Today, we will continue discussing the propagation
of electromagnetic waves in a magnetize plasma.
We will discuss longitudinal electromagnetic
wave propagation in the direction of magnetic
field. We will discuss cutoffs, resonances
and Faraday rotation.
Well, with respect to right circularly polarized
mode and left circularly polarized mode that
we introduced yesterday, we will discuss these
resonances and cutoffs. Then, we will discuss
the lower frequency waves called whistler
waves and cyclotron waves and Alfven waves.
We will discuss the phenomenon of Faraday
rotation and cyclotron resonance heating of
electrons and ions.
The references for today’s presentation
would be three books by Ginsburg, Stix and
Krall and Trivelpiece.
Yesterday we were talking about the waves
travelling in the direction of dc magnetic
field. So, this was the direction of dc or
esthetic magnetic field B s which we took
as z axis and we considered the propagation
of wave with wave vector k parallel to
B s. And we found that the value of k depends
on the estate of polarization of the wave.
If the wave is right circularly polarized,
then k was equal to k r k r which we defined
as omega by c and 1 minus omega p square;
omega p is the electron plasma frequency divide
by omega into omega minus omega c, where omega
c is the electron cyclotron frequency.
Then, there was a term here; plus i times
the collision frequency nu. This is the electron
contribution to k and the ion term was omega
p i square which is ion plasma frequency square
divided by frequency of the wave omega into
omega plus omega ci plus i times the collision
frequency of ions to the power half. This
was the wave value of wave vector when the
electric field of the wave was some amplitude
A and x cap plus i y cap exponential minus
i omega t minus k z. So, this implies that
Ey is equal to i times Ex. This is called
right circularly polarized wave RCP.
We noted that this expression for k as a resonance
at omega equal to omega c because usually
nu is very small as compare to cyclotron frequency.
So, this term is very small, an order of magnitude
is smaller than this, and hence in the vicinity
of omega equal to omega c, two things happen:
Number 1, this term reverses the sign; for
omega bigger than omega c, this term is negative
of this negative sign, but if an omega becomes
less than omega c this term becomes positive,
if I ignore nu. There is no reversal of such
a sign in the case of ion response, but the
electron response certainly shows a change
of sign. And in the vicinity of omega equal
to omega c, one would except k r to be very
large.
And we should always keep in view that that
the pointing flux of a wave or intensity of
radiation is equal to modulus of e square
upon twice mu 0 omega and real part of k,
where mu 0 is the free space magnetic permeability.
So, you may note here that if I am sending
a wave of a given intensity, then if k r increases,
then correspondingly amplitude of the wave
decrease.
So, I would say that for a given value of
intensity; pointing this is magnitude of this
quantity is called intensity of the wave.
So, for a given intensity, modulus of e square
or wave amplitude square goes as 1 upon k
r; the real part of k. For the sake of simplicity,
we can easily ignore nu to get some insight
into the wave propagation. And let us consider
the frequency is to be much bigger than ion
cyclotron frequency. In that case, the ion
motion is neglected. So, ion terms are neglected,
negligible and your expression for k square
becomes or k becomes omega by c 1 minus omega
p square upon omega into omega minus omega
c.
Now, you may note to the power half that because
of this sign reversal, there will be a frequency
for which this term is exactly equal to 1.
This entire this negative term besides a negative
sign. So, when this quantity is equal to 1,
then this k becomes a 0 and on one side, k
will be imaginary because this quantity will
be negative. On the other side will be positive
or real.
So, let me find out what is the cut off. Cut
off means when k becomes 0, the frequency
of the wave for which k becomes a 0 is called
a cut off frequency.
So, cut off and because we are talking about
the right circularly polarized wave, so, we
will call the right handed cut off.. So, I
am looking for k r equal to 0 which implies
that omega into omega minus omega c should
be equal to omega p square. And when I am
bring this, I can re-write this equation as
omega square minus omega omega c minus omega
p square is equal to 0, giving two roots and
a root the positive this frequency would be
omega equal to 1 upon 2 omega c plus under
root of omega c square plus 4 omega p square.
