Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 1

Table of contents
- a 1
  - Solution a
- b 1
  - Solution b
- c 1
  - Solution c
- d 1
  - Solution d
- e 1
  - Solution e
- f 1
  - Solution f

a

Solution

Use Gaussian units, plane waves proportional to

e^{i k \cdot r - iω t}

, and the signed cyclotron frequency

Ω_{s} = q_{s} B_{0} / (m_{s} c)

. Write

Ω_{e} = e B_{0} / (m_{e} c) > 0

for its electron magnitude, so the signed electron value is

- Ω_{e}

. Define the species plasma frequency by

ω_{p s}^{2} = 4 π n_{0 s} q_{s}^{2} / m_{s}

. The linearization of the cold Lorentz force equations gives

- iω u_{x s} - Ω_{s} u_{ys} = \frac{q _{s}}{m _{s}} E_{x}, Ω_{s} u_{x s} - iω u_{ys} = \frac{q _{s}}{m _{s}} E_{y}, - iω u_{zs} = \frac{q _{s}}{m _{s}} E_{z} .

Solving this linear system and using the species current density

j_{s} = n_{0 s} q_{s} u_{s}

gives the conductivity tensor

σ_{s} = \frac{n _{0 s} q _{s}^{2}}{m _{s}} \frac{iω}{ω ^{2} - Ω _{s}^{2}} \frac{Ω _{s}}{ω ^{2} - Ω _{s}^{2}} 0 - \frac{Ω _{s}}{ω ^{2} - Ω _{s}^{2}} \frac{iω}{ω ^{2} - Ω _{s}^{2}} 0 00 \frac{i}{ω} .

The magnetized cold-plasma dielectric tensor is

ϵ = I + (4 πi / ω) \sum_{s} σ_{s}

. Thus

ϵ = S - i g 0 i g S 0 00 P, S = ϵ_{⊥} = 1 - \sum_{s} \frac{ω _{p s}^{2}}{ω ^{2} - Ω _{s}^{2}}, P = ϵ_{∥} = 1 - \sum_{s} \frac{ω _{p s}^{2}}{ω ^{2}}, g = - \sum_{s} \frac{ω _{p s}^{2} Ω _{s}}{ω ( ω ^{2} - Ω _{s}^{2} )} .

At the stated electron frequency scale, the ion contribution is smaller by

O (Z m_{e} / m_{i})

for

q_{i} = Z e

and quasineutrality. To this accuracy,

S = 1 - \frac{ω _{p e}^{2}}{ω ^{2} - Ω _{e}^{2}}, P = 1 - \frac{ω _{p e}^{2}}{ω ^{2}}, g = \frac{ω _{p e}^{2} Ω _{e}}{ω ( ω ^{2} - Ω _{e}^{2} )} .

These expressions require negligible collisions and the cold ordering

k v_{th, s} ≪ ∣ ω ∣

; sufficiently close to a cyclotron resonance, thermal broadening or collisions must also be retained.

b

Solution

The Faraday law and Ampere-Maxwell law give

δ B = (c / ω) k \times δ E

and

k \times (k \times δ E) + \frac{ω ^{2}}{c ^{2}} ϵ \cdot δ E = 0.

With refractive index

n = k c / ω

, the linear system is therefore

S - n^{2} cos^{2} θ - i g n^{2} sin θ cos θ i g S - n^{2} 0 n^{2} sin θ cos θ 0 P - n^{2} sin^{2} θ δ E = 0.

R = S - g, L = S + g, R L = S^{2} - g^{2} .

Here

R

labels the cold-plasma right-handed wave and

L

the cold-plasma left-handed wave, with their polarizations fixed explicitly in d. In the electron-only response,

R = 1 - \frac{ω _{p e}^{2}}{ω ( ω - Ω _{e} )}, L = 1 - \frac{ω _{p e}^{2}}{ω ( ω + Ω _{e} )} .

N = n^{2}

, expansion of the determinant yields the dispersion relation

A N^{2} - BN + C = 0, A = S sin^{2} θ + P cos^{2} θ, B = R L sin^{2} θ + PS (1 + cos^{2} θ), C = PR L .

Equivalently,

P (N - R) (N - L) cos^{2} θ + (SN - R L) (N - P) sin^{2} θ = 0.

tan^{2} θ = - \frac{P ( n ^{2} - R ) ( n ^{2} - L )}{( S n ^{2} - R L ) ( n ^{2} - P )} .

