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PROFESSOR: Welcome back.
And today we are going to
look at a harder situation.
At oscillations waves in
the electromagnetic field.
Why I say it's harder,
for many reasons.
First of all, so
far we've always
considered situations which we
could either visualize or had
some sensual way of
getting a feel for what
the physical situation is.
When it comes to the
electromagnetic field,
as you well know, we can't
see it, sense it, at all.
And the only way to
describe it is, in fact,
in terms of mathematics.
So there isn't, first,
a word-- a description
by analogy with
what we see around.
Secondly, it's more complicated.
These are oscillations
in three dimensions.
And, as you well know, there
both electric and magnetic
fields.
Overall, it is just much
more difficult situation.
So first of all, I start by
a mathematical description
of this system.
Because, as I say,
there is no other way
we know of discussing it.
And the mathematical description
of the electromagnetic field,
as you all know, are the
so-called Maxwell's equations.
I've written here the four
Maxwell's equations for vacuum.
So this is what the
electric and magnetic fields
have to satisfy.
And I'm just reminding
you that the definition
of what electric
and magnetic field--
the operational definition
comes from the Lorentz force.
Basically, this is just
quickly to remind you,
if I have a charge in vacuum
and if it experiences a force,
I know there is an
electric field there.
On the other hand,
if it experiences
a force when it's
moving, then I know
that there is a magnetic field.
So this tells us that here,
although we can't see it,
there is an
electromagnetic field.
If one looks at these equations
and plays around with them,
one find that the
electromagnetic field actually
satisfy wave equations.
This is the wave equation for
the-- three-dimensional wave
equation for the electric field.
And this is for the magnetic
field, where c is the phase
velocity as always, and in
the case of electromagnetism
c is given by that.
That's the speed of
light, or the speed
of electromagnetic waves.
Now, so what this tells
us, is that in vacuum, you
can have excitations,
oscillations
of the electromagnetic
and magnetic fields,
which propagate.
And we have all of
the wave phenomena
we've learned for other systems.
The thing to keep in
mind is that whatever
the solution of the system
is, whatever is propagating,
it must satisfy all
of these equations.
Not every situation
has to satisfy this.
This is a subset of the
infinite possibilities that
are allowed by
Maxwell's equations.
OK.
So now, instead
of doing solutions
to some specific situations with
a specific boundary condition,
et cetera, since it's
already much more difficult,
all I will do
today is see how we
can identify solutions
of these equations.
What kind of waves
they correspond to.
Or vice versa, if
you want to describe
in terms of mathematics
some particular wave,
how do we do that?
That is the kind of problems
I will be discussing today.
So, let me come to
the first problem.
And probably using the
word problem is a misnomer.
The description.
I'll consider first
progressive wave
solutions of these equations.
Suppose we know that there
is an electric field, which
is a propagating electric
field, sinusoidal.
All right?
I assure you, this does not
contradict Maxwell's equations.
You can try it.
All right?
It's not complete, as
you'll see in a moment.
The question here is, if
you have an electric field
like that, can we
describe as well as
possible in words, what kind
of a wave this corresponds to?
And secondly,
answer the question
if this is a real
electromagnetic wave
in the vacuum, what must be the
corresponding magnetic field?
By itself, this
equation does not
satisfy all the
Maxwell's equations.
You need a corresponding
magnetic field.
So, let's look at that.
First of all, we know
that any function, which
is a function of x plus
or minus vt describes
a progressive wave.
It satisfies the
classical wave equation,
you can try it and see.
If these two terms--
the x and the t terms--
are of opposite sign, then this
describes a progressive wave,
which goes in the
plus-x direction.
If they are the
same sign, then it
goes in the opposite direction.
And again I say, plot
any function like this,
and see what happens
as you change t.
The shape of the
function will not change.
But it will move either to
the left or to the right
as you change the time.
So we immediately
see, since this
is a cosine of this, which is
of this form, if I divide by a,
this is x minus b over a.
And I could take
the a outside that.
So this is a progressive wave.
These two have opposite signs.