And we call this frequency at which k r becomes
0 is a right handed cut off. So, we denote
by omega r.
So, this is a important frequency. If you
have a wave of frequency higher than omega
r, the wave will travel. If the frequency
is less than omega r, it will not travel,
but this is not true for all low frequency,
all the low frequencies because k r has a
resonance at omega equal to omega c 
at which the plasma term goes to infinity
and reverse a sign.
So, if I plot. So, at this point k goes to
infinity, let us see how it happens. It will
be better to plot this expression k which
is k r for the right circularly polarized
wave as omega by c 1 minus omega p square
upon omega into omega minus omega c under
the root if I plot this, this usually plotted
as omega versus k omega here and k here.
So, omega I am having say suppose omega c
is somewhere here and right hand cut off is
somewhere here. So, what we are doing here?
When we have ignored the ion motion, then
this equation is true and also when we have
ignored the collisions. In that case, at omega
higher than or rather omega less than omega
c, certainly all frequencies are possible
because this entire quantity becomes positive.
So, when omega is less than omega c, this
is positive and k r is certainly is possible
for all frequencies.
So, let me this goes like this because has
omega approaches omega c from below, then
k r goes to infinity; positive infinity; however,
when omega becomes slightly more than omega
c, then this quantity with this turn without
this negative sign is negative is positive,
but there is a negative sign.
So, when omega is slightly more than omega
c, this entire quantity is highly negative
and k r is imaginary. Wave does not travel.
It will travel only when this quantity becomes
1 or less than 1. So, then the mode goes like
this. So, there is a gap here between these
two frequencies; electron cyclotron frequency
and right handed cut off that there is no
propagation. This is the forbidden frequency
band. There is no right circularly polarized
wave propagating in this frequency band.
It is a important consequence; however, to
appreciate the significance of this resonance,
let us see what will happen? At the resonance,
k r will increase. So, wave amplitude, as
I mention to you, that for a given intensity,
wave amplitude goes as or square of this goes
as 1 upon on k r. So, around the resonance,
you expect that k r becomes very large. So,
this becomes a small. This goes as because
k r goes as, if I ignore this 1, then this
goes as omega minus omega c inversely. So,
this goes as omega minus omega c to the power
half. So, e square goes like this.
However this resonance can have very important
implication for the heating of electrons;
obviously, this equation is not valid exactly
at omega equal to omega c because when this
becomes 0, the collision term that he had
we had here, you can ignore it. So, nu has
to be retained. So, this is cannot be exactly
zero.
Now, let us examine the electron velocity
around a resonance. The electron velocity
is governed by the equation of motion which
is m delta v by delta t the v dot del v term
I will ignore due to linearization, and this
is equal to minus e E minus e v cross B s;
the esthetic magnetic field minus m nu v.
This is the equation of motion and delta delta
t if I because my electric field is the source
and v everywhere is the cos; is the perturbed
electron velocity. So, if I replace this by
delta delta t by minus i omega, this equation
takes the following form. Minus i omega plus
i nu; when I combine this term with this after
dividing by m, you get v plus v cross omega
c is equal to minus e E upon m.
It will be useful to write down the x and
y components of this equation because electric
field has x and y components. x component
would be minus i omega plus i nu v x plus
omega c v y is equal to minus eEx upon m.
And the y component would be minus I omega
plus i nu into vy minus omega vx is equal
to minus eEy upon m, but Ey is equal to iEx.
So, I can write down this minus i eEx upon
m. So, what I am saying is that if I multiply
this equation by i, this should b equal to
this equation. So, multiply this by i and
equate to this. You will immediately get that
v y turns out to be equal to i times v x.
These two equations; the first one when you
multiply by i, the right hand side becomes
equal in the two equations. So, multiply the
first equation by i and subtract the second.
You will immediately see v y is equal to i
v x and the interesting consequences that
once you get this if I put v y is equal to
i v x here, then v x can be explicitly written
in terms of Ex and the result is that velocity
turns out to be simply equal to eE vector
upon m into i omega minus omega c plus i nu.
So, for a right circularly polarized wave,
the electron velocity has a resonance denominator
because nu if nu I ignore, then there is a
resonance in the velocity at omega equal to
omega c. That is a very interesting thing.
