NE528 Fall 2009 HW#1 (total 90 points)
Due Wednesday, September 9 by class time
1.
(10 points)
Derive the expression of the Debye length for a pure hydrogen plasma in a steady-state situation,
where electron and ion temperatures are not equal, i.e.
e
T
T
i
≠
, and both ions and electrons
follow the Boltzmann relation, i.e.
,
e
e
k T
k T
e
i
e
o
i
o
n
n e
n
n e
Φ
Φ
−
=
=
2.
(20 points)
When solving Poisson’s equation using Boltzmann’s distribution for both electrons and ions, as per
Problem # 1, a solution for the Debye length for 2-temperayture plasma was obtained. The Boltzmann’s
distribution has the potential
Φ
in the exponent in which no fluctuations appear on the potential.
N
ow
we have a situation in which the potential has fluctuations in the form
(
)
Φ + Φ
±
, which will affect
the Boltzmann’s relation for both electrons and ions and hence the Boltzmann’s relation for electrons and
ions will be
(
)
exp
e
n
n
e
o
kT
e
Φ + Φ
=
±
and
(
)
exp
i
o
i
e
n
n
kT
Φ + Φ
=
−
±
, respectively.
Thus, for fully ionized pure hydrogen plasma in which electrons and ions are at different temperatures,
what is the effect of this potential fluctuation on the Debye length?
Work out the solution starting from
Poisson’s equation.
3. (20 points)
It was shown that the Boltzmann distribution could be derived from argument of particle
dynamics, which gives a distribution for electrons and ions in presence of bulk motion, and
hence the electron and ion densities can be expressed by
2
1
2
exp
e
e
e
o
e
m v
e
n
n
kT
− Φ
=
−
and
2
1
2
exp
i
i
i
o
i
m v
e
n
n
kT
+ Φ
=
−
, respectively.
Derive the expression of the Debye length in presence of bulk motion.
How this Debye length differs from that in absence of bulk motion?
4. (20 points)
The Boltzmann relation in presence of bulk motion, as derived from argument of particle
dynamics, shows that the time-independent solution gives a Boltzmann distribution for electrons,
with “v” is the velocity of the bulk of electrons, as:
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
Φ
−
−
=
e
o
e
kT
e
m
n
n
2
e
v
2
1
exp
In obtaining this relation, the force equation assumed the right hand side (RHS) composed of 2
terms, electric force term and pressure term

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