It's a function of x and t.
So this is a progressive wave,
which is moving or progressing
in the x direction.
They're opposite signs, so
in the plus-x direction.
So immediately I know that
this is a progressive wave.
It is a sinusoidal one.
Well, this is a cosine
function, right?
It's a sinusoidal wave.
Now we know if I divide by
a, I get minus-B over a t.
So it becomes of this form.
So the phase velocity of
this wave will be B over a.
And this we call
normally, by the letter c,
that's the phase velocity
of electromagnetic waves
in vacuum.
Or commonly known
as speed of light.
That identifies it so far,
as best as we can in words.
This is, as I say, a progressive
sinusoidal electric field
moving in the plus-x direction.
What are the a's and b's?
By how much must you change
x so that the wave gets
the same amplitude
as where you started?
And the answer of that
is that, of course,
a must be 2 pi over lambda.
because then if x
changes by lambda,
your cosine changes by 2 pi.
So a in that equation
must be 2 pi over lambda.
That quantity is normally
given the symbol k,
it's called the wave number.
Similarly, if I look
at the time turn,
b must be equal to 2 pi
divided by the period.
Because if t changes
by the period,
then that cosine-- the
angle of that cosine,
the phase of that
function-- changes by 2 pi.
And you're back
where you started.
So b must be 2 pi over t.
So this tells you for
that particular wave
what the a must
be, what the b is.
2 pi over t is, of course,
the same as 2 pi times
the frequency, which
we normally call
the angular frequency, omega.
So a is k, and b is omega.
Next.
I said that any
solution that is real
of the electromagnetic
field must
satisfy Maxwell's equations.
So the same must
be true of this.
If this is the wave of the
electron, electric field,
there must be associated with it
a magnetic field such that all
of Maxwell's equations
are satisfied.
In particular, if we take
this one-- Faraday's law--
we know that the rate of
change of the magnetic field
must be equal to minus the
curl of the electric field.
This you can look up in
books of mathematics.
If you look at all the
components, the way I always
remember it, it
is the determinant
where here you have the
unit x direction yz.
This is dx, dy, dz.
And here is the x component of
electric field, y component,
and z component.
For our particular
electric field,
I only have the z component.
And it's only a function of x.
So most of the terms
of this expansion
are 0, except the one-- the
rate of change of with x of Ez.
And that will be
in the y direction.
So this db dt must be equal
to that if that is a solution
the Maxwell's equations.
If I take the x
derivative of E up there.
I end up-- and you
could almost do it
in your head-- db dt
is minus this quantity.
But if this is the
rate of change of t,
I can integrate this.
And if I integrated it, B
must be equal to-- the a
comes from here, a
minus a, a over-- sorry.
The b comes from
here, I misspoke.
That comes out, and
the integral of sine
gives you cosine, so
that must be satisfied.
But since we integrated
B, there will
be a constant of integration.
So if I add to this
any constant B,
this will still
satisfy this equation.
All of this is telling
me is that if I
have that electric field--
propagating electric field--
I must simultaneously have this
propagating magnetic field.
And on top of that, I can have
any constant magnetic field.
It means that is a more
general situation where
this electric field and
these magnetic fields
can exist with any constant
B. I'll just call it 0.
It's not an interesting
part of this,
it's not a propagating field.
And so we end up that if
you have that electric field
propagating, and in with
this magnetic field,
then that system satisfies
all Maxwell's equations.
Both the E and B will
satisfy these wave equations.
Try it for yourself,
and you'll see.
So the answer to
this is, what this
is, this is a polarized--
plane-polarized electromagnetic
wave, where we identified the
wavelength, the frequency,
it's propagating
in the x direction.
And the electric field is
polarized in the z direction.
One of the things we will learn
from this so we don't have
to repeat over and over again
when we're looking at different
formulae, which describe ways to
help us to identify , them is--
notice that what we have
found was that the electric
and the magnetic fields are
perpendicular to each other.
The electric field
in the z direction,
the magnetic in the y direction.
