Solutions to Chen's Plasma Physics

The guiding center moves a distance r 1 − r 2 :
r 1 − r 2 =
√ √
eE 2E 0 1 2E 0
E 0 ω c m ω c m
= 2eE
mω 2 c
(28)
The velocity of the guiding center is
v gc = 2 r 1 − r 2
T
= 2 ω c
2π (r 1 − r 2 ) =
4eE
2πmω c
=
since ω c = eB/m. This is a pretty good approximation.
2eE = 2E
πmω c πB ≈ E B ✷ (29)
2-5. Suppose electrons obey the Boltzmann relation of Problem 1-5 in a cylindrically
symmetric plasma column in which n(r) varies with a scale length λ; that is
∂n/∂r = −n/λ.
a) Using E = −∇φ, find the radial electric field for a given λ.
b) For electrons, show that the finite Larmor radius effects are large if v E is as large
as v th . Specifically, show that r L = 2λ if v E = v th .
c) Is (b) also true for ions?
Hint: Do not use Poisson’s equation.
a) We simply solve for φ from the Boltzmann relation for electrons.
Therefore,
n = n 0 e eφ/kTe ⇒ φ = kT e
e ln( n n 0
) (30)
E = −∇φ = − ∂φ
∂r ˆr = −kT e n 0 1 ∂n
e n n 0 ∂r ˆr = kT e ˆr ✷ (31)
eλ
b) We start with the definitions of v E , v th , and r L :
√
v E = E B , v 2kT e
th =
m ,
So, calculating the magnitude of v E :
r L = mv ⊥
eB
v E = kT e
eλB = mv2 th 1
2 eλB = r Lv th
2λ
where in the last step I have assumed that the perpendicular velocity is the thermal velocity. Now,
setting v E = v th , it is easy to see that
r L = 2λ ✷ (34)
c) Sure, why not?
(32)
(33)
2-6. Suppose that a so-called Q-machine has a uniform field of 0.2 T and a cylindrical
plasma with kT e = kT i = 0.2 eV . The density profile is found experimentally to be of
the form
n = n 0 exp[exp(−r 2 /a 2 ) − 1] (35)
Assume the density obeys the electron Boltzmann relation n = n 0 exp(eφ/kT e ).
a) Calculate the maximum v E if a = 1 cm.
b) Compare this with v E due to the earth’s gravitational field.
c) To what value can B be lowered before the ions of potassium (A = 39, Z = 1) have
a Larmor radius equal to a?
We solve for φ:
n 0 exp[exp(−r 2 /a 2 ) − 1] = n 0 exp(eφ/kT e ) ⇒ φ = kT e
e (e−r2 /a 2 − 1) (36)