Embed Size (px)
DESCRIPTION
Waves in cold field-free plasma. General dispersion-relation for electrostatic and electromagnetic waves in a cold field-free plasma. Assumptions i) No external fields. ii) Cold plasma T=0, p=0 iii) Ions stationary. High frequency waves-> only the electrons can follow iii) Small amplitudes. - PowerPoint PPT Presentation
Citation preview
Waves in cold field-free plasma
00 00 BE
General dispersion-relation for electrostatic and electromagnetic waves in a cold field-free plasma
Assumptions
i) No external fields
ii) Cold plasma T=0, p=0
iii) Ions stationary. High frequency waves-> only the electrons can follow iii) Small amplitudes
BBBBEEEE ~~~~00
General dispersion-relation for cold plasma waves
• Equation of motion for electrons
• Linearisation->neglect quadratic terms in the amplitude
0 0 0( ) ( ( )) ( ) ( )en n m n n e n n et
u u u E u B
0 0 0( ) ( ( )) ( ) ( )en n m n n e n n et
u u u E u B
Waves in cold plasma
0 0en m n et
u E
0 0 0 0 0 0 0( )n n et t
E EB j v
0 0 0 0n et
EB u
Next consider Ampere-Maxwells equation
After linearisation the equation of motion becomes
Linearisation ->
Next take the time derivative and use Faradays law and the equation of motion above, then we have
2 2 2
0 0 0 0 0 0 0 02 2( )e
en e nt t t m t
B u E EE E
Waves in cold plasma2 2
0 0 0 0 2( )e
ent m t
B EE E
2 22
0 0 0 0 2( ) ( )e
enm t
EE E E E
Rewriting the cur curl term using the BAC-CAB rule
For the case of no space charge separation, this equation reduces to
22 2 22 0
0 0 0 0 0 0 0 02 20e e
n eenm t m t
E EE E E
i.e. a wave equation, where we note the plasma frequency2
2 0,
0p e
e
n em
Waves in cold plasma
0 exp( )i t i and therefore we may use the followingrules
i it
E E k r
k
22 2
0 0 , 0 0 2( ) p e t
EE E E
Now consider the possibility of space charge separation
To analyse this equation consider a time and space dependence as
Eq* then becomes2 2
0 0 , 0 0( ) ( ) p ei i i i k k E k k E E E
*
We may now have essentially two possible directions of the electric field. It may be parallel or perpendicular to the wave vector k
Waves in cold plasma2 2
0 0 0 0( ) ( ) p k k E k k E E E
22
2 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) 0 ( )pk z k z E z k z k z E z E zc c
k k E k k E
First let’s consider the case when the electric field is parallel to the wavedirection, then we have
Case i)
and therefore for an electric field different from zero we must have
2 2,p e
This means that we recover the plasma oscillations (not a wave) forwhich the electrons oscillate back and forth in the direction of theelectric field
Dispersion-relation for plasma waves
2
22
2 2
ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( ) ( )
ˆ( )p
k z k z E x k z k z E x k E
E xc c
k k E k k E
2 2 2 2,p e k c
Next let’s consider the case when the electric field is perpendicular to the wave direction, let say that the electric field is in the x-direction and the wave propagates in the z-direction
Case ii)
For non-zero electric field we then find the dispersion-relation
Transverse elctromagnetic wave in cold plasma
Compare EM-waves in vacuum where
2 2 2 2
0 0
1k c k
Group velocity
0 0( , ) exp( ) exp( ( ))v E z t E i t i k z E i k z tk k
( )k
The phase velocity of a wave is defined as
From the dispersion relation we have in general
The phase velocity is then( )kvk
So in general the phase velocity depends on the wavenumber k (or wavelength), meaning that different wavelengths propagate with different velocity. -> Dispersive waves.To find the propagation of a wave-packet, we therefore have to consider a sum(integral) of harmonic waves, a Fourier series or Fourier integral
