 10⁻²⁷)(3.0 × 10⁶)/[(1.60 × 10⁻¹⁹)(0.50)] = 0.063 m = 6.3 cm
T = 2πm/(qB) = 2π(1.67 × 10⁻²⁷)/[(1.60 × 10⁻¹⁹)(0.50)] = 1.31 × 10⁻⁷ s = 131 ns
Key insight: Period is independent of velocity! This property is exploited in cyclotrons.
If a particle enters at an angle to the field:
Result: Helical (spiral) path around field lines.
This is why charged particles in Earth's magnetic field spiral along field lines from pole to pole, creating auroras.
qE = qvB → v = E/B
Crossed electric and magnetic fields can filter particles by velocity. When forces balance, particles pass straight through undeflected.
Example: A velocity selector has E = 2.0 × 10⁴ V/m and B = 0.10 T. What velocity passes through?
v = E/B = (2.0 × 10⁴)/(0.10) = 2.0 × 10⁵ m/s
How it works:
Radius: r = (1/B)√(2mV/q)
For singly-charged ions: r ∝ √m
Example: ²⁰Ne and ²²Ne isotopes accelerated through 1000 V, then enter B = 0.080 T. Find separation.
r₂₀ = 0.255 m, r₂₂ = 0.268 m
Separation = 2(r₂₂ - r₂₀) = 2.6 cm
Operation: Alternating electric field + constant magnetic field accelerate particles in a spiral path.
Key insight: Period T = 2πm/(qB) is independent of velocity — always in sync!
Maximum kinetic energy: KE_max = q²B²R²/(2m)
Example: A cyclotron with R = 0.50 m and B = 1.2 T accelerates protons.
KE = q²B²R²/(2m) = 17.3 MeV (mega electron volts)
| Situation | Path Shape | Energy Change | Key Formula |
|---|---|---|---|
| Uniform E (parallel to v) | Straight line | Increases/decreases | qΔV = ΔKE |
| Uniform E (perpendicular to v) | Parabola | Increases | a = qE/m |
| Uniform B (perpendicular to v) | Circle | No change | r = mv/(qB) |
| Uniform B (at angle) | Helix | No change | Circular + linear |
| Crossed E and B | Straight (if v=E/B) | Depends | Velocity selector |
When current flows through a conductor in a magnetic field perpendicular to the current, charges deflect to one side, creating a voltage across the conductor (Hall voltage).
Applications:
Understanding particle motion in fields bridges theory and application. The same physics that explains electron behavior in atoms powers the technologies defining modern civilization.
From the screens you watch to the medical diagnostics that save lives, from discovering the Higgs boson to exploring the outer solar system with ion drives — it all comes down to controlling how charged particles move through electric and magnetic fields.