The following content is
provided under a Creative Commons license. Your support will help
MIT OpenCourseWare continue to offer high quality
educational resources for free. To make a donation or to
view additional materials from hundreds of MIT courses,
visit MIT OpenCourseWare at ocw.mit.edu. YEN-JIE LEE: Welcome
back, everybody to 8.03. So today we are going
to continue discussions on the examples which we started
the last time, sound waves, and, this time EM
waves, which can be described by wave equations. So far, what we have
learned is that there are three different kinds
of systems we've discussed in a lecture or in a textbook. And the first one is actually
a string, a very long string, system with constant tension
and mass on the string. And the behavior of
the string obey wave equation, and can be
described by a wave equation. We also can produce a
density wave with a spring. And basically the
density wave or spring can also be described
by wave equations. So that's as you
described in the textbook. Finally, last time we actually
discussed sound waves. We have an open
pipe, and then we can have air inside the pipe. And the behavior of the
air, or the molecules inside the open pipe, can be
described by wave equations. Crashes So what we
are going to do today is to discuss with you
a special kind of wave, which is electromagnetic waves. And that's actually
slightly different from what we have learned in
the last few lectures. And we see what this is
different today in the lecture. All right. So this essentially
is a reminder of Maxwell's equations. So basically what
is written here is the differential form
of Maxwell's equations. So the first law is Gauss law. It says should the divergence
of e, the electric field, is equal to rho,
which is the charge density as a specific point,
divided by epsilon zero, which is actually a constant. We'll call it permittivity
of this constant. OK? Which should relay
the divergence of e and the density of the charge
at this specific point. And the second law is actually
Gauss law for magnetism. This is actually the
divergence of b equal to 0. So divergence b is always equal
to zero because we haven't yet discovered the
magnetic monopole yet. Right? So maybe you have discovered
it one time, at some time, in your experiment. Please tell me now. I want to be the first with
who knows how to do that. [LAUGHS] All right? So promise me. The third one is Faraday's law. It's curve of e equal to minus
partial e partial t and the b, as a reminder, is a
magnetic field vector. And in the last law is
actually Ampere's law. It's actually the curve
of b equal to mu 0. Mu 0 is actually a
constant, permeability. Which would lay the current
and displacement current. Epsilon 0, partial e,
partial t, to the curve of b. OK? And I would like to draw
your attention to these term. This very important term is
actually Maxwell's addition. OK? Without Maxwell's
addition, there would be no
electromagnetic wave. Then you could not see me. OK? [LAUGHS] All right. So, what we are going
to discuss today is a simpler case
at the beginning. So what will happen
if we go to a vacuum? Going to vacuum means there will
be no material charges floating around, and that means
rho will be equal to 0. Therefore, the divergence
of e will be equal to 0. And also in the last
question Ampere's law, say which is that the current
density will be equal to 0. Therefore, the
function of curl of b equal to mu 0, epsilon
0, partial e, partial t. OK? So before we go
into the discussion of Maxwell's
equation's implication, I would like to remind you
about some mathematics which will be used in this lecture. I hope you have seen this
in other courses or 8.02. So as you can see, we use
del here, which is a vector. This vector is defined
as partial x, partial y, and partial z, in the
x, y, and z direction. OK? So this is actually
some kind of operator. You see that again. A lot more operators in 8.04. And we make this definition
because I'm lazy. Because I don't want to write
so many partial, partial x, partial, partial y, partial,
partial z again and again. Therefore we define
del, which is like this. Looks really crazy,
but it really makes our lives much easier. OK? So that's the whole reason. As a physicist. And as we discussed
before, we have divergence, which is defined here: del times
a, both of them are vectors. Y is the operator
vector, the other y is actually really a vector. You basically get partial ax,
partial x, plus partial ay, partial y, plus
partial az, partial z. So basically you just multiply
them like a normal operation. And you can actually
get this question. OK. Then finally there's curl. Curl is actually del cross a. So basically, maybe
in the past you see this complicated formula. You know, it had maybe
no meaning to you. And one easy way to
remember this curl is to not care about this. Don't look at the
right hand side part. But just remember that you can
actually construct this curl by determining a matrix. In the matrix, I can
fill the first row by x, y, and z unit vector. And the second row, I filled
it with the counting of del. Finally I feel the content
of the matrix with a vector. Then you will be
able to calculate the determine of
this back matrix, then you naturally would
get this very long formula. So you don't really need
to remember the formula, but you will be able to know how
to calculate it really easily. OK? OK. So we talk about divergence. We talk about curl. What does that mean? Divergence, curl,
what does that mean? So divergence is actually
some kind of measure which measures how
much the vector v spreads out, or diverges,
from a point of interest. So in this example,
this vector field-- vector field means at any
point in the space which I am discussing, there is a
