Today's lecture is going to be
about electromagnetic radiation
which is one of the highlights
in 8.03 and one of the great
victories of 19th century
physics.
So far we have dealt with
mechanical waves and we have
dealt with sound waves but today
we are going to enter the domain
of electromagnetic waves.
Radio waves,
radar, infrareds,
visible lights,
ultraviolets,
gamma rays, x-rays,
electromagnetic radiation.
And electric field,
which is associated with a
magnetic field,
one cannot be thought of
without the other,
propagates through space,
even in empty space,
even in vacuum.
And I will start with Maxwell's
equations, and then you will see
a little bit more math which I
do to brush up on what you may
have forgotten.
So let's start with Maxwell's
equations.
01:30
And then we have the curl of E
equal minus dB/dt,
Faraday's Law.
And then we have the curl of B
equals mu zero times J,
J being current density that is
in vacuum.
02:30
If you apply the dell vector on
a function phi,
this is called the gradient of
phi, that is a vector.
That is d phi/dx,
x roof plus d phi/dy y roof
plus d phi/dz z roof.
And then we have the divergence
of a vector A.
That itself is a scalar.
And that is dAx/dx plus dAy/dy
plus dAz/dz.
I will raise this again later
but I want you to see this.
The curl of A,
the way I remember it,
and you probably do the same,
x, y, z, d/dx,
d/dy, d/dz.
And then you get your Ax,
Ay, Az.
And that then becomes dAz/dy
minus dAy/dz.
And that is in the x direction.
And then you get dAx/dz minus
dAz/dx, and that is in the y
direction.
And then you would get dAy/dx
minus dAx/dy,
and that is in the z direction.
And let me check that because
it is so easy to make a little
slip, and that is very awkward
for me and for you also later.
That is fine.
And then we get dAy/dx minus
dAx/dy, that is fine.
There is one,
and only one,
vector manipulation with the
dell vector that I want you to
know.
I don't want you to remember
it.
I certainly don't remember it,
but we need it today.
And that is the curl of the
curl.
The curl of the curl of A,
and I will show you the result
without proof,
is the gradient of the
divergence of A.
Minus dell dot dell vector A.
And this we often write as
simply dell squared.
For this we write dell square
of A.
And this is also called the
Laplacian.
And so the dell square of a
vector is d2A/dx2,
the whole vector plus d2A/dy2
plus d2A/dz2.
And this has nine terms because
this has three terms,
this has three terms and this
has three terms.
I will do the first one,
and then you have to do the
others.
I will only do the component in
the x direction which then
becomes d2Ax/dx2 plus d2Ax/dy2
plus d2Ax/dz2.
And that is then the component
in the x direction.
And there is also one in the y
direction and there is one in
the z direction.
This first term comes from this
one, the second term comes from
this one, and the third term
comes from this one.
Now we are in a good position
to start to use this on
Maxwell's equations.
I will raise this again,
I will lower this so you can
see both, and I will start
working on the center board.
What I am going to do now is
take the curl of the curl of E.
And I know that the curl of the
curl of E is minus d/dt because
the curl of E is minus dB/dt.
And so it is minus d/dt times
the curl of B.
Now I use the only thing that
we should remember,
and that is this identity,
that the curl of the curl of A
is the gradient of the
divergence of A.
But the divergence of E is zero
in empty space,
you see that there,
so this first term is zero.
I only have this term which is
then minus the Laplacian
operator on the vector A.
So this is also minus dell
squared E.
Now, the curl of B is epsilon
zero mu zero dE/dt.
So when I take the time
derivative, I get minus epsilon
zero mu zero times --
-- d2E/dt2, so that is this
part.
And now I get on the right
side, I get minus d2E/dx2 plus
d2E/dy2 plus d2E/dz2.
And this minus sign and this
minus sign eats each other up.
And this equation that you are
looking now at is a milestone in
the history of mankind.
This equation changed our whole
way of thinking about the world
and even about the universe.
This was the great victory of
Maxwell.
This is a wave equation in an
electric field in vacuum.
So this tells you that you must
be able to create electric
fields which move with speed v,
for which we always write c,
which is one over the square
root of epsilon zero mu zero.
Because, remember,
this is a wave equation.
And what you see here is always
one over v squared.
And it was Maxwell,
of course, who was the first to
recognize that,
because this was only possible
because he added this term to
[UNINTELLIGIBLE] equation.
He was a genius.
Now, you can go through a
similar reasoning.
Instead of taking the curl of
the curl of E,
you could take the curl of the
curl of B
And then you will find that
there must be an associated
magnetic field for which you get
that the dell square B equals mu
zero epsilon zero times d2b/dt2.
One cannot exist without the
other.
To put it even more bluntly,
one is the other.