But the sinusoidal part and the
phase velocity and everything
else-- wavelength, frequency--
are exactly the same and phase.
This is completely in general.
If you have a progressive
electromagnetic wave in vacuum,
you find that the
only way it can
exist if you have
simultaneously an electric
and a magnetic
field propagating.
They are always at right
angles to each other.
This is the electric field,
this will be the magnetic field.
If it's propagating
in that direction.
It's always from e to b
in a clockwise rotation,
if they're propagating
in that direction.
So I drew a general sketch here.
This is true for any
progressive wave,
electromagnetic
progressive wave.
And you have the electric field,
magnetic field perpendicular
to it, and the two
propagate in that direction,
given by this vector equation.
Furthermore, if they satisfy
Maxwell's equation the ratio
of E to B, the
magnitude, is equal to c.
This is completely general.
It is worth
remembering when we're
analyzing different situations.
So that I went
slowly through this,
but that is one
example where we see
this mathematical description
of something which
we can recognize
what it is, and which
is a solution to Maxwell's
equations in vacuum.
What actually happens in
the physical situation
depends, as always, on all
the boundary conditions,
the initial
conditions, et cetera.
This doesn't address
all those questions.
All this says is this is one of
the infinite possible solutions
of Maxwell's equation.
In other words, for
electromagnetic fields
corresponding to the plane wave
propagating in one direction.
Let's take a harder example.
The question is the following.
Can we now do the opposite?
Not someone tells
us the equation.
Can we actually describe
in mathematical forms
a electromagnetic
wave whose properties
we know what we
want and would like
to write it mathematically.
And I took a
slightly harder one,
so I said we would
like to describe both
the electric and
the magnetic fields,
which describes a monochromatic
electromagnetic wave--
monochromatic means a single
frequency, single wavelength--
with wavelength lambda,
which propagates now
not along the x or y or z
axes that makes life easy.
Let's say it goes at some angle.
It goes at 45 degrees to
the x-axis and y-axis.
And the z is out of the board.
So the wave-- we
want the wave, which
is propagating like this, where
the wave front is-- let me come
to it in a second--
where the vector
perpendicular to the wave
front is at 45 degrees
to both the x-axis and y-axis.
We want it
plane-polarized, meaning
that the electric vector
is always in a plane
and it's linearly
polarized so it's
in the same direction
in the x-y plane.
So how can we translate
that into mathematics?
Well, we'll use some
of the knowledge
we've just gained before.
First of all, we know
from what I discussed
about the electric and
magnetic field being
perpendicular to each
other and perpendicular
to the direction of propagation
that if the propagation is
in this direction, then we
know that the plane in which
the electric and magnetic
fields find themselves
are perpendicular to that.
Since this is
propagating like this,
the distance between the planes
of equal phase will be lambda.
That's the meaning
of the wavelength.
Once you've gone the
distance of 1 lambda,
the magnitude and
direction is back
to what it was before for
both the electric and magnetic
fields.
So that's what it
will look like.
So the electric vector
will be in this plane,
but we are told
furthermore it's in the xy,
so it will be in this direction.
If it's like this,
and in this plane,
so this must be the direction
of the electric vector.
So let's give it a
magnitude E-zero.
And what is this unit vector?
Well, clearly that is
in the x direction.
It has a component like
this, and in the y direction,
it has a component like that.
The magnitude of
the components is
the same, because
of the 45 degrees.
But for the x, it'll be
negative, and for the y,
positive.
So the unit vector
in the direction
of the electric vector
will be minus x-hat
over root-2 plus
y-hat over root-2.
This is a unit vector, you
can check for yourself.
If you take square this, square
that, take the square root,
you get 1.
So this is a unit vector in this
direction where we wanted it.
So if I write this
as the amplitude
and the direction of
the electric field,
I do have a field which is
linearly polarized always
in the same direction.
We'll put a sine
or cosine there,
because we're talking about a
monochromatic electromagnetic
wave with the wavelengths, so
it's a sinusoidal function.
Where I put the sine or
cosine or any other phase
just determines
where time equals 0.
