Motion of Charged Particles in a Uniform Magnetic Field

When a charged particle moves through a uniform magnetic field, it experiences a magnetic (Lorentz) force that is always perpendicular to its velocity. This force does no work on the particle — it changes only the direction of motion, never the speed.

Think of a ball tied to a string and whirled in a circle. The string always pulls the ball toward the centre (never speeds it up or slows it down) — exactly how the magnetic force acts on a charged particle. This is why circular/helical paths, not straight-line acceleration, result.

A charge q moving with velocity v at an angle θ to a uniform field B experiences a force:

This is the magnitude form of the Lorentz force law, \[\vec F\] = q(\[\vec v\] × \[\vec B\]).

Velocity Parallel to Field (θ = 0° or 180°)

• F = qv B sin⁡(0°) = 0
• No magnetic force acts on the particle.
• Result: The particle continues in a straight line, undeviated, at constant speed.

Velocity Perpendicular to Field (θ = 90°)

• F = qvB acts as the centripetal force.
• The particle moves in a uniform circular path in the plane perpendicular to B.

Charge sign effect:

• Positive charge → curves one way (say, clockwise, per right-hand rule).
• Negative charge → curves the opposite way (anticlockwise).

Radius in terms of other quantities:

• r = \[\frac{mv}{qB}=\frac{p}{qB}=\frac{\sqrt{2mK}}{qB}=\frac{1}{B}\sqrt{\frac{2mV}{q}}\]

where p = momentum, K = kinetic energy, V = accelerating potential.

Velocity at an Angle θ to Field (Oblique Entry)

• Velocity is resolved into two components:
  Parallel: v_∥ = v cos ⁡θ → produces uniform straight-line motion along B
  Perpendicular: v_⊥ = v sin⁡ θ → produces circular motion
• Combined effect: the particle traces a helical path along the direction of B.

• Cyclotron: Uses the circular motion of charged particles in a magnetic field to accelerate them to high energies.
• Mass Spectrometer: Uses the radius formula r = \[\frac {mv}{qB}\]​ to separate ions by mass-to-charge ratio.
• Van Allen Belts: Charged particles from solar wind spiral helically along Earth's magnetic field lines — a real-world helical motion example.