Today, we will discuss electromagnetic wave
propagation in magnetize plasmas. Here, we
shall discuss the anisotropic nature of a
plasma response to an electromagnetic field,
and we shall talk about conductivity tenser,
effective plasma permittivity tensor, and
then we shall discuss a special case of electromagnetic
wave propagation along the dc magnetic field.
We shall discuss the propagation of the right
circularly polarized wave, then left circularly
polarized wave, then the phenomenon of Faraday
rotation, and we shall also talk about electron
cyclotron heating.
We have already seen the need of wave propagation
in magnetize plasmas when we were talking
about tokomak heating. Then there are other
devices also, in a mirror machine also where
the magnetic field lines of force are like
this; these are the lines of force in a magnetic
mirror, then either you launch wave from here
or you may launch from here. In either case,
the wave is propagating in magnetized plasma
either perpendicular to the line of force
or along the line of force or opposite to
the line of force.
In general, wave propagation in any arbitrary
direction to magnetic field is important.
In tokomak of course, we have seen that the
lines of force is my tokomak for instance,
then there are two kinds of magnetic field;
one of them is a toroidal magnetic field and
then there was a poloidal magnetic field like
this. Due to the toroidal current, and if
you want to heat the tokomak, then you need
to launch waves from either here or from here.
So, they are either coming perpendicular to
the line of force or along the line of force
or some angle.
Then, there are other devices smaller devices
like q machine or a beam plasma system, where
you have a cylindrical plasma column with
an axial magnetic field, this kind of situation.
So, if you want to heat it or you want to
study the interaction of a large plasma chamber
with an electromagnetic wave, then obviously,
we should take into consideration the presence
of magnetic field.
Even in plasmas where you do not apply externally
any magnetic field, just like laser produced
plasma, laser produced plasma, there also
you observe a strong magnetic fields. For
instance, you can choose a thin foil, metal
foil on which you shine laser, then it forms
a large plasma plume and in the plasma, people
have observed strong magnetic fields, multi
mega gauss magnetic fields have been observed.
These days there is lot of work going on plasmas;
if you call as gas jet target plasmas? So,
this is a gas chamber, gas cylinder or gas
container and the gas is released through
a small nozzle and forms a plum here. You
shine a laser pulse on the plasma. So, a plasma
is created here and people have observed mega
gauss, hundreds of mega gauss magnetic fields
in such a situation when the laser intensities
are around 10 to 18 watt per centimeter square
or more. So, here again, the problem becomes
important that wave has to travel either along
the magnetic field or at an angle to magnetic
field and magnetic field can have very significant
influence on wave propagation.
So, today we shall look into the effect of
magnetic field on wave propagation. A few
things I would like to emphasize right in
the beginning; the major effects that we anticipate
from the presence of magnetic field is as
follows.
Suppose this is the direction of magnetic
field, static magnetic field which we normally
take to be the z axis. You may note that if
I have a plasma in which magnetic field axis,
then if I apply the electric field to a plasma,
alternating electric field like this, oscillating
electric field in this direction, I do not
expect much of influence of magnetic field
on the wave on the plasma response because
under the influence of the electric field,
when the electrons move in the direction of
magnetic field, the v cross B or Lorentz force
on them due to magnetic field will be 0.
However, if I have a situation where the electric
field is acting perpendicular to the line
of force like in this direction, it is oscillating
like this, the electric field… In this case,
when the electrons move in the direction of
electric field, the Lorentz force v cross
B force will immediately start acting on the
particle and that will be perpendicular to
the velocity and magnetic field. So, that
will be perpendicular to this plane. The force
will be in this direction. So, particle will
acquire a velocity in that direction.
What a situation that the electric field is
applied in one direction, electrons acquire
a drift in that direction as well as in a
direction perpendicular to electric field.
Consequently, the current that you see here
is no longer parallel to electric field. So,
J normally we write J equal to sigma e which
implies that J and e are in the same direction.
But as I mentioned to you that if the electric
field as a component perpendicular to magnetic
field, then J and e will be in different directions
and hence this relation is not valid.