You cannot think of them as
being separate.
One cannot exist without the
other.
So we are going to have
three-dimensional wave
equations, and the
electromagnetic waves are then
characterized by two
interdependent oscillations,
one in the E field and one in B
fields.
And the speed with which they
propagate in vacuum is this.
That follows immediately from
Maxwell's equations.
Untouched by human hands,
so to speak.
And, if you use the value for
epsilon zero and mu zero,
you will find that this is
extremely close to 3.00 times 10
to the 8 meters per second.
And the amazing thing is that
epsilon zero can be measured
without any time variability.
Epsilon zero follows from
Coulomb's Law,
as static as you can have it.
And mu zero can be measured
without any time variability.
Mu zero follows
[UNINTELLIGIBLE]'s Law.
And so it was these two,
which I call static quantities,
Maxwell was able to demonstrate
that they dictate the speed of
light in vacuum.
Mu zero is called the
permeability of free space.
You can look up what it is.
It is 4pi times 10 to the minus
7.
And epsilon zero is called the
permittivity of free space,
8.8 times 10 to the minus 12.
Maxwell knew the speed of
light.
It was known way before
Maxwell.
In 1676, Romer,
in a brilliant way using the
eclipsing times of the moons of
Jupiter derived the speed of
light.
In 1676.
And he came up with 214,000
kilometers per second.
And the only reason why he was
on the low side is that it
wasn't well-known in those days
what the distances were between
planets.
And then in 1728 James Bradley
used another brilliant
technique.
I won't go into the details,
but it is called stellar
aberration.
And he used that technique to
determine that the speed of
light was 301,000 kilometers per
second.
And then in the late 19th
century, in fact,
in 1849 both Foucault and
Fizeau measured the speed of
light in their laboratory in
France.
One used the rotating mirror.
The other used the rotating
disk.
And so they were even able to
do it in the laboratory.
And they found values which
were within 5% of 300,000
kilometers per second.
So Maxwell knew the answer.
And so when he saw this,
he immediately realized,
he postulated that light is an
electromagnetic phenomenon.
And in 1865 he laid the
foundation of the very famous
Electromagnetic Theory of Light
which changed the way that we
look at the world.
So now comes the question what
are the solutions to this wave
equation.
Well, I will start with a
simple one, and the simple one
actually holds all the key
information that you need.
But then we will build it up
and make it a little bit more
complicated.
I start with a xyz coordinate
system.
And whenever in physics you
deal with cross products.
Always make your coordinate
system right-handed.
Don't even think of left-handed
coordinate systems.
And a right-handed coordinate
system is a coordinate system
whereby x roof crossed with y
roof equal z roof.
I assume a simple case that the
E vector only exists in the x
direction, so it has a certain
amplitude.
I call it E zero x.
And I make it a traveling wave,
of course, cosine omega t minus
kz.
So Ey is zero and Ez is zero.
Only E vector in the x
direction.
The complicated
three-dimensional wave equation
collapses now to a
one-dimensional wave equation.
Of the nine terms that you have
her, only one term survives,
and that is d2Ex/dz2.
And that then becomes mu zero
times epsilon zero.
And, when you take the d2E/dt2
only one term survives because
there is no Ey,
there is no Ez,
and so this now becomes
d2Ex/dt2.
So this is now a
one-dimensional wave equation.
I started easy.
Now, remember that the curl of
E is minus dB/dt.
So the curl of E,
in this case,
and you can check that because
you know what the curl is,
here is the curl.
The curl of E,
there is only one term that
survives.
I will write down here the curl
of E.
There is only one term that
survives, and that is dEx/dz in
the y direction.
And that now must be equal to
minus dB/dt because that is
Maxwell.
I will lower this later.
I will raise it so that you can
see this, but it is important
that you can see this above my
head.
My question now is what is B?
This is the traveling wave in
the z direction which has an E
vector only in the x direction,
as of now, and I want to know
what the associated B field is
Well, for one thing,
you can already see that the B
field is going to be in the y
direction.
If we take dEx/dz --
So this is the Ex.
And we take the derivative
against z, we get a minus k,
then you get to E0x.
And then the cosine becomes
minus sine, so this becomes a
plus times the sine of omega t
minus kz.
And that now equals minus
dB/dt.
We are almost there.
All we have to do now is do an
integration in time.
It is important that we get the
y direction.
That is very important.
Remember this y is very
important.
That is going to be the
direction of the B field.
Don't forget that y.
Now you bring the [dt?] here
and you do an integration in
time.
And so now you get the B field
that is going to be associated
with that traveling wave
electric field.
If you do an integral,
the omega pops out below here,
so you get kE0x divided by
omega.
The sign becomes minus the
cosine, but I also have a minus
sign here so those two minus
signs cancel.