vector associated with that. I call it vector field. We know scalar field very much. For example, the temperature
as a function of position is a scalar field, right? So, every point you have scalar
corresponding to that point. And in the case vector
field, every point you have a vector
connected to that point. And if I arrange the
vector field like that, each arrow is actually
a straight dimensional the vector. Then if I evaluate
the divergence and you see the heart, it
looks like something is really spreading out from the
center of that graph. And that will give you
positive divergence. OK? So that's the physical
meaning of this formula. And the second formula which
we discussed is the curl. So curl is actually del cross a. It's a measurement of
how things are curling around a point of interest. OK? So you can see that if
I arrange my vectors in a space like
that, then you will see that something
is really rotating around that specific point. Therefore, if we
evaluate the curl, you would get the nonzero value. So that's actually the
physics intuition which we can or the mathematics
intuition which we can actually get before
the discussion of Maxwell's equations. So if you accept
those ideas, let's take a look at what we
have here, especially in the vacuum case. OK, so in a vacuum case, you
have a curl of e equal to minus partial p, partial t. What does that mean? That means, if you
change the magnitude of the magnetic field,
now we introduce a curling around thing in the e field. So if you change the
size of the b field, then the e field will start
to curl around, doing this. And on the other hand, if you
change the electric field, that will do something, which is
curling around in the b field. All right, so do you
have any questions? I hope everybody's familiar
with this notation. So from here actually
Maxwell, see the light. [LAUGHS] Can you see it? Maybe not yet. Maybe we are slightly
slower than Maxwell, but we will see that
together in this lecture. For that I would need as usual
help from the math department. So we are going to
use this identity. This identity is curl
of curl of a would be equal to del, divergence
of a, minus del dot del, a. So this is an identity
which we learned from the math department. And of course, if you
are patient enough, you can actually
expand all those terms and compare the left hand side
of the formula and right hand side of the formula. And you will see that
really this works. So I'm not going to do
that here in front of you. So if you accept this is an
identity, and then usually when we have del times del,
we call it Laplace. And usually we write
it as del squared. With this formula, I can
now put my electric field into this formula. assuming that I am working
in a situation of a vacuum and I plug in my electric
field into that formula. Then this is actually
what I am going to get. Curl of e. And this will be equal
to del, divergence of e minus del squared e. OK? And based on the four
Maxwell's equations, we can immediately recognize
that divergence of e is equal to 0, because I
don't have charges around. Therefore you, cannot
not introduce a gradient or divergence. You can introduce
positive divergence in the electric field. Therefore, when you
evaluate the divergence of the electric field,
that is equal to 0. According to that
formula, Gauss law. And you can also
take a look here. We have curl of e,
according to this formula. Basically you can
conclude that this will be equal to minus
partial b, partial t according to Faraday's law. So if I look at
the left hand side, that would be equal to the curl
of minus partial b, partial t. And this will be
equal to basically, I can take the minus sign out
and take the partial partial t out. And basically you
have curl of b. And according to
Ampere's law, this would be equal to minus mu 0,
epsilon 0, partial square e, partial t. OK, everybody is following? So basically what
I have been doing is copy the left-hand side and
make use of the Ampere's law. And basically you get minus mu,
epsilon zero, partial square e, partial t squared And this thing,
the left hand side, is equal to the right hand side. On the right hand
side, what is left? This is equal to 0. So this is gone. This is equal to
minus del squared, e. I can cancel the minus sign. Then basically what I am
going to get is del squared e. And this will be equal to mu
zero epsilon 0, partial square e, partial t squared. Wow, this is what? This looks like, what? Wave equation again. Oh my god. [LAUGHTER] But there is some difference. This is different from what
we've seen before, right? Before, the wave equation only
has partial squared partial x squared. This time, you have
this del square. Very strange, right? So what is this? Del square is actually
partial square, partial x squared cross partial
square partial y square, plus partial square, partial z
square is the operator, which will have three components. And basically, if you
do this calculation, you are going to
have how many times? If you do this del square
e, how many times you have? You have how many? Any anybody help me? Yeah, you have three
times in x direction, you have three times
in y direction, you have three times
in z direction. Therefore how many times? You have nine times,
because each operator is acting out the vector. OK. So it's very important because
this is a common mistake. So you have nine times. and it looks really
like the wave equations it tastes like wave
equation, it looks like equation, it feels like
equation - that wave equation - and therefore is really
the wave equation, right? [LAUGHTER] OK so this is a
three-dimensional wave equation. Very cool. So we are increasing
the dimension. So I can write it
down more explicitly. So basically what I'm
getting is partial square e partial x squared, partial
