2.2 Single-particle motion in electromagnetic fields

Charged particles in electromagnetic fields dance to the tune of the Lorentz force. This fundamental concept shapes how plasmas behave, from the lab to space. Understanding single-particle motion is key to grasping larger plasma phenomena.

In this topic, we'll explore how particles move in electric and magnetic fields. We'll cover trajectories, drifts, and adiabatic invariants - essential building blocks for comprehending plasma physics and its wide-ranging applications.

Charged Particle Motion in Fields

Lorentz Force and Equations of Motion

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• Lorentz force equation $F = q(E + v × B)$ describes force experienced by charged particle in electromagnetic fields
• Combining Lorentz force with Newton's second law $F = ma$ derives equations of motion for charged particles
• Equations of motion expressed as system of coupled differential equations in three dimensions
• Particle motion governed by electric field E and magnetic field B
• Charge-to-mass ratio (q/m) significantly influences particle trajectory
• Principle of superposition allows analysis of complex field configurations by summing effects of individual components
• Time-varying electromagnetic fields incorporated using Maxwell's equations

Factors Influencing Particle Behavior

• Particle charge determines direction of force (positive charges move with E, negative charges move against)
• Particle mass affects acceleration and trajectory curvature (heavier particles experience less deflection)
• Field strength directly proportional to force experienced by particle
• Field geometry shapes overall trajectory (uniform fields vs non-uniform fields)
• Initial conditions (position and velocity) crucial for predicting particle path
• Relativistic effects become important for particles moving at high speeds (close to speed of light)

Trajectories in Uniform Fields

Motion in Electric Fields

• Uniform electric field causes charged particles to follow parabolic trajectories due to constant acceleration
• Acceleration direction parallel to electric field lines for positive charges, antiparallel for negative charges
• Trajectory shape independent of particle mass, but acceleration magnitude inversely proportional to mass
• Examples: electron beam in cathode ray tube, ion propulsion in spacecraft
• Equipotential surfaces perpendicular to electric field lines useful for visualizing particle energy changes

Motion in Magnetic Fields

• Uniform magnetic field results in helical particle paths, combining circular motion perpendicular to field with uniform motion parallel to field
• Radius of circular motion (gyroradius) determined by particle velocity, mass, charge, and magnetic field strength
• Pitch angle defined as angle between particle's velocity vector and magnetic field direction
• Right-hand rule determines direction of circular motion (clockwise for positive charges, counterclockwise for negative charges)
• Examples: cyclotron particle accelerator, magnetron in microwave ovens

Combined Electric and Magnetic Fields

• E × B drift occurs in presence of perpendicular electric and magnetic fields
• Drift velocity $v_d = E × B / B^2$ perpendicular to both E and B, independent of particle properties
• Guiding center approximation simplifies analysis by focusing on average position of particle's gyration
• Special cases like perpendicular or parallel field orientations lead to distinct trajectory patterns
• Applications: mass spectrometry, plasma confinement in fusion reactors

Gyroradius, Gyrofrequency, and Mirrors

Gyroradius and Gyrofrequency

• Gyroradius (Larmor radius) given by $r = mv_⊥ / |q|B$, radius of circular motion in uniform magnetic field
• Gyrofrequency (cyclotron frequency) expressed as $ω = |q|B / m$, angular frequency of circular motion
• Magnetic moment of gyrating particle $μ = mv_⊥² / 2B$, an adiabatic invariant
• Gyroradius inversely proportional to magnetic field strength (stronger field results in tighter gyration)
• Gyrofrequency independent of particle velocity, only depends on charge-to-mass ratio and field strength
• Applications: cyclotron resonance mass spectrometry, electron cyclotron resonance heating in plasma physics

Magnetic Mirrors and Particle Confinement

• Magnetic mirrors use non-uniform magnetic fields to reflect charged particles under certain conditions
• Mirror force arises from gradient in magnetic field strength, given by $F = -μ∇B$
• Loss cone region in velocity space where particles escape confinement instead of being reflected
• Mirror ratio (ratio of maximum to minimum magnetic field strengths) determines confinement effectiveness
• Examples: Van Allen radiation belts, magnetic bottle for plasma confinement
• Pitch angle changes as particle moves through varying magnetic field strength
• Applications: fusion reactor designs, space weather studies

Particle Drifts and Adiabatic Invariants

Types of Particle Drifts

• Gradient B drift in non-uniform magnetic fields, perpendicular to both B and ∇B
• Curvature drift from centrifugal force experienced by particles moving along curved magnetic field lines
• E × B drift causes bulk plasma motion in crossed electric and magnetic fields, independent of particle properties
• Polarization drift results from time-varying electric fields, depends on particle's charge-to-mass ratio
• Gravitational drift in presence of gravitational field and magnetic field
• Drift velocities typically much smaller than particle's thermal velocity
• Applications: plasma diagnostics, space plasma phenomena (magnetospheric convection)

Adiabatic Invariants

• First adiabatic invariant: magnetic moment μ, constant in slowly varying magnetic fields
• Second adiabatic invariant J associated with longitudinal motion of particles bouncing between magnetic mirrors
• Third adiabatic invariant Φ related to magnetic flux enclosed by particle's drift shell in dipole-like magnetic field
• Adiabatic invariants conserved when field changes occur slowly compared to particle's periodic motion
• Used to analyze particle behavior in complex, slowly varying electromagnetic field configurations
• Examples: radiation belt dynamics, plasma confinement in tokamaks
• Breakdown of adiabatic invariants leads to particle energization or loss in space plasmas