What you require? If you want to allow a different
direction for J in comparison to e direction,
then this coefficient cannot be a scalar quantity
because whenever this scalar quantity, it
implies that J and e are in the same direction.
So, then J becomes can be written as something
like a tenser; sigma becomes a tenser. We
have talked about this earlier in our early
lectures that the plasma response to a magnetic
field is such that in general, Ax component
of electric field will produce J not only
in the x direction, but in y and z direction
also. And electric or conversely, current
in x direction is not only produced by e in
the x direction, but E y and E z as well.
So, this is a conductivity tenser and we have
deduced an expression for this and I just
want to tell you that if my magnetic field
is chosen in the z direction, in that case,
conductivity tenser which was sum of two terms;
the electron conductivity tenser plus ion
conductivity tenser sigma i plus not, this
i is not there. This is the electron conductivity
tenser plus ion conductivity tenser. They
were having similar forms and I like to write
down only one; other is very similar.
So, it turns out it was something like this
which was derived by solving the equation
of motion. Sigma was equal to this component
of sigma was sigma xx, this was sigma xy and
sigma xz was 0. This coefficient was sigma
yx which is equal to sigma xy is a negative
sign, this was equal to sigma xx, this was
0, and this was 0, this was 0, and this was
sigma xx and sigma zz.
So, conductivity tenser is like this; means
these two terms are 0, these two terms are
0, these two diagonal terms are equal, and
these two are equal in magnitude but opposite
in sign. Now for electrons conductivity, the
xx component is n 0 e square upon m. This
is omega plus i nu square minus omega c square
and here you had a term i omega plus i nu.
and sigma Exy is n 0 e square upon m omega
plus i nu square minus omega c square and
here was omega c; where omega c is the electron
cyclotron frequency defined as the magnitude
of electron charge into magnetic field upon
mass of the electron. And nu is the collision
frequency of electrons, m is the electron
mass, n 0 is the density of plasma electrons,
and I think everything else…
You can define similar terms for ions. So,
sigma ions is equal to sigma of the electrons
with following changes; omega c going to minus
omega ci; ion cyclotron frequency which is
defined as omega ci is equal to e B s upon
mi, where e is the magnitude of ion charge
mi is the ion mass. And nu to be replaced
by ion collision frequency and the well is
probably this is all. So, with these changes,
conductivity for ions can be written. Once
conductivity is known to us, it is easy for
us to study the propagation of electromagnetic
waves.
So, let me write down the Maxwell’s last
Maxwell equation where it is important to
combine the conduction current and displacement
current. Curl of h we know is equal to J plus
delta D by delta t. And in plasmas, D is equal
to epsilon 0 into e, where epsilon 0 is free
space permittivity. So, if I put this back
in here and assume that the fields that I
am considering are of this form, A exponential
minus i omega t minus k dot R in general.
This is called the plane wave solution.
So, if I put this in this equation, what do
I get? I can replace delta t by minus i omega
and forget the left hand side for a moment,
retain it as such. So, curl of h becomes is
equal to conductivity tenser dot e minus i
omega epsilon 0 into e. You can combine these
terms as taking minus i omega epsilon 0 common,
and recognizing that e can also be written
as a unit matrix dot e, where this unit matrix
is a matrix with 1 0 0 terms there, and 0
1 0 there, and 0 0 1 there.
So, if I use this, then this equation becomes
i from here, then plus i sigma tenser divided
by omega epsilon 0 dot e. This is a nice equation
because I can combine the conduction term;
this is the current density conduction part,
this is the displacement part; together and
this becomes a tenser becomes like i omega
epsilon 0 and I can call this combination
of these two terms as effective relative permittivity
tenser of the plasma.