square e, partial y square, plus partial square
e, partial z square. And this is equal to mu zero,
epsilon zero, partial square e, partial t squared. OK? So Maxwell sees this when he
adds this additional term here. As you can see, if I don't
have this additional term, the displacement of
current from Maxwell, what is going to happen? This curve of b will
be equal to zero. So what is going to
happen to this identity? This left hand side part
will be equal to zero. There will be no
electromagnetic waves. OK? So that's really thanks
to Maxwell's work. And this is actually
really an equation which changed the world,
because that actually gave us a lot of insights about
how we can send energy, how we can actually understand
the phenomenon related to light. So what is the velocity
of this wave equation? The velocity, Vp, would be
equal to what we usually call c. Because you have been using
this constant for a long time. And that will be equal to 1 over
square root of mu zero epsilon zero. And to measure the
speed of light, it takes a long time
to achieve that. Let's take a look
at the history. So the first attempt was
done by Galileo so 1638. He was doing an experiment,
and that he was trying to track the speed of light. But he was not super successful. So his conclusion was
that if the speed of light is not instantaneous,
then it is super fast. He says it's at least 10 times
faster than the speed of sound. He said OK, this is
super awesome, very fast. OK. So that's what he found. And later Romer actually made
use of the orbit of Jupiter. Basically he use Jupiter
and Jupiter's satellite to measure the speed of light. So when the earth is
closer to Jupiter, then somehow the
satellite of Jupiter appears faster than when
the earth is actually away from Jupiter. Because that light have to
travel through additional time. Two times the radius
of the orbit the Earth. Basically that's the
math that he was using. He is actually making the
first computative measurement of the speed of light. And what number he
found is 2 times 10 to the 9 meters per second. Then finally, again
using the star the observation as a tool
to actually calculate the speed of light. James actually nailed it. He found a value
which is really close to the current understanding
of the speed of light, which is 3 times 10 to
9 meters per second. Therefore, if you calculate that
using all those constants here, you will be able to see
that, indeed, from Maxwell's equation, you get-- oh, it should be 3 times
10 to the 8, not to the 9. I was saying 10 to 9. It should be 3 times 10 to
the 8 meters per second. So indeed, this
equation is actually predicting the speed of light
to be 3 times 10 to the 8 is matching the
experimental result. So that is pretty nice. And you may ask a question-- so wait a second. You said this is actually an
electromagnetic wave, right? So that's actually what
I was talking about. But this equation only
talks about electric fields. What is happening to
the magnetic field? What happened? Can we actually choose
arbitrary magnetic fields? Is a magnetic field
also described by the three dimensional
wave equation, right? The answer is that
indeed you can actually do the same exercise. You can now instead of
plugging in electric field, you can plug in
a magnetic field. And you will extract
exactly the same conclusion. You will conclude
the del square B, will B equal to Mu 0,
epsilon 0, partial square, B partial to square. OK, so it is actually
very important to see that the magnetic field
also obey this wave equation. OK? And also from
Maxwell's equation, you can see that the
changing electric field will produce a curling
around a magnetic field. The same thing
also happens here. A changing magnetic field
also produce a curling around electric field. So what does that mean? That means E, electric
field create magnetic field. Magnetic field create
electric field. And this happens all the time. Therefore, one cannot
live without the other. They are living together. They are all together, forever. All right, so what is
actually oscillating is actually both electric field
and the magnetic field, right? So you may ask, OK, we
are talking about vacuum. Vacuum means there is
no material, no charge, no whatsoever in vacuum. So what is actually oscillating? Who is oscillating? Is the electric field
and the magnetic field. This so-called field,
all those vectors-- which are actually oscillating--
it's not the material, but all those vectors
associated with the space, which is actually
oscillating up and down. All right, so
originally I would like to show you a pulse of
light in front of you. And show that it's
moving, but it's too fast. So I couldn't do that. [LAUGHTER] Fortunately, we have photos. Photos are actually collected
the recorded photons. Emitted from the
object of the interest. So this is actually how
we make applesauce at MIT. We shoot-- bullet through the
apple then we have the sauce. [LAUGHTER] But not sure if that's
tasty enough or not. But that's how we do it in MIT-- MIT style. And the good thing is that
this kind of technique is improved dramatically
in these days. I would like to show you a
short video, which is actually recording a video of-- it's recording experiment, which
you shoot some beam of light through some plastic container. And the speed of this
recording corresponds to one trillion
frame per second. So this is super fast recording. And they can actually
reconstruct the propagation of light through this bottle. The credit is actually to the
Media Lab Camera Culture group. And let's take a
look at the video. Just one second. OK, so this is actually
recording at one trillion frame per second. So you can see that
there's a light pulse-- a very short pulse created. And is really pass
through the bottle. And it can be recorded with the
