Solutions to Chen's Plasma Physics

) We can solve this expression:
L ˙v || = −v || ˙L ⇒ dv|| L = 2v || v m dt (67)
2-14. In plasma heating by adiabatic compression, the invariance of µ requires that
kT ⊥ increases as B increases. The magnetic field, however, cannot accelerate particles
because the Lorentz force qv × B is always perpendicular to the velocity. How do the
particles gain energy?
Maxwell tells us that an electric field will be induced by a changing magnetic field. The induced
electric field is what accelerates the particles. ✷
4-1. The oscillating density n 1 and potential φ 1 in a “drift wave” are related by
n 1
= eφ 1 ω ∗ + ia
n 0 kT e ω + ia
(68)
where it is only necessary to know that all the other symbols (except i) stand for
positive constants.
a) Find an expression for the phase δ of φ 1 relative to n 1 . (For simplicity, assume that
n 1 is real.)
b) If ω < ω ∗ , does φ 1 lead or lag n 1 ?
a) Solving for φ 1 leads to
φ 1 = ω + ia n 1 kT e
ω ∗ + ia n 0 e = n 1 kT e
n 0 e
(ω + ia)(ω ∗ − ia)
ω∗ 2 + a 2 = n 1 kT e ωω ∗ + a 2 + i(aω ∗ − aω)
n 0 e ω∗ 2 + a 2 (69)
Now, in a drift wave we can suppose that φ 1 ∼ exp(iδ), which in turns tells us that tan(δ) =
Im(φ 1 )/Re(φ 1 ). We have
Thus,
Re(φ 1 ) = n 1 kT e ωω ∗ + a 2
n 0 e ω∗ 2 + a 2 ; Im(φ 1) = n 1 kT e a
n 0 e
ω ∗ − ω
ω∗ 2 + a 2 (70)
δ = tan −1 (a ω ∗ − ω
ωω ∗ + a 2 ) ✷ (71)
b) For ω < ω ∗ , δ > 0. We can set the phase of n 1 to be 0, since all that matters is a phase difference.
Thus, φ 1 lags n 1 . ✷
4.2 Calculate the plasma frequency with the ion motions included, thus justifying our
assumption that the ions are fixed. (Hint: include the term n 1i in Poisson’s equation
and use the ion equations of motion and continuity.
We will use Gauss’s Law, Fourier transforming the field and charge perturbations into plane waves
of the form x = x 0 + x 1 , where x is any quantity, vector or scalar. The subscript 0 indicates the
equilibrium value, and the subscript 1 indicates the perturbation. We only keep terms of to first
order in small quantities.
∇ · E = ρ ɛ 0
⇒ ik · E 1 = e(n i − n e )
ɛ 0
(72)
Similarly, the equation of motion for the electrons,
ions,
m e
dv e
dt = −eE ⇒ iωm ev e = eE 1 (73)
m i
dv i
dt = eE ⇒ iωm iv i = −eE 1 (74)