So, the total response of electrons to the
right circularly polarized electromagnetic
field is to acquire a right circularly polarized
drift velocity with resonance at omega equal
to omega c.
Now, let me calculate the heating rate. Heating
rate of electrons is sorry heating rate is
a scalar quantity. This is equal to work done
by the electric field which is equal to the
force the electric force on the electron is
minus eE into the displacement per unit time
which is called velocity. So, when you take
the dot product of these two quantities; real
part of this electric force with the real
part of velocity, you will get the heating
rate 
of electrons by the electric field per second.
And this by using the complex number identity
becomes this is equal to this into real part
of minus eE star dot v. This is called time
average.
So, when I take time average of h, then I
get a simple expression, and if I substitute
for v and take the real of this product, it
turns out to be e square upon 2 m omega minus
omega c whole square then nu eE star. That
is a interesting expression which express
that it shows you that the heating rate is
scales as 1 upon omega minus omega c square
though as omega approaches omega c, e decreases;
we had just seen, but this e E star goes as…
we have just seen that eE star or modulus
of e square goes as omega minus omega c to
the power half. So, when I put this in here,
this entire quantity goes as omega minus omega
c to the power minus three by two.
So, the heating is very strongly influenced
by cyclotron resonance. That is something
very important and if you want to find out
the rise in a electron temperature because
of this wave which is traveling along the
magnetic field, that is not difficult to calculate.
It turns out to be heating rate if I have
to equate; with the heat loss rate.
Now, in a elastic collisions, suppose like
plasma is having only elastic collisions,
then in a elastic collision, the fraction
of energy that the electron loses per collision
is 2 times the electron mass upon ion mass
into the average kinetic energy of the electron
minus the average kinetic energy of the ion.
This is called the axis energy of the electron
over the ion energy when constant is hidden
in temperatures.
So, this is the difference in average kinetic
energy of an electron and an ion, and in each
collision, this fraction of energy is lost.
So, if there are new collisions per second,
then the loss rate is so much, heating rate
is h. Equate the 2. This nu will cancel out
with the nu in h and you will get the rise
in electron temperature over the ion temperature
in terms of e square and certainly that will
show a resonance at omega equal to omega c.
So, the heating rate one has… So, the electron
temperature rise is equal to the initial temperature
or ion temperature plus a quantity which turns
out to be equal to e square, which turns out
to be actually A square of this wave divided
by something like I think 3 into 2; this becomes
like I think 3 m square m i divided by omega
minus omega c whole square. And this term,
as I mentioned to you, because A decreases
A square decreases as omega minus omega c
2 half. So, this goes as 1 upon omega minus
omega c to the power 3 half. That is something
beautiful here.
And; obviously, if the electrons are heated
by the wave, then the wave must damp out because
of collisions. So, if you retain nu in the
expression for k, you get an expression for
the dumping rate, and let me just mention
that expression. Before I actually do that,
let me tell you the relevance of this kind
of heating scheme.
We have learnt in a mirror machine that the
lines of force go like this. The magnetic
field is center of the mirror is small and
large at the throats. These are the lines
of force. And you can typically choose B z
of a mirror machine; axial magnetic field
on the axis on this axis; this is z equal
to 0, this is z is equal to minus L by 2;
L is the length of the machine. So, L by 2
is the half length and v z I can write down
B minimum into 1 plus mirror ratio m r minus
1 into z square by L square by 4. And similarly,
density has a maximum here magnetic field
has a minimum on the axis; density has this
kind of profile. I can write down n 0 is equal
to n 0 0 1 minus z square upon L square by
4.
So, the density of plasma is maximum here.
If I launch an electromagnetic wave that RCP
wave right circularly polarized wave here
with frequency omega less than the omega at
the throat, then this wave will travel and
reach a point somewhere omega becomes equal
to omega c locally. Beyond this point, this
wave will not travel. And at this point that
the cyclotron resonance condition is met and
hence very strong heating will take place.