technique created by Media Lab. So you can see that the pulse is
really propagating through it. And the reason why
we can see the pulse is because there
are air, there are material which will
actually change the direction of the light. And therefore, those are
recorded by the camera. And they take trillions
of frames of this thing, and put them together. Then basically-- and they
take many, many frames, and they put them together
to reconstruct this movie. So as you can see
that indeed you can see the propagation
of the light through this kind of video. So I hope that we
enjoyed this video. And let's actually take a
look at some concrete example which make use of the
wave equation, which we did right here. So let's consider a
plane wave solution. Things we are entering a
three dimensional world. So that's actually consider
so-called a plane wave. So in this example, I am
considering the electric field that's actually equal to the
real part of E0 exponential i, kz minus omega t. And this electric field
I actually consider here is in the x direction. And if I write all the
terms from this expression expressively, that's actually
what I'm getting is-- x component will be E0, cosine
kz minus omega t, 0 and 0. So what does this mean? What is actually a plane wave? The plane wave
basically is actually fielding the whole space. What I mean by plane wave
is I feel the whole space with electric field. This electric field
only have one-- only one direction have
non-zero value, which is x direction in this example. And then the other
direction, there's no-- the magnitude is
actually equal to 0. So that's actually what
I mean by plane wave. And also the electric field
is filling a whole space in the discussion-- in the
example which I discussed here. And if I define my
coordinate system like this, x is in the
horizontal direction. Then that means
everything is actually-- all the electric
field is actually pointing toward the x direction
in this coordinate system. So we have discussed progressing
wave in the past few lectures. Can somebody actually tell me
the direction of propagation of this plane wave? So the hint is that this
is actually equal to E0, that the magnitude
of the x component is equal to E0, cosine
kz minus omega t. What is actually the
direction of propagation of this electric field? AUDIENCE: z. YEN-JIE LEE: It's
in the z direction. Yeah, very good. Because we know that
this is actually going in the
positive z direction. Because this is
actually kz minus-- there's a minus sign-- omega t. So therefore it's going toward
the positive z direction. Not x direction. x direction
is where the electric field is pointing to. And the direction of propagation
is toward the z direction. So there's a difference. So first thing which I would
like to do is to check if this so-called plane wave solution
actually satisfy the equation-- the wave equation
which we derive here. Del square E equal to Mu 0
epsilon 0, partial square E, partial t square. So I can now plug that
in to that equation. I can now plug in
to this equation. If I plug in the wave-- the
plane wave solution, which I have here to that
equation-- basically, I can get the left-hand side. The left-hand side
of the equation, you will get minus E0. Only one term
which contribute is the partial square E partial z
square term which contribute. Right? Because the magnitude
of the electric field only depends on z and t. Therefore, you get minus E0, k
square, cosine, kz minus omega t in the left-hand side
of the wave equation. How about the right-hand side? Right-hand side actually you
are taking partial derivative, which is fed to t. Basically, you get minus Mu 0-- Mu 0, epsilon 0-- I copied from that
formula there. And you basically
get omega square out of it because of the
partial square, partial t square operator. And then you basically get
cosine kz minus omega t. And of course I
missed the E0 term. E0 should be copied
from-- on there. So now I can show that-- OK, this cancel. Basically, this is the same
cosine kz minus omega t. And E0 also cancel. And I can cancel the minus sign. What I'm going to
get is k square is equal to Mu 0,
epsilon 0, omega square. So that means there should
be a fixed relation between k and omega, which is
actually omega over k will be equal to 1 over
square to the Mu 0, epsilon 0. And this is equal to c. If this is satisfied,
then the plane wave is a solution to
the wave equation-- only when this is
actually satisfied. Otherwise we can write
arbitrary plane wave equation, but they are not the
solution of that equation from Maxwell's equation. So now, I have derived
the electric field and also know the relation
between omega, the angle frequency, and the
k, the wave number. And now, what about
magnetic field? So I just mentioned before,
magnetic field cannot live without electric field. And electric field cannot
live without magnetic field. So what is actually
responding magnetic field? We can actually evaluate that. So now, the question is what
is actually the magnetic field? And how is that vary
as a function of time and as a function of
position in the space? So we are facing a choice. So there are two
equations, which relate electric field
and the magnetic field. It is actually very important
you make the right choice when you start your calculation. So we can use Faraday's law. We can also use Ampere's law. But there's only one, which
is actually much easier to derive a solution, which is
the choice of Faraday's law. If you choose to use
Ampere's law to evaluate B, then you are going
to get a really super complicated
problem to solve. But on the other
hand, if you choose to use Faraday's law
to solve this problem, then you can see that the
unknown is the magnetic field-- the field which I