So, this is related to conductivity through
this relation and let me write this. So, I
am defining effective plasma permittivity
is equal to I unit tenser plus i sigma upon
omega epsilon 0. Please remember that in an
unmagnified plasma, permittivity is a scalar
quantity and you normally had epsilon is equal
to 1 plus i sigma upon omega epsilon 0, but
sigma was scalar. So, this is a tenser version
of that. And if you put the expressions for
sigma and i here, then this takes the following
form. This coefficient this first term is
epsilon xx, this is epsilon xy, this is 0,
this is minus epsilon xy, this is epsilon
xx, this is 0, 0, 0, epsilon zz.
So, permittivity tenser has non-equal diagonal
terms; two of them are equal, but third one
is not equal, and it has half a diagonal terms
which are equal to magnitude, but opposite
in sign and other diagonal terms are half
diagonal terms are 0.
Now, let me explicitly write down this expression
for epsilon xx including the motion of ions
together, it is equal to 1 minus omega p square
multiplied by omega plus i nu divided by omega
into omega plus i nu square minus omega c
square. This is the electron term and similarly
I must write down the contribution for ion
term which is ion plasma frequency square
omega plus i nu I; the ion collision frequency
upon omega into omega plus i nu square minus
omega c i square ion cyclotron plasma frequency
ion cyclotron frequency square. This is the
expression.
And epsilon xy turns out to be… this is
i omega c upon omega ion cyclotron electron
cyclotron frequency upon omega into omega
p square upon the same factor here, omega
plus i nu minus omega c square and the ion
term which is minus i times omega c of the
ion in ion cyclotron frequency upon omega
into ion plasma frequency square upon omega
plus i times nu i; this is also nu i here
square minus omega c i square. This is the
ion term, this is the electron term.
Well, I forgot to mention that sigma tenser
had a sigma xx term and sigma xx here also.
These terms are called Pedersen conductivity,
and the half diagonal terms of sigma are called
hall conductivity in plasma jargon. Here,
one may note that it depends on the frequency
whether if omega is small or large that the
relative importance of these terms will is
realized.
First of all at high frequencies, when omega
is comparable or of the order or of larger
omega c, what will happen? You may note that
the ion term because omegas c i is very small.
So, you can ignore this as compared to omega;
omega is too large, then this term is smaller
as compared to this. Why? Because please recognize
that omega p i which is equal to omega p i
square rather is n 0 e square upon m i epsilon
0 and omega p square which is equal to n 0
e square upon m epsilon 0.
So, you may note here that this electron plasma
frequency is much bigger than the ion plasma
frequency and hence these terms are unimportant
as compared to electron terms. So, ion motion
is unimportant. So, for omega bigger than
omega c, ion motion is unimportant. Ion motion;
ions unimportant. You can forget them contribution.
When omega is less than omega c, then also
you can ignore the ion motion. Ion motion
becomes important when omega approaches omega
ci, the ion cyclotron frequency which is a
very low frequency as compared to omega c.
So, when omega approaches omega ci which is
defined as e B s upon mi, the ion motion becomes
very important. Ion motion becomes very important,
becomes dominant. This is a very important
characteristic of a magnetize plasma. In an
unmagnified plasma, the ion response was unimportant
because mass was too heavy, but what really
happens that when you apply a magnetic field,
the electrons cannot respond to an electric
field perpendicular to magnetic field because
they gyrate about the line of force. So, if
you had a magnetic, if you have an electric
field applied, this is my static magnetic
field and whenever you apply an electric field
in this direction, the electron motion is
inhibited by cyclotron motion because the
electron will like to go round and round about
the magnetic field.
So, magnetic field inhibits the particle moving
away from the line of force and hence ion
motion and electron motion can be comparable
or at times ion motion can dominate over the
electron motion. So, that is a very important
characteristic of magnetized plasma that the
ions start playing a very important role.
And we have already seen that magnetized plasma
is a medium in which J and e are in different
directions. And as a consequence of that,
we will learn that in general, the wave propagation
depends on the polarization of the wave.
In unmagnetized plasma, whether it is a circularly
polarized wave or linearly polarized wave,
velocity of propagation is same; k is the
same for a given omega, but here we shall
learn that it will become polarization dependent.
