Download Waves in cold field

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
E0  0 B 0  0
~ ~
E  E0  E  E
  ~ ~
B  B0  B  B
ii) Cold plasma T=0, p=0
iii) Ions stationary. High frequency waves-> only the electrons can follow
iii) Small amplitudes
General dispersion-relation for cold
plasma waves
• Equation of motion for
electrons u
(n0  n) me (
t
 (u u))  (n0  n) e E  (n0  n) e u  B
• Linearisation->neglect
quadratic terms in the
amplitude
u
(n0  n) me (  (u u))  (n0  n) e E  (n0  n) e u  B
t
Waves in cold plasma
After linearisation the equation of motion becomes
u
n0 me
 n0e E
t
Next consider Ampere-Maxwells equation
 B  0 j  0  0
E
E
  0 (n0  n) e v  0  0
t
t
Linearisation ->
 B   0 n0 e u  0  0
E
t
Next take the time derivative and use Faradays law and the equation of
motion above, then we have
B
u
 2E
e2
 2E

  ( E)   0 n0 e  0  0 2  0 n0
E  0  0 2
t
t
t
me
t
Waves in cold plasma
B
e2
 2E

  ( E)  0 n0
E  0  0 2
t
me
t
Rewriting the cur curl term using the BAC-CAB rule
e2
 2E
  (  E)  (  E)   E  0 n0
E  0  0 2
me
t
2
For the case of no space charge separation, this equation reduces to
2
2
2
2
n
e
e

E

E
 2E   0 n0
E  0  0 2  0 0 0 E  0  0 2
me
t
 0 me
t
i.e. a wave equation, where we note the plasma frequency

2
p ,e
n0 e2

 0 me
Waves in cold plasma
Now consider the possibility of space charge separation
(  E)   E  0 0 
2
2
p ,e
 2E
E  0  0 2
t
*
To analyse this equation consider a time and space dependence as
E  E0 exp( it  i k  r )
and therefore we may use the following
rules

i
t
   ik
Eq* then becomes
i k (i k  E)  (i k  i k ) E  0 0  p2,e E  0  0  2E
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 plasma
k (k  E)  (k  k ) E  0 0  p2 E  0  0  2E
First let’s consider the case when the electric field is parallel to the wave
direction, then we have
Case i)
k (k  E)  (k  k )E  k zˆ (k zˆ  E zˆ)  (k zˆ  k zˆ) E zˆ  0  (
2
c
2

 p2
c
2
) E zˆ
and therefore for an electric field different from zero we must have
 2   p2,e
This means that we recover the plasma oscillations (not a wave) for
which the electrons oscillate back and forth in the direction of the
electric field
Dispersion-relation for plasma
waves
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 zdirection
Case ii)
k (k  E)  (k  k )E  k zˆ (k zˆ  E xˆ )  (k zˆ  k zˆ) E xˆ  k 2 E 
(
2
c
2

 p2
c
2
) E xˆ
For non-zero electric field we then find the dispersion-relation
 2   p2,e  k 2c2
Transverse elctromagnetic wave in cold plasma
Compare EM-waves in vacuum where
 2  k 2c 2  k 2
1
0 0
Group velocity
The phase velocity of a wave is defined as
v 

k
E ( z , t )  E0 exp(i  t  i k z )  E0 exp(i k ( z 

k
t ))
From the dispersion relation we have in general
   (k )
The phase velocity is then
 (k )
v 
k
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
A wave packet can be represented by a Fourier integral over k
1
E ( z, t ) 
2

 A(k ) exp(i  (k ) t  i k z ) dk

Consider an initial wave-packet of the form E ( z , 0)  exp(a z 2 ) exp(i k0 z )
The corresponding Fourier transform is
z
E( k ) :=
1

 cosh
2
2


 1 k k 0 2 





k0 k   4
a


e

a 
a
Group velocity
1
E ( z, t ) 
2

Fourier transform of
Initial wave packet
 A(k ) exp(i  (k ) t  i k z ) dk

We put t=0 in this formula so that the
initial condition is given by
1
E ( z , 0) 
2

