Magnetic fields and charged-particle motion
A magnetic field changes the direction of a moving charge without changing its speed when no electric field or other force does work.
Lorentz force
For charge q moving with velocity v in field B, F = qv × B and magnitude F=|q|vB sinθ. The force is perpendicular to both v and B. Use the right-hand rule for a positive charge and reverse the direction for a negative charge. A stationary charge feels no magnetic force.
Paths in a uniform field
If v is perpendicular to B, the magnetic force supplies centripetal force: |q|vB=mv²/r, so r=mv/(|q|B). The period is T=2πm/(|q|B), independent of speed in the non-relativistic model. A velocity component parallel to B remains unchanged, producing a helix. Kinetic energy stays constant because F·v=0.
Crossed fields and devices
In a velocity selector, perpendicular electric and magnetic forces cancel for v=E/B. A mass spectrometer then uses r=mv/(|q|B) to separate mass-to-charge ratios. A cyclotron alternates acceleration across a gap while a magnetic field bends particles into growing semicircles; relativistic effects eventually limit the simple resonance condition.
- Draw v, B and charge sign.
- Find force direction and magnitude.
- Identify straight, circular or helical motion.