would like to evaluate. And the expression for the
B is actually rather simple. It's actually just a partial
derivative, partial B, partial t. So it's pretty simple
and you can actually evaluate the known part. This curl looks
pretty complicated. So you can actually evaluate
that because you know what is the electric field. On the other hand, if
you will use Ampere's Law then you will be in
trouble because you don't know what is a B, xBy and Bz. And you have to evaluate curl. And you get a lot of
terms, and that is actually equal to something
from-- the information from the electric field. And that would be very
difficult to evaluate. So therefore, what we are going
to do is to use Faraday's law, curl of E will be equal to
minus partial B, partial t. So basically, as I
mentioned in the beginning, we can make use of the equations
the determinant of matrix to evaluate the curl. So therefore, I am
going to use that. And then what I'm going to get
is x, y, z unit vector for fill the first row. And the partial partial
x, partial partial y, partial partial z, which fill
the second row of the matrix. Then I get Ex, 0, 0 because
the electric field is only in the x direction. And this will be equal to-- only two terms survive
because of these two 0's. So all other terms are killed,
and only two terms are now 0. The first term is
actually partial Ex, partial z in the y direction. And the second term is
actually minus partial Ex, partial y in the z direction. Any questions? Am I going too fast? All right, so you can see
that the electric field only depends on the position z. It's independent of y. Therefore, partial Ex, partial
y, is actually equal to 0. Wow, this become
much, much easier because there's only one
term which is surviving. This is a operator. Then basically what
we're going to get is I can now calculate
partial Ex, partial z based on that equation, E0
cosine kz minus omega t. Then basically, what
I can get is minus-- I get a K out of it, E0
sine kz, and it's omega t. So this is actually the
result of the left-hand side. The right-hand side of that
equation of the Faraday's law is minus partial B, partial t. So this will give you equal
to minus partial B, partial t. So very important-- I don't
want to drop the y direction. So this is just y direction. And this is actually a vector
and this is also a vector. So what I could do is to
do a integration over t. And those will cancel
the minus sign. So if I integrate over t, then
basically what I'm going to get is K over omega is 0,
cosine kz minus omega t in the y direction only. So I'm doing a integration
of t, cancel the minus sign, then this is what
you want to get. And of course, k/omega
is actually 1/c. So therefore, you have E0-- you can actually
simplify this fraction-- and this is actually equal to E0
over c, cosine, kz minus omega t in the y direction. OK, look at what we
have learned from here. What we have learned
from here is that-- I got started with a plane wave
solution of the electric field. And I can show that
only when omega over k is equal to the speed of
light this is actually a solution to my wave equation. And also because the electric
field and the magnetic field have to satisfy the Maxwell's
equation all the time-- because that's the
fundamental law-- therefore, I can
use those equations to evaluate and to
find what is actually the corresponding
magnetic field. And using Faraday's
law and plugging in and the solving the
question, I will be able to figure out that
B is also what kind of wave? B is also what kind of
wave I was talking about-- also? AUDIENCE: Plane wave. YEN-JIE LEE: Plane wave, right? It's also plane wave. You see? So if I got started
with a plane wave in the electric field
side, and I also get the plane wave in
the magnetic field side. They are proportional
to each other. Originally, the magnitude
of the electric field is E0. The corresponding
magnetic field-- the magnitude is
proportional to E0. But there's a factor
of 1/c difference between the magnetic
field amplitude and the electric
field amplitude. The third thing which
we learned from here is that electric field is
actually in the x direction. B field is actually
not in the x direction, it's in the y direction. What we learn from here is that
the direction of the B field can be determined by
a simple calculation. So basically, the
B is proportional-- the magnitude of
B is proportional to the electric field. But you have to multiply
the magnitude by 1/c. And also this is
actually not correct because the B is actually
in the y direction. So the original direction
of the electric field is in the x direction. Also we know the
direction of a propagation is in the z direction. Therefore, if I
take unit vector K-- K is actually the wave number,
but now I make it a vector and I take the unit
vector is equal to z-- so direction of propagation. If I make this definition then
I can now rewrite this relation. Basically, I can express
the magnitude of B by K hat, which is the direction
of propagation cross the E field. And we can check this. And then basically what you
are going to get is z cross E-- then actually really
z cross x, you are going to get y direction. And that is actually
telling you that B and the E have a rather simple relation. And also you don't
really need to go through all those calculation
again because now you can see that if you know
the direction of propagation and you know the direction
of the electric field, then you can already evaluate
what will be in the B field. So we will take a
five minute break. We'll come back in 29, and we
will continue the discussion of this solution. Let me know if you have any
questions about the content we discussed. Welcome back, everybody. So we will continue
the discussion of what we have learned
from the wave equation. So basically we start with plane