So, the problem of wave propagation along
the magnetic field lines is relevant to heating
of a mirror machine. And please note that
at one tesla magnetic field, omega c upon
2 pi is 30 gigahertz. So, suppose I have a
mirror machine that has a magnetic field of
the order of say 0.3 tesla, suppose my B field
is around 0.3 tesla or 3 kilogauss, then corresponding
omega c upon 2 pi would be 9 gigahertz. At
lower value of magnetic field, it will be
smaller; like suppose I choose a magnetic
field of about 0.1 tesla, then this will be
like 3 gigahertz, but that is and that is
microwave frequency.
So, if I launch a microwave through mirror
throat into the mirror machine axially, then
electron cyclotron resonance will take place
somewhere between the throat and the center.
And you can match the frequency of the wave
to the cyclotron frequency in the center,
and then you can heat the center plasma in
the center of the machine. So, you have a
great advantage in picking the location where
you would like to deposit the energy of your
electromagnetic wave and this is an important
scheme.
And then, I was talking about the imaginary
part of propagation constant and if I include
the collision frequency in the expression
for k, but still presume that for nu is still
less than omega minus omega c. In that case,
k r turns to be k r I can write down is equal
to real part of k plus i times the imaginary
part of k. k real part is roughly the same
as before which is omega by c 1 minus omega
p square upon omega into omega minus omega
c to the power half and k i turns out to be
equal to nu upon 2 c omega p square divided
by omega minus omega c whole square which
is very strong dependence on difference between
the omega and omega c divided by 1 minus omega
p square upon omega and omega minus omega
c to the 1 half. This is very strongly dependent
on cyclotron resonance. And this factor within
this under root with including the under root
sign is called refractive index of the plasma.
So, I can write down this is equal to omega
by c; refractive index for right circularly
polarize wave. So, this quantity within the
bracket and under root is called refractive
index of the plasma for right circularly polarized
wave. This is bigger than 1, if omega is less
than omega c.
So, we have learned something about the relevance
of these waves for plasma heating in mirror
machine. These waves are also important for
plasma heating in the ionosphere.
Ionosphere is, suppose if this is my earth,
and suppose this is North Pole of the earth,
this is south pole of the earth. The lines
of force go like this. These are the lines
of force. So, if you are in the in the polar
regions in the pole, this region is called
polar region and you launch a wave in this
direction or at some angles like a small angles,
few degrees, then this is my direction of
k and if I launch a right circularly polarized
wave, then obviously, plasma is not there
near the surface of the earth, but when you
go to a height of about 90 kilometers and
above, then, there exists a region here; ionosphere
and then your wave can have cyclotron resonance
somewhere there and it can heat the electrons.
Now, let me just mention, in the, this is
called the equatorial region and these are
the pole; this is called polar region polar
poles polar region.
So, in polar regions, electron cyclotron resonance
is certainly possible and in the magnetic
field of the earth in the equatorial region
is around 0.3 gauss and it is higher near
the poles. Suppose I choose like 1 gauss;
1 gauss is around 10 to the power minus 4
tesla.
So, omega c upon 2 pi will correspond to 30
gigahertz into 10 to minus 4 which is like
3 megahertz. So, if you are launching a wave
of about 3 megahertz, it will meet a cyclotron
resonance in the polar region of slightly
lower frequency because magnetic field will
be may be slightly less. So, if you are in
this region, this in this region, if you are
having a transmitter somewhere here and launching
a wave vertically upwards, then you can heat
the plasma via cyclotron resonance heating,
ionosphere plasma.
Now, well let me summarize what I have really
discussed about heating. I said that if you
want electron cyclotron resonance heating,
then you should have omega close to omega
c and the wave should be circularly polarized.
So, two conditions are important: omega should
be around omega c and second, the wave should
be RCP. Only then the cyclotron resonance
will be there. They are the two conditions
for resonance cyclotron heating of electrons.
Now, let me look into the waves of lower frequencies.
So far, we have ignored the ion motion. We
still can ignore the ion motion if omega is
omega less much less than omega c, even then
you can ignore ion motion, if this is still
bigger than omega c i; the ion cyclotron frequency.
This is called intermediate frequency range.