And so I get the cosine of
omega t minus kz.
But omega divided by k is c.
That is the speed of light.
And so I can also write this
now B in the y direction to
indicate that it is in the y
direction, E0x divided by c,
because omega divided by k is
c, times the cosine omega t
minus kz.
And, just to remind you,
here is your y again.
But, of course,
the By already indicates that.
And so now you have found the B
field that is associated with
this E field.
As I used to say,
and I said earlier,
one is the other.
One cannot exist without the
other.
If we compare the two now,
you see several things which
holds in general.
And that is you see the
magnitude of the B field,
you can call this B0y,
is c times lower than the
magnitude of the E field.
See the c here?
Notice also that the E field
and the B field are in phase
with each other.
You have here cosine omega t
minus kz and you have the same
here.
There is no phase difference
between them.
That means if one reaches a
maximum the other reaches a
maximum.
And if one is zero the other is
zero because they oscillate with
that frequency omega.
Notice also that B is
perpendicular to E because E is
in the x direction and B is in
the y direction.
Notice also that each one of
them is perpendicular to the
direction of propagation.
E is perpendicular to the z
direction and B is perpendicular
to the z direction.
And all this follows from
Maxwell's equations.
Now I would like to make a
sketch of what this
electromagnetic wave looks like,
and I will make an attempt to
make you see it.
It is not so easy,
but I will make an attempt.
Here is the x direction and
here is the y direction and let
this be the z direction.
I pick a particular moment in
time.
I know that the E vector is a
cosine function in z because I
pick a particular moment in
time.
And so I will try to put in
here the cosine function.
If you think it is a sine
function that,
of course, is the same thing.
And so at this frozen moment in
time the E vector would be like
this, like this,
like this, like this and like
this.
And this value here would be
E0x.
Associated with that E field is
a B field that is in the y
direction.
It is like this.
It is in the yz plane.
And so the B vector is like so,
like so, like so,
like so, here,
here, here, here.
And this value then is B0y.
And this whole pattern moves
with velocity c in this
direction, and the two are
married together.
They are stuck together.
And, if you take any plane
perpendicular to the z axis,
a plane that is infinitely
large in this direction,
infinity large in this
direction, infinitely large in
that directly and infinitely
large in that direction,
at that moment in time the E
vector is everywhere in that
plane exactly the same value.
Look at the equation.
There is no dependent on y and
x.
That is why we call them plane
wave solutions.
How realistic they are is
another matter,
but they are consistent with
Maxwell's equations.
And the same is true for the B
field.
Any plane that you take
perpendicular to z,
someone is sitting 25 miles
away from you,
but on that same plane at any
moment in time you will see the
same E vector,
all of you, and the same B
vector.
And that whole pattern then
moves with velocity c in space.
And so if I am standing here in
one of the many planes
perpendicular to the z axis and
the electromagnetic wave comes
to me and there is another
person standing there in the
same plane, we will see the E
vector go like this and we see
the B vector go like this,
but they go in unison.
When the E vector is maximum
here the B vector is maximum
there.
And then they reach both zero
at the same moment in time.
And then the E vector goes
negative and then the B vector
does that.
So that is the way the
oscillation works.
I would like to summarize for
you what is actually the bottom
line of all this.
And when I have to solve B
fields for given E fields or E
fields for given B fields,
that is really the only way
that I used to think.
So we have a traveling
electromagnetic wave.
What follows only holds for
traveling electromagnetic waves,
not for standing
electromagnetic waves which
comes in the future.
Traveling electromagnetic
waves.
The E vector is perpendicular
to the direction of propagation.
The B vector is perpendicular
to the direction of propagation.
E and B are in phase.
If one reaches zero the other
reaches zero.
E is perpendicular to B.
And, as a result of the fact
that they are both perpendicular
to the direction of propagation,
it follows immediately that E
cross B is in the direction of
propagation.
And then last,
but not least,
I have no room for it,
I will put it here,
that the magnitude of B at any
moment in time is the magnitude
of E divided by c.
If the E vector is only in one
direction, as it is here in the
x direction, we call that
linearly polarized radiation.
The word speaks for itself,
linear.
Now, of course it is entirely
possible, which is also a
perfect solution to Maxwell's
equations and to wave equations,
you could easily have an E
field which is in the y
direction.
At this moment in time,
it would be coupled to a B
field which is in the minus x
direction, so that E cross B is
still in the direction of
propagation.
And that would also be a
perfect solution to the wave
equation.
And so the sum of the two or
linear combination of the two
should also be a solution to our
wave equation.
Therefore, I can write now in
somewhat more complicated form
for traveling wave in which I
give it both a component in the
x direction, as well as a
component in the y direction.
