11.3 Motion of a Charged Particle in a Magnetic Field

Charged particles in magnetic fields move in fascinating ways, creating circular or helical paths depending on their initial velocity. The magnetic force acts perpendicular to both the particle's motion and the field, causing cyclotron motion without changing the particle's speed.

Understanding these motions is crucial for many applications, from particle accelerators to plasma confinement in fusion reactors. The radius and period of the circular path depend on the particle's properties and the magnetic field strength, while helical trajectories occur when particles enter at an angle.

Motion of a Charged Particle in a Uniform Magnetic Field

Motion of charged particles in magnetic fields

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• Charged particle experiences magnetic force when moving through magnetic field
  • Magnetic force is perpendicular to both particle's velocity and magnetic field (right-hand rule)
  • Force acts as centripetal force, causing particle to move in circular path (cyclotron motion)
• Direction of circular motion depends on sign of particle's charge
  • Positive charges (protons) circle counterclockwise
  • Negative charges (electrons) circle clockwise
• Particle's speed remains constant as magnetic force does no work on particle
  • Magnetic force only changes direction of particle's velocity, not magnitude (kinetic energy conserved)

Circular path calculations for charged particles

• Radius of circular path depends on particle's mass ($m$), charge ($q$), velocity ($v$), and magnetic field strength ($B$)
  • Radius ($r$) given by: $r = \frac{mv}{qB}$ (also known as gyroradius)
    • Larger mass (heavy ions) or velocity results in larger radius
    • Larger charge (alpha particles) or magnetic field strength results in smaller radius
• Period ($T$) of circular motion is time for particle to complete one revolution
  • Period given by: $T = \frac{2\pi m}{qB}$
    • Period depends on particle's mass, charge, and magnetic field strength
    • Period independent of particle's velocity (frequency of revolution constant)
  • The inverse of the period is known as the cyclotron frequency

Helical trajectories in uniform magnetic fields

• When charged particle enters magnetic field with velocity not perpendicular to field, path becomes helix
  • Particle's velocity decomposed into components parallel and perpendicular to magnetic field
  • Perpendicular component causes particle to move in circular path (as described above)
  • Parallel component unaffected by magnetic field, causing particle to move along field lines (uniform motion)
• Resulting motion combines circular motion in plane perpendicular to magnetic field and constant velocity along field lines
  • Combination creates helical path (spiral motion)
  • Pitch of helix depends on ratio of parallel velocity component to perpendicular velocity component
    • Larger parallel component results in longer pitch (stretched helix)
    • Larger perpendicular component results in shorter pitch (compressed helix)
  • The angle between the particle's velocity vector and the magnetic field lines is called the pitch angle

Particle behavior in non-uniform magnetic fields

• In non-uniform magnetic fields, particles can experience magnetic mirroring
• Magnetic moment of a particle is conserved in slowly varying magnetic fields
• As a particle moves into a stronger magnetic field region, its pitch angle increases
• If the magnetic field strength increases sufficiently, the particle can be reflected back towards weaker field regions