So let's put sine.
It's going to be propagating
in this direction, plus k,
so these two will
have opposite sign.
This will be the frequency--
angular frequency--
of oscillations of this.
And here we must
describe a plane.
Because along this plane,
the phase has to be the same.
That's what we
mean by wavefront.
Vectorially, how do
we describe a plane?
Well, we will have the plane
which is perpendicular to k
if we take k dot
product of the vector r.
r is the vector.
Here is the vector
r, from the origin
to a point on the plane
which I want to describe.
So this is k dot r.
So this now, we'll have k,
which is the wave number,
and this whole thing
is called the k vector,
will have a magnitude
which is 2 pi over lambda.
Same as in the other problem.
But now it's pointing in
this direction, which again,
by analogy, how we calculated
that is the unit vector
x over root-2 plus unit
vector y over root-2.
So this is k.
r is nothing I want to
describe this point.
I have x in the x direction,
y in the y direction,
z in the z direction.
So that describes any
point on that plane.
If I take the dot
product between them,
I will get then a wave which
is moving the the k direction.
And this describes the
position on the wavefront.
So putting it all together,
this electric field
at every point of x, y,
and t will have a magnitude
is E-zero times this
direction, the direction
of polarization of the
electric field, times sine.
This is now telling me it's
propagating in this direction.
And with angular
frequency omega.
So that describes the
electric part of this wave.
How about the magnetic one?
Well, we could do
the same as before.
The magnetic part is
determined by this,
because all Maxwell's
equations have to be satisfied,
including Faraday's law.
But I told you, so it saves me
doing it over and over again,
we've learned once and for
all, for a progressive wave
the e and b are
perpendicular to each other,
and the ratio between them is c.
So since I know what E is, the
magnitude of the magnetic field
is E-zero over c.
It'll be at right
angles to this direction
and to the propagation,
and therefore it
will be out of the board.
So that from E-cross-B, the
vectors are in the k direction.
So the b will be out
of the board, which
is easier this time.
That's in the z direction.
And it will be, as I said,
exactly in phase in time
and space with the
electric field.
The two are coupled together.
So that now describes
it entirely.
So this is, in fact, the
answer to our question.
It describes an electric
or magnetic field
which is monochromatic.
It's an electromagnetic wave.
It has wavelength lambda.
It propagates at 45
degrees to x and y axes,
and is plane-polarized.
e is always in the same
direction and in the xy plane.
So this is the answer.
See, notice.
In the past when we
were doing problems,
we focus more on
things like what
is the wave equation
for this string?
Or for a pipe with a gas in it?
Or a transmission
line, et cetera.
Here, even guessing what
solutions we're interested in,
what kind of solution,
it's already hard or even
to describe the wave
we're interested in.
So this, for the
other situations,
this would have
taken a few minutes.
Here it needs a fair
amount of analysis.
And it takes much longer.
Let me take one more case.
The last case I'm going to
exhibit is the following.
Again the issue will be,
there's this particular wave
we want to produce.
We know what we
want, and we want
to know how to describe
it mathematically.
So once again, we want
to find a solution
of our Maxwell's
equations, which
have the following property
that correspond to a circularly
polarized electromagnetic
wave which
is propagating in y direction.
And it just says "any." so
any, any circularly polarized
electromagnetic wave which
is propagating in the minus y
direction.
First of all, what we mean
by circularly polarized wave?
A circularly polarized
wave is that, if I
took a snapshot, if I could,
at a given instant of time,
one would find that the electric
vector along the propagation
direction is rotating
like this on the spiral.
If that wave is
moving towards you,
what you would see in any plane,
a rotating electric field.
And associated with a magnetic
field at right angles to it.
It doesn't tell
us whether we want
a left-handed or a
right-handed rotated field.
So just arbitrarily take one.
And by the way, if ever
you're interested in the left-
and right-handed and
figuring out which is which?
It's a mess.
Different communities use
different definitions,
what they mean by
right- and left-handed.
So I won't try to confuse
you more than that.