And I can even change the phase
between the two.
In other words,
the following E vector is
perfectly kosher.
E0x times the cosine omega t
minus kz.
That would then be in the x
direction.
And then I would have another
E0y times the cosine omega t
minus kz.
And I can give it any random
phase angle delta.
And that would then be in the y
direction.
And so clearly this satisfies
the wave equation because each
separately satisfies the wave
equation.
Now we make delta equal zero.
You still have linearly
polarized radiation.
Look at the xy plane.
This is x and this is y,
and the radiation is coming
straight to you.
At a certain moment in time,
the E vector in this direction
reaches a maximum E0x.
And, if delta is zero,
that is the moment in time that
the E vector in the y direction
also reaches a maximum because
delta is zero.
So this one is 0y,
of course, at the same moment
as that.
So what is the net E vector
that is the factorial sum of the
two?
The E vector is this.
This is E total which is the
square root of E0x2 plus 0y2.
And, if you see this coming to
you, you will see an electric
field going like this,
linearly polarized.
No longer linearly polarized in
the x direction,
no longer in the y direction,
but in this direction.
I can also make the phase angle
between the two 90 degrees.
I could make delta pi over two.
Now you get something very
interesting.
This is now x and this is now y
so that the radiation comes to
you.
X cross y is a new direction.
And now I pick a moment in time
that E0x is maximum here,
so this is the vector,
but now E0y,
the E vector in the y direction
is now zero because they are 90
degrees out of phase.
And so, therefore,
if this one reaches a maximum,
this one is zero.
A quarter period later,
this one becomes zero.
But now you have here E0y.
A quarter period later,
this one is back to zero and
now this one is here.
And a quarter period later,
this one again is zero and this
one is here.
Now what you are going to see,
there is never a moment that
the E vector is zero,
but E vector rotates around in
an ellipse.
It goes like so,
like so and so on.
And so it rotates around like
this, and we call that
elliptically polarized
radiation.
There is nothing very special
about it.
It is a perfect solution to
Maxwell's equations.
You have one component in the x
direction, another in the y
direction, and you offset them
by 90 degrees.
You can choose this angle any
value you want to.
If E0x is the same as E0y,
it is a circle.
And then we call it circularly
polarized radiation.
In this case,
it is going clockwise.
But, of course,
if you make delta minus pi over
two it will go counterclockwise.
Now, suppose you were asked to
calculate the associated B
field, that is a piece of cake
because you just follow these
simple rules.
You take the component in the x
direction and you calculate the
associated traveling wave in B.
And then you do for the y
direction and you calculate the
associated traveling B wave.
And you add them up.
That gives you then the
solution in B.
This situation is nice and
simple in 2D.
But I think I owe you a more
general description to widen
your insight.
And I want to,
at least in terms of the math,
go in 3D.
So we now have an xyz
coordinate system.
And now we want the option of
having the E vector,
not just in the xy plane or not
in the xz plane but in a random
direction.
37:33
It is now a vector.
It is not in x or y or in z.
It is in three-dimension.
So this E0 has an x component,
a y component and a z
component.
And now I can write down here
the cosine of omega t minus k
dot r, which is now the most
general way that I can write
this electromagnetic wave.
Whereby, k is kx in the x
direction --
-- plus ky in the y direction
plus kz in the z direction.
And k, as you will see shortly,
is the direction of
propagation.
And the magnitude of k is
always 2pi divided by lambda.
And that magnitude of k is the
square root of kx2 plus ky2 plus
kz2.
And now I would like to give
you some insight in the meaning
of this k dot r.
And for that I need a little
bit more space.
And now we have to make a touch
decision what we are going to
kill.
I am going to kill this.
To make you see what a
geometric meaning is of k dot r,
I will go first
two-dimensional,
and then you will immediately
get the picture what it is in
three-dimensions.
I have here the x direction and
I have here the y direction.
Here I have the k vector.
And so this is kx and this is
ky.
It is a vector.
I am going to draw a line
perpendicular to the k vector.
40:30
You can see that immediately.
If this is my vector r,
which is the position vector in
space given by the relation that
you see there,
then this angle here between k
and r, I call that theta.
Then k dot r is a scalar,
is the magnitude of k times the
magnitude of r times the cosine
of theta.
And what is r cosine theta?
That is this here is r cosine
theta.
And any vector r that ends on
this red line will have the same
value for r cosine theta.
So, therefore,
on the entire red line,
which in this two-dimensional
plot is just a line,
anywhere on this line the value
for k dot r is a constant,
is the same.
Now, to give a moment in time,
t equals zero,
we have an electromagnetic wave
traveling in the direction of k.
And this line happens to be the
line where the E field,
at this moment in time,
is the maximum and pointing in
your direction say.