So, the waves of frequency bigger than ion
cyclotron frequency but less than the electron
cyclotron frequency is still ion motion can
be ignored. And if I do this, in that case,
the propagation constant for the right circularly
polarized wave becomes omega by c this becomes
1 plus omega p square upon omega omega c to
the power half. In most plasmas of interest,
this quantity is much bigger than 1. You can
ignore unity. This becomes equal to omega
p by c and omega upon omega c to the power
half.
This is a wave which has a phase velocity
v phase which is defined as omega upon k r,
and that would be c, just divide this, you
will get this is equal to c under root of
omega omega c upon omega p. The phase velocity
increases with frequency as under root. It
is a dispersive medium and how about the group
velocity which is defined as delta omega by
delta k r. It turns out to be 2 times phase
velocity. Peculiar case when the phase and
group velocities are not equal to magnitude.
In an unmagnetized plasma, we had seen that
group velocity is always less than the phase
velocity, but in a magnetized plasma, at low
frequencies, in this frequency band, phase
velocity is less than the group velocity.
You can just check it. From here you can check
that omega is proportional to k square. It
is square both sides and we will see that
omega goes as k square. Well a special kind
of wave where omega goes as k square is called
as whistler wave.
In early fifty’s, this wave was observed
during thunder storms. So, when there is lightning
during rainy season, then these waves are
produced somewhere; suppose this is my earth,
these are kind of lines of force. Suppose
I have produced some event has taking place
and suppose this is the wave has been produced
here by in some thunder storms some wave is
generated, this wave has been found to be
travelling, entering the ionosphere and travelling
along the field lines like this, and can be
observed here somewhere here or can leak in
any point.
So, what is happening that these waves have
a tendency to travel along the field lines.
So, the field lines are curve, they also bend
to travel along the field lines and can travel
almost half the hemisphere from North Pole
to South Pole, they can travel half the earth
curvature. Very thousands of kilometers they
can travel and with very little attenuation.
The attenuation of these waves is characterized
by a parameter called nu upon omega c. In
the ionosphere, nu is around 10 to 3 10 to
4 collisions per second. Omega c is around
10 to 7 radian per second. So, this ratio
is about 3 or 4 orders of magnitude small
as compare to unity and consequently the imaginary
part of k which is responsible for damping
is very weak. And these waves have been used
for exploration of the ionosphere; upper regions
of ionosphere as well as magnetosphere. They
are very useful waves.
Well, these waves have been found to be very
useful in case of metals also. We have seen
that wave propagation in a unmagnetized plasma
is very similar to wave propagation in a metal
because effective plasma permittivity that
we define in plasma in a unmagnetized plasma
was 1 minus omega p square by omega square,
and for a metal this 1 was replaced by what
we call as epsilon L; lattice permittivity
unmagnetized.
But if you take a metal metal rod for instance,
and put it inside a solenoid or coil that
carries current so that there is a magnetic
field produced in this direction. And you
can launch a wave through the metal by using
another coil, RF coil. So, if you launch a
wave here of frequency omega, in the presence
of magnetic field, the effective plasma permittivity
which is relevant is epsilon plus which is
epsilon xx plus i epsilon xy and this turns
out to be equal to epsilon L minus omega p
square upon omega into omega minus omega c
plus i nu.
So, this is a quantity which is very similar
to plasma except that this lattice permittivity
appears in place of 1. So, your wave propagation
constant for the right circularly polarized
wave would be omega by c epsilon plus to the
power half. So, when you are talking of frequencies
much less than omega c and less than omega
p, then this quantity is really unimportant
as compared to this entire term. Then the
character becomes very similar to in the case
of plasma and this wave; this whistler wave
in a metal is called helicon wave.
So, this has a frequency much less than electron
cyclotron frequency and usually in metals,
this is less than omega p much less than omega
p indeed. So, k is very large because epsilon
plus is very large, under root of this quantity
which is called the refractive index of the
right hand circularly polarized wave; this
is much bigger than 1. This can be 10 to the
power 4, 10 to the power 5.
So, these waves travel with a very low velocity.