So here we want any
wave, which corresponds
to circular polarization,
and is moving in the minus y
direction.
So if it's moving in plus
or minus y direction,
we know that the electric
field will be in the xz plane
at every instant of time.
If it's circularly
polarized, we know
that the magnitude of the
electric field at all locations
of x, y, and z at all
times will be the same.
It does not change.
It's a constant.
Now so how do we
create such a thing?
Well, if we stop and
think for a second,
if we superimpose two
solutions-- suppose
we have one solution, which is a
plane-polarized electromagnetic
wave going towards
you, and I superimpose
on that another one which is
out of phase with it and at 90
degrees, then at every
location in space,
I'll have two components.
If I make those components
change, but in such a way
that the vector
addition of the two
gives me a unit vector,
a constant vector,
I will have achieved
what I wanted to do.
So here is a equation which
satisfies everything I've said.
Let's consider an
electromagnetic wave
which is the same in all
x and all z positions.
The only variable is
in the y direction.
If I write that as
the superposition
of an electric field which
is in the x direction,
and propagating as a sine--
it's a sinusoidal wave--
and I add to it a cosine,
which is at right angles.
Furthermore I'll use
the other information.
It's going in the
minus y direction.
So I'll make these two
opposite sign-- sorry,
I make them the same sign,
it is in minus-y direction.
If it was in the plus-y, they
would have opposite signs.
If it's minus-y, this
would have to be the same.
So this is a
sinusoidal wave moving
in the minus-y direction.
It'll have the wave number
k, this is 2 pi over lambda.
And this is 2 pi, the frequency
or 2 pi over the period.
Omega over k has to be c, the
speed of electromagnetic waves.
If I add to this, the
resultant electric vector
everywhere in space
has a magnitude E-zero.
I can check it.
The magnitude of E
is the square root
of the x component
of this squared
plus the z component
of this squared.
So it's E-zero, the
x component squared--
the sine squared of this.
The z component is the cosine,
so the squared is that.
For all values of x,
y, and z at all times,
if I add these and take
the square root, I get 1.
And so this is E-zero.
So this propagating wave
does satisfy my requirement
that everywhere is
magnitude E-zero.
It is a propagating wave.
Each one of these
are propagating
with the speed of light
in the direction of y.
I'm sorry, forgive me, can't
copy from one line to the next.
This is plus, this is plus.
All right.
It's moving in the
minus-y direction.
The way I had it, it was
going in the plus-y direction.
I corrected it.
This is in the
minus-y direction.
All right?
And this is what was required.
OK.
So this mathematical description
of the electric vector,
how it's propagating.
And now we want to know what
the magnetic one is doing.
Well, again, we could
go back and make sure
that Maxwell's equations
are completely satisfied.
And you'll find that here,
in order for Faraday's law
to hold, I have to have also
a changing magnetic field.
But instead of doing
that, I'll make
use of what we learned
by the previous examples.
We know that this
is a superposition
of two progressive waves.
Each one of these is a
solution of Maxwell's wave.
I don't need both of them.
I only needed both to get a
circularly polarized wave.
Each one of these has to
satisfy Maxwell's equation.
So associated with each
of these components,
I must have a magnetic field
which satisfies the requirement
that there is an electric
vector and magnetic vector
at right angle to each
other moving together
in the direction of propagation
in phase and in time.
So for each one of
these, I will find
the corresponding
magnetic field,
the magnitude will
be E-zero over c,
because we know that the
ratio of the electric field
to the magnetic field is
always equal to c in vacuum.
It's at right angles.
This was in the x direction,
this is in the z direction.
And in this case,
then add this one.
Here, this was plus-z
and this is minus-x.
And you can draw
yourself a little picture
to make sure you get
everything right.
Let me just talk
about, say, this one.
The second component.
What I have in the
[INAUDIBLE], this
is moving there in minus-y.
This component is
in the z direction,
so it's over here,
coming out of the board.
If it's in this direction,
moving down here,
then the b must be
in that direction.
So it must be in
this direction, which
is minus-x, which is correct.