So, there are some interesting features that
are we going to notice, and I have written
the two components of permittivity tenser;
the third in finite component is epsilon zz
which remains unmodified by magnetic field
and is given by 1 minus omega p square by
omega into omega plus I nu. The ion contribution
to this is negligible and all other components
are 0.
So, once we have constructed the plasma effective
permittivity tenser, if you really go over
to revisit the Maxwell’s equations, you
will recognize that even the first Maxwell
equation which is divergence of d is equal
to rho. If you view this equation in conjunction
with the equation of continuity, delta rho
by delta t plus divergence of J equal to 0.
For currents that are time dependent, charged
density will also be time dependent in general.
So, if I replace this by minus i omega delta
delta t, then this equation gives me minus
i omega rho; the perturbed charged density
is equal to from here minus divergence of
J. J is equal to sigma e. So, if I substitute
the value of rho from here in this equation,
and this D is simply epsilon 0 into e in plasma.
In that case, you can combine these two terms
because rho is in terms of divergence of some
quantities, bring this back on the left hand
side.
And when you add these two terms, this simply
become divergence of epsilon 0 into epsilon
dot e equal to 0. So, the first Maxwell equation
has become similar to the one that you encounter
anisotropic dielectrics of permittivity epsilon.
So, the entire information about anisotropy
or plasma response is contained in one quantity;
that is, plasma permittivity tenser. And the
relevant equations for wave propagation are
the third and forth Maxwell equations which
we write as curl of e is equal to minus delta
B by delta t and curl of h; that we have just
written, is equal to minus i omega epsilon
0 epsilon tenser dot e and this also becomes
equal to I, if I replace delta delta t by
minus i omega, so i omega and B as mu 0 h.
Well, to obtain a dispersion relation or understand
the propagation of waves, I must replace del
operated by i k because I have already presumed
for a plane wave solution that my e goes as
A; some amplitude vector, exponential minus
i omega t minus k dot R in general, if my
wave is going in any general direction k.
So, just replace del operator by i k vector,
you get these equations as follows: the first
equation becomes k cross e is equal to omega
mu 0 h and the forth Maxwell equation becomes
k cross h is equal to minus omega epsilon
0 epsilon tenser dot e.
Now, you multiply this equation by k cross
on both sides. Then this equation becomes
k cross k cross e 
is equal to omega mu 0 k cross h. Use the
k cross h equation here, and this becomes
minus omega becomes omega square, mu 0 epsilon
0 is 1 upon c square, then relative permittivity
tenser dot e, and this I can write down as
k vector k dot e minus e into k square. This
is the equation vector equation which is three
components that governs the polarization and
propagation characteristics of the electromagnetic
wave in a magnetize plasma in general travelling
in an arbitrary direction to magnetic field.
And formally, I can write down this equation
as this: k square this i term I can write
down first and this later, I can write down
this k square i dot e this term as such because
e i can write down as i dot e, this is unit
matrix, this is minus k dot e and this term
as minus omega square by c square epsilon
dot e is equal to 0.
What I can do? Formally, I can take e outside,
and the remaining quantities I can call as
some quantity say a tenser new tenser let
me call this d dot e equal to 0. Here I am
calling d tenser as k square i minus kk and
then this is minus omega square by c square
epsilon. So, this d is a tenser three by three
tenser, and if this equation is to have non
trivial solutions, then determinant of d must
vanish. Here there are two solutions that
one e itself is 0, and then this whole equation
is everything is 0, so, but then there is
no physics.
But if you want an electromagnetic wave to
have finite electric field and it is travel
in plasma, then determinant of D must vanish.
D may have individual components finite, but
the determinant of this is 0. This is an important
condition and that is called the dispersion
relation. So, dispersion relation of an electromagnetic
wave in plasma in general is the determinant
of D equal to 0; that is a formal dispersion
relation.
So, let me write down the dispersion relation.
The determinant of D is equal to 0. Well I
think before I delve into the general case
of wave propagation at arbitrary angle because
this is a little complicated thing, I would
like to go step by step. Today I like to discuss
the propagation of wave along the direction
of magnetic field.
