Download EM Wave in plasmas (note12)

§3. Electromagnetic Waves
§3.5. Plasma waves
A plasma is an ionized gas consisting of charged particles (e.g., electrons and ions).
Various waves can be excited easily in a plasma. Wave phenomena have been an important
subject in the plasma research community.
~ = 0 still holds. However, neither
The plasma is nearly charge neutral. So the ∇ · E
the conduction current nor the displacement current can be ignored. Waves in plasma is
different from the waves in vacuum and in conductors.
§3.5.1 Effective Permittivity in a Plasma
In vacuum, the phase velocity of an EM wave is:
c=
1
ω
=√
k
²0 µ0
This can be generalized for phase velocity of waves in matter:
v=
1
ω
=√
k
²µ
In plasma µ = µ0 , but ² 6= ²0 .
Electric field, magnetic field, and generally any quantity in plasma can be divided into
two parts: DC part that does not depend on time and parts that associated with waves:
Qtotal = QDC + Q(~r, t)
We study now only the part associated with waves, Q, and assume
~
Q = Q0 ei(k·~r−ωt)
In plasma, the current is predominately carried by electrons, as in a conductor, because
the mass of a electron is small compared with that of ions. The electrons in plasma experience
the electric force and suffer from collision with ions. The velocity of electrons has been derived
before (when we study waves in conductors):
~v = −
~j =
e
~
E
m(ν − iω)
ne2
~
E
m(ν − iω)
~ term in the equation of motion).
if there is no DC magnetic field (So we can ignore ~v × B
Phys 463, E & M III, C. Xiao
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In most cases, the electron density, and thus the collision frequency ν, are much smaller
than in conductors. So:
2
2
~ = i ne E
~
~j ' − ne E
miω
mω
The 4th Maxwell’s equation becomes:


Ã
!
2
~
~ = µ0 i ne E
~ + ²0 ∂ E 
∇×B
mω
∂t
ne2 ~
~
= µ0 i
E − iω²0 E
mω
Ã
!
ne2
~
= −iω²0 µ0 1 −
E
m²0 ω 2
Let
ωp =
s
ne2
m²0
Plasma frequency
~ = B/µ
~ 0 , we find
and recall H
~ = −iω²0
∇×H
Ã
!
ωp2 ~
1− 2 E
ω
For low frequency wave, ωp À ω, conduction current dominates (like waves in a conductor).
For high frequency wave, ωp ¿ ω, displacement current dominates (like waves in vacuum).
If we define an effective permittivity in plasma,
²(ω) = ²0
Ã
!
ωp2
1− 2 ,
ω
The 4th Maxwell’s equation for a monochromatic plane wave in plasma becomes
~
~ = −iω²E
~ = ² ∂E ,
∇×H
∂t
similar to the equation for a monochromatic plane wave in the vacuum, except ε0 → ε.
Therefore, all the results for waves in vacuum can be used in plasma wave with the modification ε0 → ε. Note that the effective permittivity in plasma depends on the frequency
ω. Note also ² < ²0 .
§3.5.2 Dispersion Relation
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The phase velocity of plasma waves:
1
ω
=√
k
µ²
1
1
q
= √
µ0 ²0 1 − ωp2
ω2
c
= q
>c
ωp2
1 − ω2
v =
(1)
In plasma, ² depends on ω and thus the phase velocity depends also on ω. The wave in
this case is dispersive.
A square wave contains the fundamental frequency and its higher harmonics. If the phase
velocity depends on frequency, it will spread out while it propagates.
The dependence of ω on k is called dispersion relation.
We can solve eq. (1) for ω
ω2
c2
=
ω2
k2
1 − p2
Ã
ω2 1 −
ωp2
ω2
!
ω
= c2 k 2
ω 2 = ωp2 + (ck)2
Phys 463, E & M III, C. Xiao
dispersion relation in plasma
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The dispersion relation of EM waves in vacuum is a straight line (non-dispersive) with a
slope tan α = c.
The dispersion relation of EM waves in plasma is above the line ω = ck because
ω=
q
ωp2 + (ck)2 ≥ ck,
but approaches the line ω = ck when k becomes large because
ω=
q
ωp2 + (ck)2 ' ck
when k À
ωp
.
c
Propagation and Reflection of EM Waves in Plasma
Assume a plane wave propagating in +z direction.
~ =E
~ 0 ei(kz−ωt)
E
From the dispersion relation, we obtain
k=
1q 2
ω − ωp2
c
• If ω > ωp (like in vacuum)
~ =E
~ 0 ei(kz−ωt) , wave propagates without decay.
k is real , E
• If ω < ωp (like in conductor)
q
k = i 1c ωp2 − ω 2 = i|k| is imaginary.
~ =E
~ 0 e−iωt e−|k|z
E
Waves decays in z direction and will be reflected.
If ω ¿ ωp , k ' i ωcp = i 1δ
δ=
c
ωp
skin depth in plasma
Short Wave Communication
Ionospheric plasma (height: 50 km to 100 km) has a typical density of 1013 /m3 .
ωp
1
fp =
=
2π
2π
s
ne2
= 28 MHz
m²0
Short wave radio (f ∼ 10 MHz) relies on the multiple reflection between the ionospheric
plasma layer and the earth to reach a distant receiver.
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Earth is “conductor” (σ ∼ 10−2 S/m) as long as the impedance
¯s
¯
s
¯
¯
¯ −iωµ0 ¯
¯ ¿ Zair = µ0
|Z| = ¯¯
σ ¯¯
²0
¯
which requires
ω¿
or
σ
²0
σ
10−2
=
= 180 MHz
2π²0
2π × 8.85 × 10−12
For f = 10 MHz, the earth is good conductor.
f¿
§3.5.3 Group Velocity
According to Einstein’s relativity, nothing should propagate faster than the light speed
c.
In plasma, phase velocity
vp =
ω
c
=q
k
1−
ωp2
ω2
>c
But the phase velocity does not correspond to the information (energy) propagation velocity.
The information is propagating at the group velocity
dω
dk
For non-dispersive waves, e.g., EM waves in vacuum,
vg =
ω = ck
vp = vg = c.
In plasma, wave is dispersive. The group velocity is
s
ω2
dω
vg =
= c 1 − p2 < c.
dk
ω
Phys 463, E & M III, C. Xiao
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