Single Particle Motion

Field equations: Maxwell's equations & Electric Force, Magnetic - Lorentz Force:
The electric current is defined as: $j = e (n_{i} v_{i} - n_{e} v_{e})$
and the electric space charge density : $ρ = e (n_{i} - n_{e})$

Uniform magnetic field

![[plasma_lecture3.mp4]]

Gyration

Conditions: Single particle under time invariant magnetic field and $E = 0$
Equation of motion:

m \frac{d \vec{v}}{d t} = q (\vec{v} \times \vec{B})

Taking the dot product with v and using Vector relations $v \cdot (v \times B)$ :

m \frac{d \vec{v}}{d t} \cdot \vec{v} = \frac{d}{d t} (\frac{m v^{2}}{2}) = 0

The particles Kinetic Energy is constant under a magnetic field (even if it is spatially variant)
Conditions: Single particle in Uniform Magnetic field $\vec{B} = B \vec{z}$ , $E = 0$

m {\dot{v}}_{x} = q B v_{y}

m {\dot{v}}_{y} = q B v_{x}

m {\dot{v}}_{z} = 0

The component parallel to the magnetic field is constant. Taking the second derivative we get a Harmonic Oscillator:

{\ddot{v}}_{x} = - ω_{g}^{2} v_{x}

{\ddot{v}}_{y} = - ω_{g}^{2} v_{y}

where $ω_{g}$ : gyrofrequency or cyclotron frequency (charge dependent sign)

ω_{g} = \frac{q B}{m}

and the corresponding gyroradius:

r_{g} = \frac{v_{⊥}}{| ω_{g} |} = \frac{m v_{⊥}}{| q | B}

We can also derive the gyroradius from the fact that the kinetic energy as well as the parallel component of the velocity are constant, which means that the magnitude of the perpandicular velocity has to be constant too ==> circular motion - Balance with centrifugal force :

m \frac{v_{⊥}^{2}}{r_{g}} = | q | v_{⊥} B

The direction of the circular orbit is such that it creates a magnetic field opposite to the one that induced it ==> Conservation of Angular Momentum - Self-Inductance
The ratio between the perpendicular and parallel components of the velocity is called pitch angle and is defined as:

a = \tan^{- 1} (\frac{v_{⊥}}{v_{∥}})

Questions

Magnetic moment is an adiabatic invariant

Uniform magnetic and electric field

![[plasma_lecture4.mp4]]

Electric Drifts

E x B Drift

Conditions: $B = B_{z}$ , E: electrostatic field
The parallel component to the magnetic field describes an acceleration along the magnetic field lines $m \dot{v_{∥}} = q E_{∥}$ . However in geophysical plasmas most parallel electric fields cannot be sustained due to the high mobility of the electrons along the lines. Assuming perpendicular electric field: $E_{⊥} = E_{x} {\hat{e}}_{x}$

{\dot{v}}_{x} = ω_{g} v_{y} + \frac{q}{m} E_{x}

{\dot{v}}_{y} - ω_{g} v_{x}

Second derivative:

{\ddot{v}}_{x} = - ω_{g}^{2} v_{x}

{\ddot{v}}_{y} = - ω_{g}^{2} (v_{y} + \frac{E_{x}}{B})

This describes a gyration with a superimposed drift of the guiding center in the -y direction:

v_{E} = \frac{E \times B}{B^{2}}

Independent of the sign of the charge and across the ExB direction. An ion is accelerated in the first half of the motion, increasing its gyroradius and the decelerating decreasing it again => shift in the position.

This drift has a fundamental physical root in the Lorentz transformation of the electric field into the moving system of the particle. Transformation does not depend on charge -> drift is also independent.
timestamp : 6:00

Single Particle Motion

Uniform magnetic field

Gyration

Questions

Uniform magnetic and electric field

Electric Drifts

E x B Drift

Polarization Drift

Questions

Plasma confinement based on single particle motion

Questions