3.1 Charged particle motion in electric and magnetic fields

Charged particles in electric and magnetic fields dance a complex ballet. Their motion is governed by the Lorentz force, causing cyclotron motion, drifts, and helical trajectories. Understanding these movements is key to grasping plasma behavior.

This topic dives into the nitty-gritty of particle motion. We'll explore concepts like gyroradius, gyrofrequency, and guiding center approximation. These ideas form the foundation for understanding more complex plasma phenomena.

Particle Motion in Magnetic Fields

Cyclotron Motion and Gyroradius

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• Cyclotron motion describes circular movement of charged particles in uniform magnetic fields
• Particles rotate perpendicular to magnetic field lines due to Lorentz force
• Gyroradius measures radius of circular path, depends on particle mass, charge, velocity, and magnetic field strength
• Larger gyroradius indicates wider circular path, smaller gyroradius results in tighter orbits
• Equation for gyroradius: $r = \frac{mv_\perp}{qB}$
  • m: particle mass
  • v_⊥: velocity component perpendicular to magnetic field
  • q: particle charge
  • B: magnetic field strength
• Heavier particles or those with higher perpendicular velocities have larger gyroradii
• Stronger magnetic fields decrease gyroradius, resulting in tighter orbits

Gyrofrequency and Helical Trajectory

• Gyrofrequency represents number of circular rotations particle completes per second
• Inversely proportional to particle mass, directly proportional to magnetic field strength and charge
• Equation for gyrofrequency: $\omega_c = \frac{qB}{m}$
• Higher gyrofrequency indicates faster rotation around magnetic field lines
• Helical trajectory combines circular motion with linear motion along magnetic field lines
• Pitch angle determines shape of helical path, measures angle between velocity vector and magnetic field line
• Smaller pitch angles result in elongated helices, larger angles create tighter spirals
• Larmor radius equivalent to gyroradius, named after Joseph Larmor who studied charged particle motion

Guiding Center Motion

Guiding Center Approximation

• Guiding center approximation simplifies analysis of charged particle motion in magnetic fields
• Separates particle motion into rapid gyration around field lines and slower drift of gyration center
• Useful for studying long-term particle behavior in complex magnetic field configurations
• Approximation most accurate when magnetic field changes slowly compared to gyration period
• Allows focus on average particle position rather than detailed circular motion
• Simplifies calculations for particle trajectories in non-uniform magnetic fields
• Particularly valuable in plasma physics and space physics applications (magnetosphere studies)

E×B Drift and Other Drift Motions

• E×B drift occurs when electric and magnetic fields are present simultaneously
• Particles drift perpendicular to both electric and magnetic field directions
• Drift velocity equation: $v_d = \frac{E \times B}{B^2}$
• E×B drift independent of particle charge, mass, or velocity
• Results in collective motion of charged particles in plasmas
• Other drift motions include:
  • Gradient drift: caused by magnetic field gradients
  • Curvature drift: occurs in curved magnetic field lines
  • Gravitational drift: due to gravitational forces in magnetized plasmas
• Understanding drift motions crucial for plasma confinement in fusion devices (tokamaks)

Electromagnetic Forces

Lorentz Force and Its Effects

• Lorentz force fundamental electromagnetic force acting on charged particles
• Combines effects of electric and magnetic fields on moving charges
• Equation for Lorentz force: $F = q(E + v \times B)$
  • F: force vector
  • q: particle charge
  • E: electric field vector
  • v: particle velocity vector
  • B: magnetic field vector
• Electric field component (qE) acts parallel to field lines, accelerates or decelerates particles
• Magnetic field component (qv×B) acts perpendicular to both velocity and field, causes circular motion
• Lorentz force explains various phenomena in plasma physics and astrophysics (auroras)
• Applications include:
  • Particle accelerators (cyclotrons, synchrotrons)
  • Mass spectrometers for particle separation
  • Hall effect devices in electronic sensors