Fundamentals of Plasma Physics

6.2 Dielectric tensor 179

One might expect that a procedure analogous to the previous derivation of electrostatic
susceptibilities could be used to derive inductive “susceptibilities” which would then be
used to construct dispersion relations for inductive modes. It turns out that such a procedure
not only gives dispersion relations for inductive modes, but also includes the electrostatic
modes. Thus, it turns out to be unnecessary to analyze electrostatic modes separately. The
main reason for investigating electrostatic modes separately as done earlier is pedagogical

• it is easier to understand a simpler system. To see why electrostatic modes are auto-
  matically included in an electromagnetic analysis, consider the interrelationship between
  Poisson’s equation, Ampere’s law, and a charge-weighted summation of thetwo-fluid con-
  tinuity equation,

∇·E=−

1

ε 0

∑

σ

nσqσ, (6.1)

∇×B=μ 0 J+ε 0 μ 0

∂E

∂t

, (6.2)

∑

σ

qσ

[

∂nσ

∂t

+∇·(nσvσ)

]

=

[

∂

∂t

∑

σ

nσqσ

]

+∇·J=0. (6.3)

The divergence of Eq.(6.2) gives

∇·J+ε 0

∂

∂t

∇·E=0 (6.4)

and substituting Eq.(6.3) gives

∂

∂t

[

−

∑

σ

nσqσ+ε 0 ∇·E

]

=0 (6.5)

which is just the time derivative of Poisson’s equation. Integrating Eq.(6.5) shows that

−

∑

σ

nσqσ+ε 0 ∇·E=const. (6.6)

Poisson’s equation, Eq.(6.1), thus provides aninitial conditionwhich fixes the value of
the constant in Eq.(6.6). Since all small-amplitude perturbations are assumed to have the
phase dependenceexp(ik·x−iωt)and therefore behave as a single Fourier mode, the
∂/∂toperator in Eq. (6.5) is replaced by−iωin which case the constant in Eq.(6.6) is
automatically set to zero, making a separate consideration of Poisson’s equation redundant.
In summary, the Fourier-transformed Ampere’s law effectively embeds Poisson’s equation
and so a discussion of waves based solely on currents describes inductive, electrostatic
modes and also contains modes involving a mixture of inductive and electrostatic electric
fields such as the inertial Alfvén wave.

6.2 Dielectric tensor

Section 3.8 showed that a single particle immersed in a constant, uniform equilibrium mag-
netic fieldB=B 0 ˆzand subject to asmall-amplitudewave with electric field∼exp(ik·x−
iωt)has the velocity

̃vσ=

iqσ

ωmσ

[

E ̃zzˆ+

E ̃⊥

1 −ω^2 cσ/ω^2

−

iωcσ

ω

ˆz×E ̃

1 −ω^2 cσ/ω^2

]

eik·x−iωt. (6.7)