wave in the electric field. And this electric field
is in the x direction. And we evaluated the
corresponding B field which is in the y direction. And what we found is that
actually we can find a pretty simple relation between electric
field and the magnetic field, which is actually magnetic
field vector is equal to 1/c, K hat cross E. And the K hat
now which is-- you find here-- is actually the
direction of propagation. So basically, in this case in
the discussion we had before, the direction of propagation
is in the positive z direction. So if I go ahead and
visualize the whole-- solution-- plot the magnetic
field and the electric field is a function of z, x and the y-- it's a function of
z actually here. And I only evaluate the
value at x equal to 0, and the y equal to 0. And basically, this is
actually what you have. So basically, you
have two sine wave. One is actually pointing
to the x direction. And the other one is actually
pointing to the y direction, which is the B field. And those lines
doesn't mean a lot because those lines are
just connecting the end point of all those vectors. So you can see that they are
cosine wave structure when you connect all those vectors. And keep in mind that
those are evaluated at x and y equal to 0. Therefore, what we actually get
is actually a lot of vectors. So those individual
arrows are vectors. And this whole thing-- this
whole electromagnetic wave is propagating to the
positive z direction. And those electric field
and the magnetic field are propagating at the speed
of light, which you see. And also you can see
that the magnitude-- also I plotted here-- the magnitude, there's
no phase difference between electric field
and the magnetic field. This is actually
not always the case. In which we will show a example
probably later in the lecture. So in general, what
we can actually do is to write down a general
expression for the plane wave. So for example, I can have a
plane wave, which is actually propagating in some direction. Which is actually
given by this K vector. K vector is actually
giving you information about the wave number. And also the direction
of propagation. And in this case, what
I am trying to construct is a solution, which is
actually propagating along in the direction
of the K vector. And the electric
field is actually going to be pointing to
a direction perpendicular to the direction
of the K vector. So basically, what I can do
is I can write this plane wave in this functional form. E0 is actually a vector,
which is actually telling you the direction
of the electric field-- E0 vector-- is actually
have this function of form. And the K vector
is actually placed in the exponential
function-- inside the exponential function. Exponential i, k
dot r minus omega t. And what is actually r? r is actually x x hat,
plus y y hat, plus z z hat. And omega is actually
the angular frequency which we are familiar
with and that's actually equal to c times
the magnitude of K, which is actually
the wave number. And you can actually show that-- OK, indeed this
expression can satisfy the wave equation, which we did
right for the electric field. And of course there
are some requirements, which is actually that the
direction of the electric field have to be perpendicular to
the direction of propagation. Which you can
actually derive that. And finally, this expression
B field equal to 1/c. K, which is the
direction of propagation cross E field is still
valid because basically we have shown that it works
for the plane wave pointing to the x direction propagating
to the z direction. We can always redefine
the coordinate system because we can actually
rotate this coordinate system and the physics
should not change. Therefore, you must see
that this expression must be still valid. And also that the direction
of the electric field, which is actually
proportional to E0, must be perpendicular to the
direction of propagation. So that is actually
what we can actually learn a general description
of electric field pointing to some random direction. So we have talked about
the progressing wave solution and also the
plane wave and also the corresponding
magnetic field. I hope that you can
actually apply this-- the technique which
we learned here-- if you are given
a magnetic field, you must know that there must be
a corresponding electric field because they cannot be
separated from each other. And you can actually obtain the
corresponding electric field if you are given magnetic field
by using Maxwell's equations. So what is going
to happen is that now if I emit this photon-- or say this electromagnetic
wave from the light source, for example, that one-- the one of which is
pointing at my face. Basically, my face
is going to bounce some of the electromagnetic
field around. And some that actually
go out of the window. And then when they
go out the window, maybe they are lucky
they are not hitting any building in the MIT. Then what is going to
happen is that they're going to propagate forever
toward the end of the universe. Really, they are going
straight forever as you can see from this solution. It's like some kind of
wave propagating forever at the speed of light. If they don't encounter
anything before the end of life of the electromagnetic
wave, it's going to be propagating
forever toward that direction-- escaping from that window. So that is actually
fascinating and-- but we would like to
introduce some more excitement to see what is going to happen. So what I'm going to do is
now instead of only discussing about the plane wave-- what I'm going to
do is that I would like to add a perfect
conductor into the game and see what is going to happen. So what do I mean by
a perfect conductor? A perfect conductor can
be seen in a musical, like in a concert. [LAUGHTER] But the one which