So, suppose I choose my wave propagation along
the static magnetic field which is the z direction.
So, I am choosing k vector to have only z
component; means, I am allowing my wave, electric
field to be of this form AE is equal to a
vector exponential minus i omega t minus k
z. I would like to learn two things; what
is the connection between k and omega, for
a given omega what is the value of k and what
are the relationships what is the relations
between different components of A vector.
Let us see whether k depends not only on omega,
but does it also depend on the relationship
A access with ay and so on or is it independent
of this relationship. Let us examine this
issue.
So, I am going to discuss a particular case
when k is parallel to z axis; means I am considering
a magnetic field in this direction, and my
wave to traveling in the same direction, but
I would like to find out whether the wave
still maintains its transverse character or
not, whether it is a linearly polarized wave
or its characters different. Let us examine
all that issue.
The entire information is contained in this
equation, this simple equation D dot e is
equal to 0. Rather than writing the… sorry
this D is not here vector is a scale a tenser,
this equation was if I write the value of
D little more carefully, then this was simply
k square e minus k k dot e is equal to omega
square by c square epsilon dot e. This was
the explicit form of this equation. If I put
the value of D properly, then this a this
is the form.
I want to write down this in component form
recognizing that k has only z component. So,
if k has only z component. So, if I write
down the x component of this equation or y
component of this equation, then this term
will not contribute at all. So, I would like
to write down the x component of this equation,
then y component of this equation and let
us see what we get.
x component’s equation gives k square Ex
is the left hand side, right hand side is
omega square by c square epsilon dot e. I
want the x component of this quantity which
implies omega square by c square, I must keep
the first index of this product to be x and
the second index of epsilon should run and
it should be common with the electric field
index. So, if it is x here, then should be
E x. If it is xy, then the second index E
should be having y. If it is xz, then it should
be Ez; means, you have to keep the second
index of epsilon same as the index on the
electric field or the component of the electric
field. This is how tensers multiply, but as
we have seen that epsilon xz is 0 in plasma
where magnetic field is in the z direction,
this is 0. So, this gives me… remove this
term and you can take this term on the left
hand side, So, this equation becomes k square
minus omega square by c square epsilon xx
into Ex is equal to the remaining term here
omega square by c square epsilon xy Ey because
this is equal to 0, epsilon xz is 0.
And for the y component, let me write 
the equation wave equation gives you k square
Ey which is equal to omega e square by c square
epsilon dot Ey component which turns out to
be equal to omega square by c square. This
is epsilon, first index is y, second is running.
So, yx Ex plus epsilon y Ey plus epsilon yz
Ez, but we recognize that this is 0, yz is
not there, this is 0. And this is equal to
epsilon xx and this is equal to minus epsilon
xy. So, when you remove this term, combine
this Ey term with the Ey here on the left,
and then you get a equation like this.
So, this equation becomes k square minus omega
square by c square epsilon xx Ey is equal
to omega square by c square with the negative
sign epsilon xy into Ex. Always also remember
the first equation that we have derived for
Ex was like this: k square minus omega square
by c square epsilon xx Ex was equal to omega
square by c square epsilon xy into Ey.
What you can do? Multiply the left side of
this equation with the left side here, right
we have with the right side. You will get
Ey Ex here, Ex Ey there. They will cancel
out and you get a dispersion relation. The
dispersion relation is k square minus omega
square by c square epsilon xx whole square
is equal to a negative sign omega 4 by c 4
epsilon xy whole square. I can take the under
root on both sides. So, you get k square minus
omega square by c square epsilon xx is equal
to plus times or minus times omega square
by c square epsilon xy.
There are two possibilities. So, k does not
have a single value. If there were no magnetic
field, epsilon xy is 0 and this term is not
there. So, k has simply this value, but now
because of the magnetic field, epsilon xy
is finite and there are two possibilities;
either this equation can have positive sign
right side can have a positive sign or a negative
sign and corresponding to two signs, we get
two roots of this equation, two values of
k rather, one of them is called k R, the other
is called k L.