It is a crest in E.
It is a mountain coming out of
the board in maximum value.
And here I draw another line
perpendicular to r.
Here k dot r is zero.
The dot product is zero.
R is perpendicular to k to this
point.
I will also assume that the E
vector here is a maximum
pointing in your direction.
And then there is one here.
K dot r here is,
therefore, 2pi.
And k dot r here is 4pi because
this now represents a wave.
This is the fool wavelength of
the wave where E is a maximum in
your direction,
E is a maximum in your
direction, E is a maximum in
your direction.
In other words,
this here is by definition of
lambda.
And this whole thing moves out
in space with speed c.
And so you see the lines k dot
r perpendicular to k represent
the maxima of the E vector at
this moment in time.
And all that starts to move
out.
Now you go to the third
dimension.
Then k dot r is a constant,
are now no longer lines,
but they are planes
perpendicular to the vector k.
And this whole plane
perpendicular to the vector k,
in that whole plane at the
moment in time,
the E vector is everywhere the
same.
Whether it is linearly
polarized, whether it is
circularly polarized or whether
it is elliptically polarized is
irrelevant.
It is everywhere the same.
And that whole plane then moves
out with the speed of light in
the direction of k.
And so k dot r,
in three dimensions,
are all planes perpendicular to
the vector k.
If you stand anywhere in space
and you look in the direction
where the radiation is coming
from then in any plane that is
perpendicular to k going to
infinity there,
to there, to there,
at any moment in time the E
vector is the same and the B
vector is the same.
And I said whether it is
linearly polarized radiation or
circular or elliptical that is a
different matter.
It could be either one of
those.
And so this is the best way
that you can think of the
general form of an E vector
going in three-dimensions.
These are then planes
perpendicular to the direction
of propagation.
In our first case,
which was so very simple
because I wanted to warm you up
slowly, k dot r became simply
kz.
Kx was zero.
Ky was zero.
Because it was only going in
the z direction,
kz was the only one which was
not zero, which was k.
And so my dot product,
k dot r collapses into a kz.
And so the wavelength lambda is
2pi divided by this k.
But, of course,
if you have a three-dimensional
case then the situation is a
little bit more complicated
because then k is the square
root, as you see here,
of kx2 plus ky2 plus kz2.
Our first case was to make it
simple for you.
This is the right moment to
stop.
After the break we can look at
some demonstrations also.
We have to relax a little to
digest all this.
This is not easy.
And so let's start handing out
this mini-quiz.
To make you feel good about
yourself, I made it easy this
time.
Don't start yet.
Can you start handing it out?
I put here on the blackboard
the xy plane.
This is your electromagnetic
wave going in this direction
perpendicular to the direction
of k.
This is, per definition,
a wavelength.
This is where E is a maximum in
your direction and this is where
it is a maximum in your
direction, and so this is the
wavelength.
Now, look at the intersection
of this wave in the y-axis.
This wave interacts here and it
intersects there.
And so the distance from here
to here is ly,
which is way larger than
lambda.
And the same is true for the
distance in the x direction.
It is also larger than lambda.
And in the z direction,
in general, it would also be
larger than lambda.
Now, this wave moves with the
speed of c.
And when it has moved a
distance lambda in the k
direction this crest is here.
How far has it moved in the y
direction then?
All the way from here to there.
So its speed in the y direction
is larger than c.
And so that speed,
which we call the phase
velocity in the y direction is
very simply, is ly divided by
lambda times c.
And that is larger than c.
It follows immediately from the
geometry because this angle here
is the same as this angle there.
This is also k divided by ky
times c.
And that is not only larger
than c, but it can be way larger
than c.
Ky is 2pi divided by l of y.
I refuse to call l of y lambda
of y.
I do not want to have to think
in terms of a wavelength in this
direction, a wavelength in this
direction and another wavelength
in that direction.
For me there is only one
wavelength, and that wavelength
is 2pi divided by k.
I refuse to call this lambda of
y.
Ky is 2pi divided by that l of
y.
And you can do the same,
of course, in the x and in the
z direction.
And you will find then that the
phase velocity in the c
direction equals k divided by kx
times c.
And the phase direction in the
z direction is k divided by kz
times c.
And what is the phase velocity
in the direction k?
That is k divided by k times c.
That is c, of course.
In the direction of k,
it propagates with the speed of
light.
But this pattern moves way
faster than the speed of light.
And it can actually get
completely out of hand.
It can be very,
very large.
Suppose these waves travel in
the x direction.
That means these red lines will
be vertical.
That means l of y will go to
infinity.
It means that k of y goes to
zero.
It means that the phase
velocity goes to infinity.