So this is how I get this right.
If I add these, I get
the total magnetic field.
This, now, describes
one possible wave
which satisfies
this requirement.
It's a circularly polarized
electromagnetic wave
propagating in the
minus-y direction.
OK, so let me stop at these
examples of progressive waves,
and I'll move over
to standing waves.
So let's continue in
a second, thank you.
So I've now erased
the board, and I
can continue talking
about wave solutions
to Maxwell's equations.
But let's recap for a second.
What we find is the following,
that basically in vacuum
at every location in space it's
as if there was an oscillator.
It can be displaced
from equilibrium.
It can be made to oscillate.
Displacement from
equilibrium means
there is an electric
field there,
or there is a
magnetic field there.
These can oscillate.
They don't have to oscillate.
So for example, you could
have a static field,
just an electric field constant
in time everywhere in space.
That means every location space
is displaced from equilibrium.
There could be a constant
magnetic field instead,
or both constant.
Imagine how complicated this is.
At every location the
direction of this displacement
from equilibrium for the
electric and magnetic fields,
they are vectors.
There are possibility of
the electric field facing
a different directions
of the magnetic field.
What we find is that
whatever that combination is
in space and time,
that combination
has to satisfy
Maxwell's equations.
That completely describes
what happens in vacuum
at every point in
space and time.
Now there are in
particular combinations
of these displacements of
oscillations in space and time,
which satisfy the wave equation
for the electric and magnetic
fields.
It's a tiny subset of
total, but there are such.
And we are considering now
for that tiny subset what kind
of solutions exist,
how to describe them.
And even there, we're
limiting ourselves
to a tiny subset
of a tiny subset.
So far, I took the subset
where this displacement
from equilibrium of the
electric and magnetic fields
is a progressive wave.
And what we found,
in order to make sure
that the Maxwell's
equations are satisfied,
you can't have any
old electric field
wave, or any old
magnetic field wave.
There's an interplay.
There is, in reality, just
one electromagnetic field,
and that propagates.
We'll now go and look for other
solutions of these equations.
And very interesting
solutions are standing waves.
So let me take a concrete
example and discuss it.
So here is, you could
call it a problem.
Suppose that I have everywhere
in space an electric field
which consists of
a standing wave.
You can recognize this
when we were talking
about standing waves on
strings, for example.
Where you have the electric
field always pointing
in the x direction.
It's oscillating at
every point in space
with the same frequency
and phase, cosine omega t.
It's oscillating with
that angular frequency.
And spatially, it not change
in the x and y direction,
but it does in the z direction.
And that is a cosine like this.
So this is a standing
wave of electric field.
This by itself
cannot be a solution.
Is not a situation you
can have in vacuum.
It violates, by itself,
Maxwell's equation.
If you look at them, you
find that in order for this
to satisfy Maxwell's
equation, the
must be associated with
it a magnetic field that
looks like that.
And so the question,
the first thing is,
show that if you have this, you
must also have this present.
The second part is
some more discussion
about when you have
these two present, when
you have a standing
wave in vacuum
of electromagnetic
waves, for example,
then what is the energy density?
You know, in an electric
field or a magnetic field,
if you have in
space, if you take
any value inside the volume,
there will be energy.
And the energy per unit
volume per cubic meter
is the energy density.
So we're going to calculate
how much energy density there
is in this standing wave.
And another quantity, which
is for practical reasons
very important, is when you have
an electric and magnetic fields
present, actually energy
flows through that system.
And the amount of
energy per unit area
that flows-- per unit
area perpendicular
to the direction of flow-- is
called the Poynting vector.
And by the way, the
Poynting has nothing
to do with a vector that points,
it's to do with a gentleman
by the name of Poynting,
after which this was called.
So the second part
of the problem
is, once we found a standing
wave that satisfies everything
possible [INAUDIBLE] in vacuum,
for this particular case
what is the energy density, the
magnetic and electric fields,
and what's the Poynting vector?
OK, so how do we do this?
We know what the
electric field is doing,
it's the standing wave.