You can make the phase velocity of these waves
much less than c and you can employ these
waves for diagnostics like measuring the effective
mass of electrons inside a metal. In some
metals are having anisotropic mass; mass is
a tensor like bismuth. You can explore those
character is different components of mass
by using these waves.
So, helicon waves have been found to be very
useful for diagnostics of materials. These
waves also adjust in plasmas, in semiconductors,
intensive semiconductors as well as doped
semiconductors. What then you have to include
the effect of holes also, but there this is
a fascinating field of study.
Well then, now let me go over to waves of
even lower frequencies when I am talking of
omega comparable or less than omega ci. In
this case, the electron term in your expression
for propagation constant was omega p square
upon omega into omega minus omega c.
Now, because your frequency is less than ion
cyclotron frequency, it means it is much less
than omega c. So, this is of the order of
minus omega p square upon omega omega c, but
omega p square by omega c is the same thing.
As omega pi square upon omega c i is same.
Just substitute for omega p in terms of omega
p square is n 0 eE square by m epsilon 0 and
omega c is e v upon m. Just you can equate
them and you find that mass cancels out, and
they are equal exactly equal.
So, if I use this approximation for the electron
term, then my propagation constant for the
right circularly polarized wave takes the
following form.
Kr becomes is equal to omega by c into 1 plus
omega pi square by omega omega c i. This is
the electron term and ion term is omega pi
square upon omega into omega plus omega ci
to the power half. The ion term certainly
has a larger denominator than the electron
term, and consequently, the positive electron
term will be more than the ion term and because
the frequency is very low, this electron or
ion term both are really much bigger than
1. I can ignore 1. So, then add these terms
and this becomes nearly equal to omega by
c into omega pi by omega ci divided by 1 upon
1 plus omega upon omega ci.
Well, this quantity c omega ci upon omega
pi, hence the dimension of velocity and is
given a name Alfven velocity and we shall
learn its relevance in a moment. So, then
my wave number k r can be written as omega
upon v Alfven into 1 upon 1 plus; this is
the half under root here. So, 1 plus omega
upon omega ci to the power half. If omega
were much less than omega ci, then this factor
is like unity and these waves travel as if
they travel in a medium with phase velocity
equal to Va. So, then Va becomes the velocity
of the wave of low frequency and the wave
of omega much less than omega ci is called
Alfven wave.
So, this is the right circularly polarized
Alfven wave when it travels along the field
line and so let me just explicitly write this
that for omega less than omega ci, the propagation
constant of the right circularly polarized
wave becomes equal to omega upon v Alfven.
So, v Alfven is equal to omega by k r which
is called the phase velocity of the wave.
However; if a frequency is not much less than
omega ci, in that case, k r is no longer this
much. So, when omega is of the order of omega
ci, in that case, k r is equal to omega upon
VA into 1 upon 1 plus omega upon omega c i
to the power half and the wave is dispersive.
This is not a dispersive wave because the
phase velocity does not depend on frequency.
It is simply VA. So, when omega is much less
than omega ci, the wave the medium becomes
non-dispersive, but when the frequency is
comparable to omega ci or higher, in that
case, k r is this expression.
So, we have learned different characteristics
of electromagnetic wave in various frequency
domains. Let me plot a graph of frequency
versus k. I will plot omega here and k for
the right circularly polarized wave here.
What I am saying that the frequency of interest
r 1 is called ion cyclotron frequency omega
ci which is three or orders of magnitude are
more smaller than omega c. So, omega c is
somewhere here. And then there is a right
cut off; omega r.
So, at very low frequency, your wave is an
Alfven wave and its characters and its dispersant
relation would be something like this. But
as it approaches omega c, then its character
changes, and then it becomes like a whistler
wave at higher frequency than omega c, and
this goes increases as square omega goes as
k square. So, it goes as it becomes parabolic
here, and then this goes to infinity. This
is called cyclotron resonance here.
So, this is the branch here called whistler
and this is the branch in the vicinity of
electron-cyclotron resonance. This is a small
bend of spread of frequency is called electron-cyclotron
wave, and this low frequency part is called
Alfven wave here.
So, the whistler frequency starts actually
from here to here; almost close to omega c.