This pattern here,
the intersection here will then
go as a speed which is
infinitely high,
and that is no violation of
Einstein's theory of special
relativity because no energy
will flow with that speed.
And I can best convince you of
that by showing that something
similar can happen with water.
Suppose we have a shoreline and
we have some water waves coming
in like this.
Maybe I should put them in red
so you begin to make the
connection between the
electromagnetic waves and water
waves.
Here is a wave rolling in
nicely.
This is, by definition,
the wavelength lambda.
And all of that moves with
velocity v.
But the intersection of these
two waves here,
at point A and here at point B,
I call this distance l of x.
I could have called it l of y.
As I did here,
it is the intersection with
this axis that I gave the symbol
l of y.
I called it l of x.
The difference in arrival time
between wave one and wave two at
point B is the period of the
wave, which is simply lambda
divided by B.
That is trivial.
But that is not only the case
for point B.
That is also the case for point
A.
This wave one reaches A before
wave two reaches point A.
And this is the time in between
the arrival time of wave one and
wave two, which is a completely
different question from what is
the difference in arrival time
of wave one alone at A and B.
That difference in arrival time
between one wave,
between point A and B depends
on this angle theta.
And, when theta goes to zero,
that difference in arrival time
goes to zero.
Both A and B will,
at the same moment in time,
see that wave.
That means in that case,
when theta is zero,
this l of x becomes infinitely
large.
And so, if you express that in
terms of a phase velocity in
this direction,
the phase velocity then becomes
infinitely high.
It is this pattern that moves
with the velocity that can be
way larger than c,
but no water will move with
that velocity.
It is very clear there is no
water going from here to there.
And so this is not a violation
of Einstein's theory of special
relativity.
And so several very dedicated
students wrote me emails that
after last lecture they could
not sleep.
And I did not even feel guilty.
And the reason why they could
not sleep is that we had this
wonderful demonstration whereby
I have here an aluminum place
and I had another aluminum plate
and this was the z direction and
the separation between the
plates was A and we had
electromagnetic radiation going
in that direction.
And we concluded,
I will not go over the
reasoning again,
that the phase velocity in the
z direction was omega divided by
k of z and that was larger than
c.
That is why you guys couldn't
sleep.
And, in fact,
I even demonstrated that if you
make the radiation,
the frequency close to the
cutoff, that this phase velocity
even goes to infinity.
Now you know that there is no
problem.
There is no energy flowing with
that speed.
It is no different from the
water.
In fact, the energy flow is
with the group velocity which is
d omega over dkz,
and that was always less than
c, as you perhaps remember.
Not zero.
My enthusiasm is getting
carried away.
Less than c.
And remember at the cutoff
frequency where no longer
radiation would go through,
this phase velocity went to
infinity and the group velocity
went to zero.
I have fulfilled a promise to
those of you who could not sleep
to tell you that the meaning of
phase velocity is larger than c.
It is natural.
There is nothing wrong with it.
You have it with water.
You have it with
electromagnetic radiation.
You cannot transport any mass
with that speed.
You cannot transport any energy
with that speed.
There is nothing obscene about
it.
Very straight.
I now want to do a
demonstration to show you that
electromagnetic waves can be
linearly polarized.
Next lecture we will discuss
how we generate electromagnetic
waves.
We do that by accelerating
charges.
In this case,
electrons.
And I have here a transmitter.
This is an antenna through
which we are going to oscillate
a current at a frequency of 80
megahertz.
F is 80 megahertz.
The wavelength lambda is about
3.75 meters.
The wavelength is the speed of
light divided by the frequency,
which comes out to be about
3.75 meters.
And, as we oscillate 80 million
times per second back and forth
electrons in this antenna,
that means a current is going
back and forth.
Electromagnetic radiation is
produced.
Next lecture you will exactly
see how much and why.
And that electromagnetic
radiation is polarized in this
direction linearly.
That should not surprise you.
If the antenna is like this and
the electrons move like this,
it should not surprise you,
but you will see next Tuesday
why it is linearly polarized in
this direction.
Here I have a receiving
antenna.
It is a copper wire that is cut
in half, and where the two
copper rods connect there is a
light bulb.
Any current that flows in here
must go through the light bulb.
When I turn on this
transmitter, I want to show you
that as long as I hold this
receiving antenna like this that
I will see this light go on,
but when I do this I won't.
Because the electromagnetic
radiation, the E field goes like
this, is going to slosh a
current in here with a frequency
80 megahertz and the light will
go.
But when I do it like this the
electric field is like this.
There is no current flowing in
this direction,
and so this is a dramatic way
of demonstrating that there is
such a thing as linearly
polarized radiation.
What have we decided on the
lights?
We were going to TV.
We made it as dark as we
possibly could.
Here is the transmitter and
here is the receiver.
