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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.

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waves in cold field free plasma
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

general dispersion relation for cold plasma waves
General dispersion-relation for cold plasma waves

Equation of motion for electrons

Linearisation->neglect quadratic terms in the amplitude

waves in cold plasma
Waves in cold plasma

After linearisation the equation of motion becomes

Next consider Ampere-Maxwells equation

Linearisation ->

Next take the time derivative and use Faradays law and the equation of

motion above, then we have

waves in cold plasma1
Waves in cold plasma

Rewriting the cur curl term using the BAC-CAB rule

For the case of no space charge separation, this equation reduces to

i.e. a wave equation, where we note the plasma frequency

waves in cold plasma2
Waves in cold plasma

Now consider the possibility of space charge separation

*

To analyse this equation consider a time and space dependence as

Eq* then becomes

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 plasma3
Waves in cold plasma

First let’s consider the case when the electric field is parallel to the wave

direction, then we have

Case i)

and therefore for an electric field different from zero we must have

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
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 z-direction

Case ii)

For non-zero electric field we then find thedispersion-relation

Transverse elctromagnetic wave in cold plasma

Compare EM-waves in vacuum where

group velocity
Group velocity

The phase velocity of a wave is defined as

From the dispersion relation we have in general

The phase velocity is then

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 velocity1
Group velocity

A wave packet can be represented by a Fourier integral over k

Consider an initial wave-packet of the form

The corresponding Fourier transform is

z

group velocity2
Group velocity

Fourier transform of

Initial wave packet

We put t=0 in this formula so that the

initial condition is given by

We assume that the frequency w varies slowly with k around the wavenumber

and consider a Taylor expansion of w(k) keeping the first two terms

group velocity3
Group velocity

Initial shape of

wave at t=0 is

translated with group

velocity

group velocity4
Group velocity

For the example with initial wave packet

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 velocity

Another property of dispersive waves is that the shape persists but is broadened

group velocity5
Group velocity

Example 1:

Group velocity of electromagnetic wave in vacuum

Example 2:

Group velocity of transverse electromagnetic wave in cold plasma

dispersion relation cold plama waves
Dispersion-relation cold plama waves

Suppose we have a wave with the form

(1)

From the dispersion-relation we get

Together with (1) we get

Now what happens if the frequency is lower than the plasma frequency

transverse waves in cold plasma
Transverse waves in cold plasma

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

n0(x)

If there is some point x0 where

w is equal to the plasma

frequency the wave is reflected

at this point

transverse em waves in cold plasma
Transverse EM waves in cold plasma

z

Ionosphere plasma z > 80km??

Problem:The ionospheric plasma has a maximum density of about

Calculate the frequency needed for reflection

transverse em waves in cold plasma1
Transverse EM waves in cold plasma

Answer: The frequency must be greater than 9 MHz

transverse em waves in cold plasma2
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 ?