Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part I, Chapter 5

Magnetized Plasmas

Cyclotron motion, magnetic moment, and drift velocities

5.1 Magnetic Moment

A charged particle gyrating in a magnetic field has an associated magnetic moment:

$$\mu = \frac{m v_\perp^2}{2B} = \frac{E_\perp}{B}$$

This is the magnetic moment of a current loop with current I = qωc/2π and area A = πrL²:

$$\mu = IA = \frac{q\omega_c}{2\pi} \pi r_L^2 = \frac{qv_\perp}{2\pi r_L} \pi r_L^2 = \frac{qv_\perp r_L}{2}$$

Using rL = mv⊥/qB:

$$\mu = \frac{qv_\perp}{2} \cdot \frac{mv_\perp}{qB} = \frac{mv_\perp^2}{2B}$$

Key Property: μ is an Adiabatic Invariant

If the magnetic field varies slowly (on timescales >> ωc−1 and length scales >> rL), the magnetic moment μ is conserved. This is the first adiabatic invariant.

5.2 Grad-B Drift

In a non-uniform magnetic field, particles experience a gradient-B drift. Consider ∇B perpendicular to B:

As the particle gyrates, it experiences stronger field on one side of the orbit than the other. Since rL ∝ B−1, the radius is smaller where B is larger, causing a net drift.

$$\vec{v}_{\nabla B} = \frac{1}{2}\frac{mv_\perp^2}{qB^2}(\vec{B} \times \nabla B) = \frac{\mu}{q}\frac{\vec{B} \times \nabla B}{B^2}$$

Direction: perpendicular to both B and ∇B. For ∇B pointing radially outward in a cylinder, positive charges drift one way toroidally, negative charges the opposite way.

Charge-Dependent Drift

Unlike E×B drift, grad-B drift depends on the sign of the charge. This can drive currents in the plasma, which is important for tokamak physics (vertical drift causes disruptions unless controlled).

5.3 Curvature Drift

In a curved magnetic field, particles experience a centrifugal force in the guiding center frame:

$$\vec{F}_c = m\frac{v_\parallel^2}{R_c}\hat{R}_c$$

where Rc is the radius of curvature. Using the general drift formula:

$$\vec{v}_c = \frac{m v_\parallel^2}{qB^2 R_c}(\vec{B} \times \hat{R}_c) = \frac{m v_\parallel^2}{qB^2}(\vec{B} \times \vec{\kappa})$$

where κ = b̂ · ∇b̂ is the curvature vector (b̂ = B/B).

Combined Grad-B and Curvature Drift

In most configurations, ∇B and curvature occur together. The total drift is:

$$\vec{v}_d = \vec{v}_{\nabla B} + \vec{v}_c = \frac{m}{qB^2}\left(\frac{v_\perp^2}{2} + v_\parallel^2\right)(\vec{B} \times \vec{\kappa})$$

In a tokamak, this drift points vertically, causing charge separation and requiring feedback control.

5.4 Polarization Drift

If the electric field varies in time, particles cannot respond instantaneously due to inertia, causing a polarization drift:

$$\vec{v}_p = \frac{m}{qB^2}\frac{d\vec{E}}{dt}$$

Derivation: The E×B drift velocity is:

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

Taking the time derivative and using F = m dv/dt:

$$m\frac{d\vec{v}_E}{dt} = \frac{m}{B^2}\frac{d\vec{E}}{dt} \times \vec{B}$$

This inertial effect produces the drift above. Unlike E×B, polarization drift is mass-dependent, so electrons and ions drift differently.

5.5 Magnetic Mirrors

Consider a particle moving along a converging magnetic field (B increases). Conservation of magnetic moment μ = mv⊥²/2B requires:

$$\frac{v_\perp^2}{B} = \text{const}$$

As B increases, v⊥ must increase. Since total energy E = m(v∥² + v⊥²)/2 is conserved:

$$v_\parallel^2 = v_0^2 - v_\perp^2 = v_0^2 - \frac{2\mu B}{m}$$

When B increases to the mirror point B = Bm, v∥ → 0 and the particle reflects.

Mirror Ratio and Loss Cone

The mirror ratio is Rm = Bmax/Bmin. Particles mirror if:

$$\frac{v_\perp^2}{v^2} > \frac{B_{min}}{B_{max}} = \frac{1}{R_m}$$

Particles with pitch angles in the loss cone (small v⊥/v) are not confined and escape along field lines. The loss cone angle is:

$$\sin\theta_c = \frac{1}{\sqrt{R_m}}$$

Applications

  • • Earth's radiation belts: Particles bounce between magnetic poles
  • • Magnetic mirror machines: Fusion concept (abandoned due to end losses)
  • • Solar coronal loops: Plasma confined in magnetic loops

5.6 Summary of Drifts

Complete perpendicular velocity in magnetized plasma:

$$\vec{v}_\perp = \vec{v}_E + \vec{v}_{\nabla B} + \vec{v}_c + \vec{v}_p + \vec{v}_{gyr}$$

Charge-Independent Drifts

  • • E×B drift: vE = (E × B)/B²

Charge-Dependent Drifts (can drive currents)

  • • Grad-B drift: v∇B = (μ/q)(B × ∇B)/B²
  • • Curvature drift: vc = (mv∥²/q)(B × κ)/B²
  • • Polarization drift: vp = (m/qB²)dE/dt
  • • Gravitational drift: vg = (m/q)(g × B)/B²

These drifts are fundamental to understanding plasma confinement, transport, and stability in magnetic fusion devices.

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