We know that it must satisfy
all Maxwell's equations,
in particular Faraday's law.
As before, we can calculate
the curl of the electric field.
Now here, the electric field
is only in the x direction.
And it's a function of z.
And so the curl of this, to
be only just one component
of that, and that is
given by this quantity.
So this is minus
the curl of this E.
And we know by
Faraday's law that this
must equal to the rate of
change of the magnetic field
at that place of x, y, and z.
Now I can integrate
this equation,
and find what B is at every
point in space and every time.
And that's easy enough.
We just have to integrate that,
which gives you the sine here,
and the omega comes
down, and you get this.
Whenever you integrate,
there is a constant.
All it's telling us is that I
can satisfy Maxwell's equations
not only with an
oscillating electric field
present with an
oscillating magnetic field,
but I can always add a
constant magnetic field
throughout space.
I could have also added a
constant electric field.
So there's an infinite number
of solutions I can superimpose.
I'm not interested in them.
I am interested in the standing
wave, the time-dependent part.
So might as well make that 0.
And so we are essentially home.
We have found that the magnetic
field is also a standing wave.
And this, by the way,
we look at the problem,
is what we were asked to prove.
So we have proven
the first part,
that if this is the
description of the standing
wave of the electric field,
then there must be corresponding
a standing wave magnetic field.
So the two-- but notice,
unlike in the case
of progressive waves, where in
the progressive waves, wherever
you had an electric
field, the magnetic field
was at right angle to it and
in magnitude proportional
to the electric field and
in phase with it, et cetera.
Here, they're not.
Here, the electric field,
when this is cosine omega t,
this is sine omega t.
When this is cosine
kz, this is sine kz.
These two are out of
phase with each other,
both in time and in space.
I've tried to sketch it
here, it's not very good
sketch, but anyway.
Suppose at some instant of
time, if I look at these,
at some instant of time,
the electric vector--
the magnitude of it-- is
represented by this curve.
And it is in the x direction.
So the electric vector is
this, like this, and like that.
If this is the maximum, it is,
the magnetic field at that time
will be 0, if I look
at these equations.
So there'll be no
magnetic field.
Over this distance
in space, there
will be the electric
field up here, down here,
and no magnetic field.
Later on, half a
period later, what
you find is that when this
comes to 0-- it's a quarter
period-- when this comes to
0, the electric field is 0,
there will be a magnetic
field at its maximum.
But it will not be this shape.
It will be, first of all,
pointing in the y direction.
This is in the x, it
will be the y direction.
It's maximum will
be in the middle,
well here it was always 0.
And these two oscillate.
It's a standing wave.
The B does this, and the E does
this, all in the same place.
But both in space and time,
the two are out of phase
with each other.
Completely different solution.
And both progressive waves
satisfy Maxwell's equations,
and the standard waves.
So it's important to realize
there is this difference,
often it's easy to
get confused about it.
In a progressive wave, the
electric and magnetic fields
are right angle.
And as if they were
locked together,
and they move forward like this.
On the other hand, in
a standing situation,
they're still at right
angle to the other.
But when one is a maximum,
the other's a minimum.
When this one is--
They're out of phase
with each other in
both space and time.
So that's the first part.
And the next part we were asked,
now for this standing wave,
imagine this could be
inside your microwave oven.
Inside the microwave oven,
there is a standing wave.
Unless they specially make it so
it moves a little bit in space
so you cook your
meat everywhere.
But then the cheapo
microwave oven,
you have a stationary
standing wave.
And suppose this is it.
At every place in space,
there is an energy density
which actually fluctuates,
goes up and down in time
and is different
in every location.
Let's calculate that.
Well, as Professor
Walter Lewin showed,
the energy density
in an electric field,
whether it's changing
with time or not,
if I've got in space
somewhere an electric field e,
at that location, I
have an energy density.
The amount is 1 over
epsilon-zero times
the magnitude of the
electric field squared.
That is the energy density
of an electric field.
It is not a vector.
This is E-squared, the
square of the magnitude
of the electric field
energy is a scalar quantity.