So, let me write down the first case. When
I choose k square as k R square which turns
out to be equal to omega square by c square
epsilon xx plus… actually I forgot, when
I took the under root, then minus under root
will give you i also here. I forgot to write
i. So, I get I times epsilon xy, but if I
put this value of k square is equal to k R
square, this value in my any of those equations
that were relating Ex to Ey, I get an expression
for let me write down, for this case, if I
choose so much, then the equation which was
k square minus omega square by c square epsilon
xx Ey which was equal to minus omega square
by c square epsilon xy Ex.
Put the value of this expression put the value
of k square is equal to this expression. So,
when I put this value here, I get I epsilon
xy Ey is equal to minus epsilon xy Ex or Ey
is equal to I times Ex; means k can take so
much value with a positive sign here if E
y is equal to iEx.
And this has a very important message in here.
We shall discuss this. So, this particular
kind of wave with Ey having 90 degree being
90 degree out of phase with the Ex component,
this wave will have a propagation vector given
by positive sign between these two terms.
On the other hand, if I choose the negative
sign there, then this is denoted as kL square;
symbol kL square is used for this is omega
square by c square epsilon xx minus i epsilon
xy and if I use this in this equation, I get
Ey is equal to minus iEx. What a dramatic
change here? A wave with electric field Ey
leading Ex by pi by 2 k value is different
and for a wave in which Ey is behind Ex by
pi by 2, in that case, the k value is different;
means and let me first lets understand the
implication of this relation.
What is the meaning of i being here and what
is the meaning of minus i being here. We shall
just learn that this corresponds to right
circular polarized wave and a right circularly
polarized wave travels with a different wave
number and a left circularly polarized wave
travels with the different wave number, but
let us appreciate or let us recognize why
this wave is a right circularly polarized
wave. We shall discuss this.
I am saying that Ey is equal to iEx where
my electric field I had presumed to be written
as A exponential minus i omega t minus k z.
And this k, I am calling k R because for this
I have considered this or this is my wave
field. Ey is equal to iEx implies that if
I choose Ex is equal to some suppose Ax, exponential
of this quantity kRz and if I count kR to
be real, suppose there are no collisions and
k R is real, in that case, this equation simply
implies is equal to and A x is real too, then
this is simply A x cos omega t minus k R z.
Now, let me write down the y component of
this equation. The y component of this equation
would be Ey or simply Ey I can write down
from here, E y is i times E x. So, if I take
Ey i times A x exponential minus i omega t
minus k R z. Now take the real part of this
because whenever I write any field in complex
notation, its implied that the real part of
the quantity is the physical quantity. So,
if I take the real part of this quantity,
it turns out to be equal to Ax sign omega
t minus k R into z.
To appreciate this, let me plot a graph here.
If I plot Ex on this axis and Ey on this axis,
what you will note? First of all from this
itself, E x square plus E y square is A x
square 
and on this plane this for equation of a circle,
but what kind of circle? What I say is that
suppose I choose without any loss of generality,
suppose I choose z equal to 0 or some given
some value. So, at an instant, when this quantity
is 0, then Ex will be equal to Ax and Ey will
be 0 because sign of argument is 0. So, your
field will be somewhere here, this is the
field.
After a while, when time increases, when this
quantity become more than 0, then Ex will
have a value less than Ax because cos value
will decline whereas, sine value will increase.
So, your point will be located somewhere here.
At a time later further, this will move to
here. So, your tip of the electric vector
is going from here to here, then going from
here to here; means it is moving on a circle
which is moving on a clockwise sense like
this on anti clockwise sense from this figure
like this.
And if you examine this in terms of a right
handed screw, if I take a right handed screw
and rotate it like this, then if the direction
of rotation of a screw is given by this arrow
of the circle and then the direction of advancement
of the screw will give you z axis because
this x axis y axis here. What I am saying
here is that when you view in the direction
of magnetic field, suppose this is the direction
of magnetic field and if my electric field
rotates in the clockwise sense, in this sense
in the right handed screw sense where the
direction of advancement of this screw gives
the direction of wave propagation or magnetic
field, then such a wave is called right circularly
polarized wave.