I am talking about is not that one, which
is also fascinating, but this is a different system. The interesting thing is
that both the conductors in the concert and
this one is very busy. It's a very busy system. What do I mean by
perfect conductor? That means all the little
charges inside the conductor can move freely. So if they move they don't
actually cause any energy. They can move around-- all the electrons
inside the conductor can be moved freely without
costing anything, without any of this energy dissipation. So that's actually what I
mean by perfect conductor. What do I mean by
a very busy system? That means whenever
there are any distortion on the electric field-- any electric field approaching
to this conductor-- what is going to happen is
that this conductor will, oh, this is electric
field, so I have to move from some
of my electrons. Then it's going to cancel
all the electric field inside the conductor
because it cost nothing. So you have fast-- really fast the react to this
change in the electric field and they really carefully
arrange all the electrons. And so that the electric
field is canceled. Otherwise, all those electrons
will continue to move around until this happens-- this cancellation happens. So that's actually what
I mean by a busy world and what I mean by
a perfect conductor. If I put this conductor into
game, what is going to happen? What is going to happen is that
if I consider a situation-- if I have my x's defined here
pointing up to be the x-axis, pointing to the right to the
z-axis, pointing to the-- pointing toward you is
actually the y-axis. So I can now again
take the plane wave which I started with. There will be a
plane wave like this. And it's going toward a
piece of perfect conductor. What is going to happen
is that as I actually mentioned before there are many
charges all over the place. They are going to
quickly rearrange-- all those charges to
cancel the electric field. So if you have a
plane wave going toward the perfect
conductor at the surface of the perfect conductor-- the electric field
will become 0. But if you have only one
plane wave it cannot be-- the magnitude cannot be equal to
0 because I know the functional form. I know that the functional
form of that electric field is E0 cosine kz minus omega t. If I place this perfect
conductor at Z equal to 0, then I can evaluate
the electric field is not equal to 0 because
it is actually equal to E0 cosine minus omega t. So what can I do to
cancel the electric field? This is actually very
similar to the situation when you have a progressing wave
on this string hitting a wall. Because the magnitude
of the string which is fed to the
equilibrium position is actually equal to 0. That's actually
what we have learned in the last few lectures. And this is actually exactly
the same situation, right? You have a progressing wave. And there is some kind of
boundary, which is actually when this progressing
plane wave encounter this perfect conductor. There-- the electric field-- the boundary condition--
has to be E is actually-- E, x, y, 0, which is actually
the position of the z of the perfect conductor. As a function of time
will be equal to 0. The whole plane will
have 0 electric field. So that means there must
be what kind of wave? There must be a reflective
wave because of the presence of the perfect conductor. It's actually similar
to the situation which we discussed there's
a progressing wave hitting the wall. And this string
wall system-- there will be a reflecting
wave coming out of it. So therefore, what we are
expecting is some kind of-- refracting wave
which actually cancel the magnitude of the electric
field at Z equal to 0. And then this progressing wave
is going to the left-hand side direction. So now, I can actually write
down the incident wave-- expression. The incident wave-- I call it Ei, this is Ei-- is expressed as E0 over 2,
cosine kz minus omega t. This is actually what I
putting to the system. The magnitude is E0 over
2, and it's actually propagating toward
the z direction, as you can see from here. And that the direction
of the electric field is in the x direction. And of course the
E field will have a corresponding magnetic
field, which is actually-- you can actually write it down
directly using this formula-- B equal to 1 over c, K
cross E. K here is z, therefore you can
quickly evaluate and then conclude that
the magnetic field must be in the y direction. And that the magnitude
of the magnetic field would be E0 divided by 2 c. Cosine Kz and this omega t. So that is actually
the incident wave. And of course I also
need, as I discussed, there must be a
reflective wave, Er, which you actually cancel the
electric field at z equal to 0. If that cancels
the incident wave, that means the magnitude must
be in the opposite direction of the incident wave. Therefore, I can
quickly write down what would be the resulting
reflective wave that would be equal to
minus E0 over 2, cosine minus Kz, minus
omega t in the x direction. And then the
corresponding B field, I can also write it down using
exactly the same formula. And basically what I
conclude is that this will be equal to E0 over
2 c, cosine minus kz, minus omega t in
the y direction. So you can actually check this
expression after the direction. So now, I would
like to check what would be the magnitude of the
electric field at z equal to 0. So basically, at
z equal to 0, you have something which is
proportional to cosine minus omega t for
the incident wave. And then the magnitude
is E0 over 2. And if you evaluate z equal to
0, basically you get minus E0 over 2 cosine minus omega t. Therefore, they really
cancel and give you the desired boundary
condition, which is actually E equal
