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
Top images from around the web for Lorentz Force and Equations of Motion 11.3 Motion of a Charged Particle in a Magnetic Field – University Physics Volume 2 View original
Is this image relevant?
1 of 3
Top images from around the web for Lorentz Force and Equations of Motion 11.3 Motion of a Charged Particle in a Magnetic Field – University Physics Volume 2 View original
Is this image relevant?
1 of 3
Lorentz force equation F = q ( E + v × B ) F = q(E + v × B) F = q ( E + v × B ) describes force experienced by charged particle in electromagnetic fields
Combining Lorentz force with Newton's second law F = m a F = ma 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)
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 v_d = E × B / B^2 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 = m v ⊥ / ∣ q ∣ B r = mv_⊥ / |q|B r = m v ⊥ /∣ q ∣ B , radius of circular motion in uniform magnetic field
Gyrofrequency (cyclotron frequency) expressed as ω = ∣ q ∣ B / m ω = |q|B / m ω = ∣ q ∣ B / m , angular frequency of circular motion
Magnetic moment of gyrating particle μ = m v ⊥ 2 / 2 B μ = mv_⊥² / 2B μ = m v ⊥ 2 /2 B , 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 F = -μ∇B 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