Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part I, Chapter 2

Single Particle Motion

Particle trajectories in electromagnetic fields

2.1 Lorentz Force Equation

The motion of a charged particle in electromagnetic fields is governed by the Lorentz force:

$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$

Newton's second law gives the equation of motion:

$$m \frac{d\vec{v}}{dt} = q(\vec{E} + \vec{v} \times \vec{B})$$

Note that the magnetic force is perpendicular to velocity, so:

$$\frac{dE_{\text{kin}}}{dt} = \vec{F} \cdot \vec{v} = q\vec{E} \cdot \vec{v}$$

Key insight: The magnetic field does no work—it only changes the direction of motion, not the speed. Only electric fields can change particle energy.

2.2 Motion in Uniform Electric Field

With E = E0ẑ and B = 0:

$$m\frac{d\vec{v}}{dt} = q\vec{E}_0$$

The solution is simple uniform acceleration:

$$\vec{v}(t) = \vec{v}_0 + \frac{q\vec{E}_0}{m}t$$
$$\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2}\frac{q\vec{E}_0}{m}t^2$$

Particles gain energy from the electric field continuously.

2.3 Motion in Uniform Magnetic Field

For B = B0ẑ and E = 0, the equation of motion is:

$$m\frac{d\vec{v}}{dt} = q\vec{v} \times \vec{B}_0$$

Cyclotron Frequency

Decomposing into parallel (∥) and perpendicular (⊥) components to B:

$$\frac{dv_\parallel}{dt} = 0 \quad \Rightarrow \quad v_\parallel = \text{const}$$

For perpendicular motion, writing v⊥ = vxx̂ + vyŷ:

$$\frac{dv_x}{dt} = \omega_c v_y, \quad \frac{dv_y}{dt} = -\omega_c v_x$$

where ωc is the cyclotron frequency (or gyrofrequency):

$$\boxed{\omega_c = \frac{|q|B}{m}}$$

Circular Trajectory

The solution describes circular motion in the plane perpendicular to B:

$$v_x(t) = v_\perp \cos(\omega_c t + \phi_0), \quad v_y(t) = -v_\perp \sin(\omega_c t + \phi_0)$$

The Larmor radius (or gyroradius) is:

$$\boxed{r_L = \frac{v_\perp}{\omega_c} = \frac{m v_\perp}{|q|B}}$$

The complete trajectory is a helix: circular motion in the perpendicular plane combined with uniform motion along the field line.

Typical Values

  • • Electrons in B = 1 T: fce = 28 GHz, rL ≈ 0.18 mm (at 10 eV)
  • • Protons in B = 1 T: fci = 15 MHz, rL ≈ 7.2 mm (at 10 eV)
  • • Earth's field (B ≈ 50 μT): fce ≈ 1.4 MHz

2.4 E×B Drift

With both E and B present (perpendicular to each other), the equation of motion is:

$$m\frac{d\vec{v}}{dt} = q(\vec{E} + \vec{v} \times \vec{B})$$

Decompose velocity into gyration plus a drift: v = vgyro + vd. Averaging over a gyroperiod cancels the gyration term, leaving:

$$0 = q(\vec{E} + \vec{v}_d \times \vec{B})$$

Cross both sides with B:

$$q(\vec{E} \times \vec{B}) = q \vec{v}_d \times \vec{B} \times \vec{B} = -q v_d B^2 + q(\vec{v}_d \cdot \vec{B})\vec{B}$$

Since vd ⊥ B, the second term vanishes, giving:

$$\boxed{\vec{v}_E = \frac{\vec{E} \times \vec{B}}{B^2}}$$

Key Properties of E×B Drift

  • • Independent of particle mass and charge (both electrons and ions drift together)
  • • Perpendicular to both E and B
  • • No energy gain: vE · E = 0
  • • Maintains quasi-neutrality

The complete motion is: helical gyration around the field line, drifting perpendicular to bothE and B at velocity vE.

2.5 Guiding Center Approximation

When rL << L (where L is the scale length of field variations), we can separate the rapid gyromotion from the slower drift of the guiding center.

Write the particle position as:

$$\vec{r}(t) = \vec{R}(t) + \vec{\rho}(t)$$

where R(t) is the guiding center position and ρ(t) is the gyration vector (|ρ| = rL).

Guiding Center Equations

The guiding center motion is governed by:

$$\frac{d\vec{R}}{dt} = v_\parallel \hat{b} + \vec{v}_d$$

where b̂ = B/B is the unit vector along the field, and vdincludes all drift velocities (E×B, ∇B, curvature, etc.).

The parallel velocity evolves according to:

$$m\frac{dv_\parallel}{dt} = q E_\parallel - \mu \frac{\partial B}{\partial s}$$

where μ is the magnetic moment (discussed in Chapter 5) and s is the distance along the field line.

2.6 General Force Drifts

Any force F perpendicular to B causes a drift. The general formula is:

$$\vec{v}_F = \frac{\vec{F} \times \vec{B}}{qB^2}$$

Derivation: In the guiding center frame, average over a gyroperiod:

$$\langle \vec{F} + q\vec{v} \times \vec{B} \rangle = 0$$

Since the gyromotion averages to zero, v → vd in the average:

$$\vec{F} = -q \vec{v}_d \times \vec{B}$$

Cross with B and solve for vd to obtain the general drift formula above.

Examples of Force Drifts

  • • Gravity drift: F = mg → vg = (m/q)(g × B)/B²
  • • Centrifugal drift: F = mv²/R → drift in curved fields
  • • Gradient-B drift: Covered in Chapter 5
  • • Polarization drift: From time-varying E field

Important: Unlike E×B drift, these force drifts are charge-dependent, causing electrons and ions to drift in opposite directions, potentially generating currents.

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