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Astro 405/505, fall semester 2004
Homework, 5th set, solutions.
Problem 1: Electrostatic waves
The momentum conservation equation (14.5 in the notes) now reads
m
e
n
e
∂
~
V
e
∂t
+ (
~
V
e
· ∇
)
~
V
e
+
~
∇
P
=

e n
e
~
E
Be careful, because
c
s
will also be perturbed! Using
∂P
∂r
=
∂P
∂ρ
∂ρ
∂r
∂P
∂ρ
=
c
2
s
⇒
∂P
∂r
=
c
2
s
A
Z
m
p
∂n
e
∂r
we derive the perturbation equation in Fourier representation.
n
e,
0
ı ω v
1
=
e
n
e,
0
m
e
E
1
+
c
2
s
A
Z
m
p
m
e
ı k n
e,
1
which must be solved together with the three unchanged equations.
ı k E
1
=

4
π e n
e,
1
∧
ı ω E
1
=

4
π e v
1
n
e,
0
∧
ω n
e,
1
=
n
e,
0
k v
1
The solution is
ω
2
=
ω
2
p
+
c
2
s
A
Z
m
p
m
e
k
2
=
ω
2
p
+
γ
κT
m
e
k
2
so the mean thermal velocity of the electrons enters the dispersion relation as a characteristic
scale.
The phase velocity is
v
φ
=
ω
k
=
v
u
u
t
ω
2
p
k
2
+
γ
κT
m
e
'
ω
p
k
small
k
q
γ
κT
m
e
large
k
The group velocity is
v
gr
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.
 Fall '08
 RABE
 Physics, Momentum, Work

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