Just very close to omega c, the wave is given
a special name called electron cyclotron wave,
electron cyclotron wave. This is called; I
can call this ECW and for frequencies higher
than right handed cut off, the waves goes
like this. This is simply called RCP. All
these waves are RCP waves when they travel
in the direction of magnetic fields, but their
characteristics are different, phase velocities
are different; obviously, everything is right
circularly polarized.
So, we have learned something about this.
One thing more before I close my discussion
on right circularly polarized waves that the
magnetic field of these waves, if we use the
third Maxwell’s equation is given by k cross
E upon omega and if this is magnitude wise,
so this gives you B magnitude is like k magnitude
into magnitude of e upon omega.
So, when k is large, this is right circularly
polarized. So, k r. Then b is very large.
So, in the vicinity of cyclotron resonance,
when k becomes very large, then this magnetic
field of the wave is also very large as compared
to electric field. And these waves can be
employed for deflection of particles. Currently
there is lot of efforts to employ large amplitude
whistlers because they are primarily oscillatory
magnetic fields to deflect particles.
Suppose I have a space craft, and I want to
save it from some explorer outside because
when some nuclear explosion takes place in
outer space or somewhere, then it releases
very energetic particles ions. And if I want
those ions should not be entering the space
craft and should not destroy its electronics,
then I would like those particles to be deflected.
Then, what you can do?
If I have some sort of a space craft, suppose
this is my earth and suppose this is my space
vehicle somewhere here, and I want to protect
it from stream of particles that could be
produced somewhere in some explosion here.
Suppose an explosion takes place somewhere
and particles are coming, then I can produce
waves with magnetic field perpendicular to
this, then the particles which are going this
way, these particles will be deflected back.
So, these large amplitude whistlers, because
of their large magnetic fields, are very helpful
in deflecting particles and currently there
is significant amount of efforts being put
in to generation of very high frequency waves.
And in order to stimulate the conditions in
the ionosphere or outer space rather, much
above the ionosphere, one what people do?
They are doing large scale experiments in
laboratories to launch high frequency whistlers
and examine the particle dynamics. So, this
is about the right circularly polarized wave.
Now, a few similar is the character of LCP;
left circularly polarized wave. The only thing
is that in this case, the k vector of this
wave is omega by c 1 minus omega p square
upon omega into omega plus omega c, and the
ion term is omega p i square upon omega into
omega minus omega c i to the power half.
So, rather than having electron cyclotron
resonance, you are expecting a ion cyclotron
resonance and similar properties occur. If
you examine the cutoff of this wave for this
wave means when k L becomes a 0, then you
will discover that k L becomes 0 when omega
turns out to be… this term is really unimportant
when you put k L equal to 0 because this just
dominates over this and you can find this.
So, the high frequency cutoff actually is
can have a low frequency cutoff as well. The
high frequency cutoff turns out to be is equal
to is given by actually omega into omega plus
omega c is equal to omega p square and it
turns out to be omega is equal to omega L
left handed cutoff is equal to 1 by 2 minus
omega c plus under root of omega c square
plus 4 omega p square. So, there is a cutoff
frequency.
But what is happening here? That a wave of
omega less than omega L will not travel. So,
omega less than omega L does not propagate.
Omega bigger than omega L propagates, but
if you make omega substantially smaller less
than omega ci, the ion term becomes change
a sign and one gets propagation again. So,
there is something spectacular happening at
omega equal to omega c i.
So, this wave has a very interesting characteristic.
It is a left circularly polarized wave with
Ey is equal to minus iEx in which ions can
be re-cyclotron resonance sensitive and ions
can be heated can be a heated via ion cyclotron
resonance and that is a very important wave.
In mirror machine, rather than launching a
RCP high frequency wave, if I had launched
a low frequency LCP wave of frequency less
than or slightly less than omega ci or comparable
to omega ci, I can heat preferentially the
ions and that is the beauty here. I think
other characteristics like Faraday rotation
and other phenomena of wave propagation in
magnetic plasmas, we shall continue or discuss
in our next lecture. I think I will stop at
this stage. Thank you very much.