And I am holding it like this.
And you see that I am receiving
a signal.
And so there is a current
sloshing back and forth and the
light is on, and now it is off.
The E field comes in like this
and the light says tough luck,
I do not see a current going
through me, but now it does.
This is sensitive to the
polarization of the incoming
signal.
And when I go a little bit to
this side then,
as we just saw,
electromagnetic radiation,
the E field must be always
perpendicular to the direction
of propagation.
The direction of propagation
now is in this direction.
That is why I hold it like this
for maximum effect,
because now the E field is like
this.
Whereas, here the E field is
like this.
Now the E field is like this.
And the light is not as bright.
And you will see Tuesday why.
But clearly when I do this
nothing because of the linear
polarization of the
electromagnetic radiation.
If I come too close to this
then I will burn the light.
Do you want to see that?
Who wants to see that?
Oh, God, you children.
There is another way that I can
show you this,
and that we do with radar,
with the same setup that we
used last time,
except we did a different kind
of experiment last time.
This is 10 gigahertz so that
produces three centimeter radar.
And this is the transmitter
where the antenna is in this
direction.
It is only a very small antenna
of only three centimeters
wavelength.
The E field goes like this and
here is the receiver.
And, if we put the antenna of
the receiver like this,
it will receive the signal.
If we put the antenna like this
it says sorry,
I cannot receive it.
And this 10 gigahertz signal we
modulate with 550 hertz which is
a triangular modulation so that
you can hear it.
And triangular audio signal are
always not very pleasant
because, if you do a Fourier
analysis of this,
you have many,
many high harmonics.
It is not just a beautiful 550
hertz sinusoidal.
You hear a very sharp tone.
But, in any case,
the period is such that the
frequency is 550 hertz.
I will make you hear it and I
will make you see it.
You can see it there.
That is the signal as it is
sent by the transmitter.
That is the triangular shape.
And now I will turn on the
receiver and you will hear the
sounds.
This is the loudspeaker.
And you will see the signal of
the receiver there.
The radiation comes in like
this.
And now I am going to rotate
this one 90 degrees,
the receiver.
And it is gone.
The same phenomenon that I just
showed you with the 80
megahertz.
How can I get it back?
One way I can get it back is
rotating this back.
But I can do something better,
something more convincing.
I can rotate this by 90
degrees.
Now the transmitter,
now I have it back.
You see here two dramatic cases
of linear polarization.
My hand absorbs it.
A nice feeling.
Make sure we have some light
back.
We will discuss extensively in
8.03, actually,
one of my favorite parts of
8.03.
How you can turn unpolarized
light into linearly polarized
light.
There are various ways that we
can do that.
The cheapest,
which is really a copout,
is to buy a linear polarizer.
You have three of them in your
little envelope.
Don't take them out yet.
This is one of them.
They were invented by Edwin
Land.
And what they do is they change
unpolarized light into 100%
linearly polarized light at the
expense of the light intensity.
An ideal linear polarizer would
reduce the light intensity by a
factor of two.
Let us first discuss what is
unpolarized light?
The light from the desk lamp
and the light from the sun is
unpolarized.
Imagine that there is a desk
lamp here and light comes
straight to you.
And here is a plane wave
solution by the E vector
oscillate linearly polarized.
It is coming straight at you in
this direction and it is
linearly polarized.
But a little later there is one
that comes in like this,
also linearly polarized.
And then one like this and then
one like this and then one like
that.
Chaos.
All possible angles of linearly
polarized light.
And we call that unpolarized
radiation.
Now, what is this magic piece
of plastic doing that Edwin Land
invented?
It is doing the following.
There is one direction,
not always indicated on the
plastic what that is,
and I will just put it
arbitrarily vertical,
but I can put it any direction.
And that is the direction that
the E vector will have when it
emerges from this linear
polarizer.
Now let us suppose that
linearly polarized light comes
in, which is one of the many
unpolarized radiation.
And this has an amplitude E
zero and it is oscillating like
this, cosine omega t,
and that this angle be theta.
Now, one way to explain the
reduction in light is to project
this E onto this preferred
direction.
It is the only direction in
which the E vector will emerge.
And so you see there is a
reduction of the E vector,
and the reduction is the cosine
of theta.
Now comes the question,
if there is a reduction of the
cosine of the theta in the E
field, what is the reduction in
light intensity?
Well, light intensity means
energy.
And energy is always
proportional with the square of
the amplitude.
If that does not convince you,
next Tuesday we will talk about
the pointing vector.
You remember from 8.02,
the pointing vector that is
responsible for energy transport
which is proportional to E cross
B, but B is proportional to E.
So E cross B is always
proportional to E squared.