So, this represents an RCP; right circularly
polarized wave. So, for a right circularly
polarized wave, wave number is k R and k R
is given by this expression.
Let me just write down the value of k R. I
wrote down this is equal to omega by c epsilon
xx plus i times epsilon xy under the root.
Put the values of epsilon xx and epsilon xy
and this turns out to be omega by c, this
one minus the electron term which turns out
to be omega p square and it turns out to be
omega plus i nu divided by omega into omega
plus i nu whole square 
minus omega c square. This was the value of
epsilon xx. Let me write down the terms.
There was similar ion term. For the moment,
suppose I ignore the ion term, this is very
similar. Let me write down probably plus ion
term. Let me just call this ion term and for
this, I substitute, I get plus i omega c upon
omega omega p square upon omega plus i nu
square nu i square omega c i square plus the
ion term due to epsilon xy to the power half.
What you see here, these two terms have same
denominator here and they can be easily combined
and this becomes like omega plus i nu minus
omega c factor there plus actually i is here,
i is there. So, this becomes minus sign actually.
this is the minus here and i is gone, there
is no i here.
So, this becomes omega plus omega c plus i
nu and you can factorize this denominator
and one of them will cancel out and it turns
out to be simply like this is equal to omega
by c, this is 1 minus omega p square upon
omega into omega plus i nu minus omega c.
This is the electron term. Ion terms also
combined, they give you omega p i square upon
omega omega plus i times nu I; the ion collision
frequency and becomes plus omega c i the ion
cyclotron frequency. Usually the collision
frequency is much a smaller than omega c in
plasmas of interest. nu i is much smaller
than omega c i of interest.
So, one may ignore for a while. They are essentially
responsible for absorption of waves and suppose
I ignore the absorption for a moment, then
one may note here that this has a resonance
denominator with omega equal to at omega equal
to omega c. This will overflow becomes very
large and it is changes sign as well. For
omega less than omega c, this term is negative
so the whole thing becomes positive, whereas;
for omega bigger than omega c, this is positive.
So, this whole thing becomes negative.
So, there is a change in the character of
wave propagation at omega equal to omega c
and at omega equal to omega c, around in the
vicinity of omega c, k R can take very large
value. So, the phase velocity wave will be
very tiny. Waves can be made to travel very
slowly and electrons can interact with them
resonantly at cyclotron resonance.
This is the ion term; the ion term does not
show any resonance. The reason is that the
electrons in a magnetic field also rotate
in the same right handed sense as the electric
field is rotating and consequently there is
a resonance. For a gyrating electron, if the
wave is going in this direction and electron
is gyrating like this, this is the direction
of electron motion and this is direction of
wave electric field.
So, if the electric field rotates in the same
sense in which any electron would have a tendency
to gyrate, then the magnetic field, then the
wave field as seen by the electron will be
shifted in frequency and the effective frequency
of the wave as seen by the particle will be
omega minus omega c and there is a possibility
of resonance when they two become equal.
In the case of ion, the ions gyrate in the
opposite sense left handed polarized sense
left handed sense and consequently for this
mode, there is no there is nothing like a
resonance. One thing more, we had seen earlier
when we were talking about wave propagation
in unmagnified plasmas that wave can travel
only when omega is bigger than omega p; however,
here because of the presence of omega c here,
since this term can change sign when omega
is less than omega c.
So, irrespective of whatever is the value
of omega p, if omega is less then omega c,
this term becomes positive and wave can penetrate
any dense plasma. What is the density is and
that is a very important characteristic of
plasmas that it can of magnetized plasmas
that a magnetic field can allow the penetration
of a low frequency wave into a plasma of any
density. And I think more such fascinating
characteristics we shall learn in more detail
in our subsequent lectures. I think I will
stop at this point. Thank you very much.