to 0 and the surface of the perfect conductor. So that's very nice. And this is actually the physics
of which we already learned from this string wall system. So what I can do
now is to calculate the total electric field
if I add them together. Basically, I would get E-- total electric field,
which is actually overlapping the incident
and the reflective wave. What I am going to
get is Ei plus Er. And basically, what I get is
E0 over 2 because the incident wave and the reflective
wave of the electric field is always in the x direction. Therefore, I only need to
take care of the x direction. So basically, I have cosine
kz minus omega t, minus-- right, because there's
a minus sign here-- minus cosine minus kz, minus
omega t in the x direction. And there should be-- And of course this is
a cosine minus cosine. So we have all
the formulas-- one from, for example, Wikipedia,
or from your textbook. So you can actually
calculate this-- rewrite this expression to
be E0 sine omega t, sine kz. And then this is actually
in the x direction. Everybody's following? I hope it's not too fast. All right, and of
course, I can also calculate the
corresponding B field. So it's actually again,
exactly the same thing-- Bi plus Br. And basically, I
will skip the step. Basically, you can add
this term and that term. And you will be able to
conclude that the B field will be equal to E0 over
c cosine omega t, cosine kz in the y direction. This is actually
pretty interesting. If you look at this result,
I have a electric field, which is proportional to E0,
the magnitude, sine omega t, and sine kz. What does that I mean? This is a special kind of
wave which we learned before. What kind of wave is this? AUDIENCE: Standing. YEN-JIE LEE: It's
a standing wave because the shape is
actually fixed, the sine kz. And the magnitude is
actually changing up and down at the angle frequency omega t. It's a standing wave. Another thing which is really
interesting is that if we look at the expression of a electric
field and the magnetic field-- if we compare that-- one is actually sine, sine. The other one is cosine, cosine. That's kind of interesting
because this is actually different from what we
actually usually learn from the progressing
wave solution, or traveling wave solution. Where the electric field and
the magnetic field are in phase. There's no phase difference. In the case of the
superposition of the incident and the reflective wave-- the solution of
a standing wave-- actually you can see that the
phase of the B field and the E field are different. Finally, very important--
you will see that-- look at this expression-- B equal to 1/c, K cross E-- that means this only
work for traveling wave. Clearly, this doesn't
work for standing waves. So very important. So don't blindly
apply this expression. This is only useful for the
traveling wave solution. And you can see a very
concrete example here. This doesn't work
for standing waves. That's kind of interesting. And if you look at
this result, you will see that if I don't
have magnetic field-- if I only have the
electric field-- there will be a instant of
time, for example, t equal to 0. When t is equal to 0,
sine is equal to 0. What is going to happen? You will have no electric field. That means electric
field completely disappear because
we are operating this system in vacuum. There's nowhere to hide. Where is the energy? The energy, fortunately--
electric field have a very good partner,
which is actually B field. All the energy's actually stored
in the form of magnetic field. You can see that now magnetic
field is reaching the maximum. So of course I can now
calculate the Poynting vector. Poynting vector is E cross
B divided by mu zero. And these will be equal
to one over Mu 0, Ex, By, and the z direction. There's only one
term which survive. So Poynting vector is
not pointing vector. It's not pointing around. There's a gentleman
who is called Poynting and he has a vector. And this vector is a
directional energy flux. It's a directional energy flux,
or the rate of energy transfer per unit area. So that is actually the
meaning of Poynting vector. And then each magnitude
is proportional to E cross B divided by Mu 0. So I can calculate that. Basically, I have
the E and the B-- Ex and By, then I can calculate. That would be equal to
E0 square, over Mu 0, sine omega t, cosine
omega t, cosine Kz, sine Kz in the z direction
because I have x cross y. And I'm going to
get the z direction. And I can simplify this. I have the sine, cosine. And also all have
cosine, and sine. Basically, you can
simplify this expression and get E0 squared divided
by 4, Mu 0c sine 2 omega t, sine 2kz in the z direction. So you can see that
the directional energy flux is in the z direction. It has a vector-- it has a wave number 2 times
of the original wave number. And it's actually going
up and down 2 times of the speed of the oscillation
of the original electromagnetic wave. And this energy is actually
vibrating up and down. And the shape of this energy
transfer Poynting vector is actually a sine wave. So that this is actually how
the microwave actually works. So basically, what
we are doing is to have generate microwave
inside your device. And in the oven this
microwave is actually bouncing back and
forth because you have metal walls,
which actually bounce the electromagnetic
field back and forth. And it really can cook the
food by vibrating the molecules inside the food back and forth. So as you can see the magnitude
of the Poynting vector is actually isolating
up and down. That actually cause additional
vibration and that heat up the food. So after this
lecture, you will be able to say proudly
that you understand the physics of microwave oven. [LAUGHTER] Thank you very much. I hope you enjoyed
the lecture today. And you have any
questions, I will be here.