Group velocity
1( , ) ( ) exp( ( ) )2
E z t A k i k t i k z dk
A wave packet can be represented by a Fourier integral over k
Consider an initial wave-packet of the form 20( ,0) exp( )exp( )E z a z i k z
:= ( )E k
cosh
12
k0 k
ae
1
4
k2 k02
a
a
The corresponding Fourier transform is
z
Group velocity1( , ) ( )exp( ( ) )2
E z t A k i k t i k z dk
1( ,0) ( ) exp( )2
E z A k i k z dk
0k
We put t=0 in this formula so that theinitial condition is given by
We assume that the frequency varies slowly with k around the wavenumber
and consider a Taylor expansion of k) keeping the first two terms
0
0 0
0 0
0 0
1( , ) ( ) exp( ( ) ( ) .. )2
1exp( ( ( ) ) ) ( ) exp( ( ))2
k
k k
dE z t A k i k i k k t i k z dkdk
d di k i k t A k i k t z dkdk dk
Fourier transform ofInitial wave packet
Group velocity
0
0 0
0 0
0 0
0 0
0 0
1( , ) ( ) exp( ( ) ( ) .. )2
1exp( ( ( ) ) ) ( ) exp( ( ))2
exp( ( ( ) ) ) ( ,0)
k
k k
k k
dE z t A k i k i k k t i k z dkdk
d di k i k t A k i k t z dkdk dk
d di k i k t E z tdk dk
( ,0)gE z v t
Initial shape of wave at t=0 is translated with groupvelocity
0
is the group-velocitygk
dvdk
Group velocity
0
20( ,0) exp( ( ) ) exp( ( ))g g g
gk k
E z v t a z v t i k z v t
dwhere vdk
20 0( , ) exp( ( ) ) exp( ( ) ))gE z t a z v t i k z i k t
For the example with initial wave packet
20( ,0) exp( )exp( )E z a z i k z
we have
The complete solution is then
We get the time average energy of the wave by multiplying the electric field with its complex conjugate-> Energy propagates with the group velocityAnother property of dispersive waves is that the shape persists but is broadened
Group velocity
0 0
1
g
k c k
dv c vdk
Example 1:Group velocity of electromagnetic wave in vacuum
Example 2:Group velocity of transverse electromagnetic wave in cold plasma
2 2 2,
2 2 2 2, ,2
2
2
2 2 2,
Index of refraction 1 in a plasma
p e
p e p e
g
p e
k c
k cv c c
k kd kcv cdk k c
cnv
Dispersion-relation cold plama waves
0 ˆ exp( ( ( ) ))E x i t k z E
2 2 2 2 2, ,p e p ek k
2 2,
0 2ˆ exp( ( ))p eE x i t z
c
E
Suppose we have a wave with the form
From the dispersion-relation we get
Together with (1) we get
(1)
Now what happens if the frequency is lower than the plasma frequency
20
,0
p ee
n em
Transverse waves in cold plasma2 2
,0 2ˆ exp( )exp( )p eE x i t z
c
E
The + sign corresponds to an amplitude increasing in the z-direction which is unphysical and the negative sign corresponds to a damping.Therefore no wave exists if the frequency of the wave is less than the plasma-frequency. This is called cut-off.
Ex: Suppose we have a plasma with density n(x) with a plasma frequency
20
,0
( )( )p ee
n x ex
m
If there is some point x0 whereis equal to the plasma frequency the wave is reflectedat this point
n0(x)
Transverse EM waves in cold plasma
Ionosphere plasma z > 80km??z
Problem: The ionospheric plasma has a maximum density of about 12 3
0,max 10n m
Calculate the frequency needed for reflection
Transverse EM waves in cold plasma
12 30,max 10n m
2 12 19 270
, 12 310
76
,
10 (1.6 10 ) 5.7 10 /8.854 10 9.1 10
5.7 10 9 102
p ee
p e
n erad s
m
f Hz
Answer: The frequency must be greater than 9 MHz
Transverse EM waves in cold plasma
Problem 4.9A space capsule making a reentry into the earth’s atmosphere suffers a communication blackout because a plasma is generated by the shockwave in front of the capsule. If the radio operates at a frequency of 300MHz,what is the minimum plasma density during this blackout ?
Transverse EM waves in cold plasma
20
,0
2 2 20 0 00 , 2 2 2
12 318 2 15 3
19 2
for black-out
the limiting density ( 2 )
8.854 10 9.1 10(3 10 2 ) 10 /(1.6 10 )
p ee
e e ep e
n em
m m mn fe e e
part m