 A(k ) exp(i k z ) dk

We assume that the frequency varies slowly with k around the wavenumber
k0 and consider a Taylor expansion of (k) keeping the first two terms
1
E ( z, t ) 
2


A(k ) exp(i  (k0 )  i

d
 exp(i ( (k0 )  i
dk
k0 )t )
k0
1
2
d
dk
(k  k0 ) t  ..  i k z ) dk 
k0



A(k ) exp(i k (
d
dk
t  z )) dk
k0
Group velocity
1
E ( z, t ) 
2


A(k ) exp(i  (k0 )  i

k0
d
exp(i ( (k0 )  i
dk
d
k0 )t ) E ( z 
dk
k0
k0 )t )
d
vg 
is the group-velocity
dk k0
(k  k0 ) t  ..  i k z ) dk 
k0

d
 exp(i ( (k0 )  i
dk
1
2
d
dk

A(k ) exp(i k (

d
dk
t  z )) dk 
k0
t , 0)
k0
E( z  vg t ,0)
Initial shape of
wave at t=0 is
translated with group
velocity
Group velocity
For the example with initial wave
packet
E ( z, 0)  exp(a z 2 ) exp(i k0 z )
we have
E ( z  vg t , 0)  exp(a ( z  vg t ) 2 ) exp(i k0 ( z  vg t ))
d
where vg 
dk
k  k0
The complete solution is then
E( z, t )  exp(a ( z  vg t )2 )exp(i k0 z  i(k0 ) t ))
We get the time average energy of the wave by multiplying the electric field with its
complex conjugate-> Energy propagates with the group velocity
Another property of dispersive waves is that the shape persists but is broadened
Group velocity
Example 1:
Group velocity of electromagnetic wave in vacuum
1
 kck
vg 
 0 0
d
 c  v
dk
Example 2:
Group velocity of transverse electromagnetic wave in cold plasma
  k 2 c 2   p2 ,e
v 
k 2 c 2   p2 ,e
k
 c 
2
 p2 ,e
k2
c
d
kc 2
vg 

c
2 2
2
dk
k c   p ,e
Index of refraction n 
c
 1 in a plasma
v
Dispersion-relation cold plama
waves
Suppose we have a wave with the form
E  E0 xˆ exp(i ( t  k ( ) z ))
(1)
From the dispersion-relation we get
k 2   2   p2,e  k    2   p2,e
Together with (1) we get
E  E0 xˆ exp(i ( t 
 2   p2,e
c
2
z ))
Now what happens if the frequency is lower than the plasma frequency
   p ,e
n0 e 2

 0 me
Transverse waves in cold plasma
E  E0 xˆ exp(i  t ) exp(
 p2,e   2
c
2
z)
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.
n0(x)
Ex: Suppose we have a plasma
with density n(x) with a plasma
frequency
n0 ( x) e 2
 p ,e ( x ) 
 0 me
If there is some point x0 where
is equal to the plasma
frequency the wave is reflected
at this point
Transverse EM waves in cold
plasma
z
Ionosphere plasma z > 80km??
Problem: The ionospheric plasma has a maximum density of about
n0,max  1012 m3
Calculate the frequency needed for reflection
Transverse EM waves in cold
plasma
3
n0,max  10 m
12
 p ,e
n0 e2
1012  (1.6 1019 ) 2
7



5.7

10
rad / s
12
31
 0 me
8.854 10 9.110
f p ,e
5.7 107

 9 106 Hz
2 
Answer: The frequency must be greater than 9 MHz
Transverse EM waves in cold
plasma
Problem 4.9
A space capsule making a reentry into the earth’s atmosphere suffers a
communication blackout because a plasma is generated by the shock
wave 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
   p ,e
n0 e 2

 0 me
for black-out
 the limiting density n0  
2
p ,e
 0 me
e2

2
 0 me
e2
 ( f 2 )
8.854 1012  9.1 10 31
15
3
 (3 10  2  )

10
part
/
m
(1.6 1019 ) 2
8
2
2
 0 me
e2