The undivided determinant equation remains valid when this rational form has simultaneous zero factors.

c

Solution

A plasma wave resonance has

n^{2} \to \infty

. At finite coefficients of the magnetized cold-plasma dielectric tensor, its generic condition is the vanishing of the coefficient of

n^{4}

:

S sin^{2} θ + P cos^{2} θ = 0, tan^{2} θ = - P / S .

A plasma wave cutoff has

n^{2} = 0

, so the constant term of the dispersion relation vanishes:

PR L = 0, that is P = 0, R = 0, or L = 0.

Both statements are branch conditions: check the undivided dispersion relation at special angles and simultaneous zero factors. In particular, the parallel plasma oscillation at

P = 0

is a longitudinal branch with arbitrary

k

in the cold limit, while the transverse parallel branches have plasma wave cutoffs at

R = 0

or

L = 0

. Poles of the magnetized cold-plasma dielectric tensor can give additional limiting plasma wave resonances, such as the parallel cold-plasma right-handed wave at

ω = Ω_{e}

; these are not obtained by assuming finite

S, P, g

.

d

Solution

For parallel propagation, use the undivided dispersion relation:

P (n^{2} - R) (n^{2} - L) = 0.

The five requested cases are:

Plasma oscillations: $P = 0$ and $δ E ∥ B_{0} ∥ k$ . With negligible ion response, $ω^{2} = ω_{p e}^{2}$ , independently of $k$ in this cold approximation.
Cold-plasma right-handed waves: $n^{2} = R$ , $E_{y} = i E_{x}$ , $E_{z} = 0$ , for propagation along $+ B_{0}$ . At a fixed point, the real electric field rotates from $+ \hat{x}$ towards $+ \hat{y}$ , in the sense of electron gyration.
Cold-plasma left-handed waves: $n^{2} = L$ , $E_{y} = - i E_{x}$ , $E_{z} = 0$ . The real electric field rotates in the opposite sense.
Ordinary plasma waves: for perpendicular propagation, $θ = π /2$ , $n^{2} = P$ , and $δ E ∥ B_{0}$ . Their dispersion relation is $ω^{2} = ω_{p e}^{2} + c^{2} k^{2}$ .
Extraordinary plasma waves: for perpendicular propagation, $n^{2} = R L / S$ , $E_{z} = 0$ , and $E_{y} = i g E_{x} / (S - n^{2})$ . In general the electric field has both longitudinal and transverse components.

For the perpendicular cases the undivided dispersion relation factors as

(n^{2} - P) (S n^{2} - R L) = 0

, up to an overall nonzero factor. This also identifies the extraordinary plasma wave plasma wave resonance at

S = 0

.

e

Solution

An electrostatic plasma wave has

δ E = - i k ϕ

. In the absence of externally imposed charge density, the Gauss law for the magnetized cold-plasma dielectric tensor is

k \cdot ϵ \cdot δ E = 0

. Consequently,

k_{⊥}^{2} S + k_{∥}^{2} P = 0.

The antisymmetric terms of the magnetized cold-plasma dielectric tensor cancel in this scalar contraction. For oblique propagation this is the leading longitudinal condition in the large-refractive index approximation; a small transverse electric field is generally required by the full Maxwell equations at finite

n

. For parallel propagation an exactly longitudinal solution exists:

P = 0

, hence

ω^{2} ≃ ω_{p e}^{2}

. It coincides with the

θ \to 0

limit of the finite-coefficient plasma wave resonance condition in c. At exactly

θ = 0

,

P

factors out of the transverse dispersion relation, leaving the independent plasma oscillation rather than an isolated divergence of the transverse branches.

f

Solution

For the ordinary plasma wave, choose

k = k \hat{x}

and

δ E = E_{z} \hat{z}

. The equations in a give

u_{x s} = u_{ys} = 0

and

u_{zs} = \frac{i q _{s}}{m _{s} ω} E_{z} .

The driven particle motion is entirely parallel to

B_{0}

: the parallel Lorentz force has no magnetic contribution. Nevertheless, the ordinary plasma wave is electromagnetic, since

δ E ⊥ k

and the Faraday law gives

δ B = - \frac{c k}{ω} E_{z} \hat{y} \neq = 0.

An electrostatic plasma wave instead has

δ E ∥ k

and zero perturbed magnetic field. Alignment with the equilibrium magnetic field does not make a wave electrostatic. At the exact plasma wave cutoff

k = 0

, this propagating-wave distinction becomes degenerate.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version