So not surprising, this is not a
vector, it's a scalar quantity.
I can now immediately
go over to what we know.
We know the electric field,
we know the magnetic field.
So I can replace E-squared by
what it is at every location.
At every position
z and every x, y.
At all times.
And this is the energy density.
You can see it does oscillate,
but there's always [INAUDIBLE].
How about the magnetic field?
The magnetic field
also has energy.
If I take it anywhere,
suppose you have a bar magnet,
one of these pocket
magnets, you hold it,
and there's a magnetic
field all around the magnet.
Take any cubic
meter of the volume,
you'll find this
amount of energy.
It's 1 over 2 Mu 0 times the
magnitude of the magnetic field
squared.
Again, I know what B is
in for my standing, wave
so I can calculate it,
and I get this answer.
So these are the two
energy densities.
Now what one finds, if one
does-- if you plot this,
or thinks about it-- that
in this standing wave,
you find that that energy
moves backwards and forwards.
At any location in
space, I can calculate
how much energy is
moving per second
per square meter-- per
unit area-- perpendicular
to the direction of
motion of that energy.
And that is what is called
the Poynting vector.
If you think, for
example, suppose
you take an electromagnetic
wave like light shining
that the wall.
It'll warm up to
the wall, I mean
there's heat being transmitted,
there's energy comes over.
At any instant of time, how
much energy per unit area
is hitting the wall?
It will be equal to the Poynting
vector at that instant of time.
And the Poynting vector
s is E-cross-B over Mu 0.
By the way, this applies to any
electric and magnetic fields,
not necessarily for
progressive waves or standing
waves, et cetera.
It's something we
want to think about
and this is very surprising.
Even if you have static electric
and magnetic fields which
are not parallel to each
other, so that this is not 0,
there is a flow of energy.
It's something we
want to think about.
But in our case, E and B are
perpendicular to each other.
The electric field
everywhere was
in the x direction, the
magnetic in the y direction.
And so they are right angles, so
it's just the x component of E
and the y component
of B. Well, they're
the only components
that are there.
So it's 1 over Mu 0.
E x times B y in
the z direction,
so this if E and B are
perpendicular to each other,
z is perpendicular
to both of those,
which is in the z direction.
If I calculate this for
this, I get this equation.
And I can rewrite it.
And I find that the
energy is some constant,
goes in the z direction, and
this looks like sine 2 omega t
times sine 2 kz.
Going back to our
diagram, what this
looks like is that-- if you
remember that E oscillates,
it's a maximum
here, maximum here,
and it oscillates up and
down, up and down, like this.
B is a maximum in the middle,
and that's going like this.
The product of the
two, it'll be 0 here,
because B is always 0 here.
It'll be here because
E is always 0.
And you cross B there for 0.
So here, here, and
here is going to be 0.
And if you look
at that function,
its actually a function
which has twice the frequency
of the electric
field oscillations
or the magnetic
field oscillations,
and also half the wavelength.
And you will find that the
maximum is somewhere here.
So if you look at where the
maximum transfer of energy is,
it's at the quarter
and 3/4 location.
And so it's consistent
with this picture.
Energy is doing this
in that situation.
And so that answers
what they were ask.
This is the Poynting
vector as a function
of-- for all positions in
space as a function of time.
This is the energy, the
electric and magnetic field,
and we found the magnetic
field corresponding
to the electric field.
So this is another example
of a possible solution
to Maxwell's equations,
this time corresponding
to standing waves.
As I mentioned before,
I'm repeating myself ,
there are infinite
possibilities of solutions
of Maxwell's equations.
So to cover them
all makes no sense.
What is important, that one
gets a good understanding
of the interesting situations.
Interesting situations are
some static solutions to, say,
magnetic fields if you
need special magnets.
Or if you have a
progressive wave,
like light, or standing
waves, like in the microwave,
for example.
And so I've taken
two cases here.
First progressive wave solution.
And then standing wave solution.
And from this, we will later
go on to some other problems.
Thank you.