And so, therefore,
the light intensity that
emerges from this plate is
proportional to cosine square
theta.
And that is the reduction of
the light intensity.
And that is referred to as
Malus' Law.
Now, if all angles are present,
randomly angles of theta then
the average value of cosine
square theta becomes one-half.
For an ideal polarizer,
which don't exist,
but an ideal polarizer would
turn unpolarized light into 100%
polarized light all in this
direction, if that is the
direction.
And then the intensity of the
light would be twice as low as
the incoming one.
In practice,
however, you may not get 50%,
but you may get 40%.
There is always some additional
absorption.
And, in problem set number
seven, you will be able to
answer some interesting
questions and use the polarizers
that you have in your envelope,
in your optics kit.
Clearly, if I have one linear
polarizer like this,
this is the preferred
direction, and the other one is
like this, for which we have a
name in physics,
we call them cross-polarizers,
then, of course,
independent of this absorption
phenomenon that I mentioned,
no light can emerge.
Because now the cosine of the
angle is zero.
Cross-polarizers will turn
unpolarized light into darkness,
and that is something that I
would like you to see now.
Let's make it a little darker.
I have here several sheets of
that linear polarizer.
And the first thing I want you
to see is this absorption
phenomenon that I just
mentioned.
Here is a linear polarizer.
The light that comes through
here now is linearly polarized.
The light that reflects off the
screen is not linearly
polarized.
That is a different thing
because it goes through a
reflection process.
This light that comes up here
is linearly polarized.
You and I have no way of
telling.
There are animals that can see
polarized light,
who can distinguish it from
unpolarized light,
and they can even see the
direction of polarization.
Bees can see the direction of
polarization.
And I have learned also to see
it under ideal conditions.
If any of you is interested,
I can teach you,
but it is not easy.
You have to come to my office.
There is a way that humans can
actually recognize linearly
polarized light and see the
direction of polarization.
But, apart from that,
we have no way of knowing that
this is linearly polarized
light.
And, if I rotate this,
the direction of linear
polarization will change,
but we have no way of knowing.
In fact, the preferred
direction, as indicated on this
sheet, is like this.
Now I take a second one,
which is identical.
And I am going to put it on top
of it so that the two directions
are aligned.
If they were ideal then there
would be no light reduction
anymore, but there is light
reduction.
Look at this middle portion
where they overlap.
There is still a further
reduction in light intensity.
Even though the direction of
polarization has not changed,
it is not the result of Malus'
Law that there is a reduction
but there is an absorption
phenomenon that I mentioned.
Only here are two plates and
here is one plate.
Now I have here a black stripe.
That is on top of the overhead.
Here now is one linear
polarizer and here comes the
other.
They are now aligned.
And I am going to rotate them.
The angle of theta is
increasing and increasing and
increasing, and I am approaching
slowly 90 degrees.
And there you are.
And it is as dark as you can
have it.
Two cross-polarizers now
through an unpolarized light,
first into linearly polarized
light, that is sheet number one,
and then sheet number two kills
it altogether.
Nothing can get through.
Cosine of the angle is zero.
Now comes the great miracle,
which is something you can also
do at home because you have
three linear polarizers in your
optics kit.
Suppose I stick a third linear
polarizer in between the two at
a random angle,
will I see light coming through
or will I not see light coming
through?
Very good.
If I do it in front of number
one, darkness will remain
darkness.
If I do it above number two,
darkness will remain darkness.
But if I do it in between one
and two, there is my line again,
light comes through and there
is an immediately consequence,
of course, of the fact that you
have to apply now the cosine
square theta twice.
But you never will see it go
down to zero.
Now I want you to open your
envelope.
Don't put your fingers on all
the beautiful pieces that are in
there that we will have to use
in the future.
Open it and get out of it one
linear polarizer.
And you will recognize them.
They are green plates and they
have this shape.
And you just get one.
Don't drop them on the floor
and don't make them too dirty.
Can you see me?
Who cannot see me?
Good.
You can rotate your polarizers
until you are purple in your
face, and you won't see much
difference.
You will see Walter Lewin no
matter what.
Use your polarizers and rotate
them around and look at my face.
It is not going to change very
much.
Now, I am going to hold in
front of my face this linear
polarizer.
Now you are looking at Walter
Lewin in polarized light.
And that now gives you the
option to make me come and go.
You can rotate in such a
direction that you say there he
is, but if you rotate it at 90
degrees you say thank goodness,
he is gone.
Now, keep in mind that if you
rotate your linear polarizers
such that you can see me that
the reverse is also true.
I can then look through your
linear polarizer and I can see
your eye.
And if somehow,
in an evil way,
you prefer 90 degrees so that
you don't see me then you have a
black eye.
And who wants a black eye?
See you next Tuesday.
