ECE3300 Lecture 28-1 Maxwell's Equations

The first thing that we want to talk about today is Maxwell's
equations. There are four Maxwell's equations. I'm going to show
them to you in point form. That means that they are correct at a
specific point in space. Here's the first one. The divergence of
the electric flux density is equal to the volume charge. What
that means is that I have a volume charged density. Let's think
of an individual charge. And it creates an individual flux line.
If I had a charge per meter cubed, I would end up with so many
electric flux lines. That's what this first Maxwell's equation
means. The second Maxwell's equation is that the curl of the
electric field is equal to the time derivative of the magnetic
field. This is the Faraday's Law equation. This is saying that a
time varying magnetic field produces an electric field. The third
Maxwell's equation says that the divergence of the magnetic field
or magnetic flux density is equal to 0. If you compare these two
electric, these two Maxwell's equations, you can see that this is
like saying that the magnetic field charge is equal to 0. That's
not really charged density, but the magnetic field charges are
equal to 0. The fourth Maxwell's equation is that the curl of the
magnetic field is equal to the conduction current density given in
amps per meter cubed, plus -- the amps per meter squared, excuse
me -- plus the time derivative of the electric flux density is
equal to the magnetic field. Here you can see that a time varying
electric field produces a magnetic field. Now, if we assumed that
all of these fields, E or H, are a function of E to the J omega T,
that these are sinusoidal fields with a specific frequency omega,
then we can see that the time derivative of E to the J omega is T
is J omega, E to the J omega T. So anyplace that we see a time
derivative, we simply substitute J omega. For instance, in here,
if we want to convert this to phaser form, we've got this minus
sign here, then there's this time derivative, which becomes J
omega and then it's B. Down here, here's our time derivative. So
this becomes JC. And then this plus J omega replacing the
derivative D. Now, there are a few other substitutions that I'd
like to make in here. The first one is that D is equal to epsilon
E and B is equal to mu H. So right here we can put mu H and right
here we can put epsilon, which is epsilon naught times epsilon R
times the electric field. The other replacement that we'd like to
make is for conduction current density. And that is that this is
conductivity times the electric field. Now we have our four
Maxwell's equations, one, two, three, four. We had them on the
time domain on the left, and we have them in the phase or
frequency domain on the right. There's one other piece that we'd
like to pull in here. And let's take a look right here at this
part of Maxwell's equations. This is sigma times the electric
field, plus J omega epsilon naught times epsilon R. All of these
times the electric field. This right here is often written as J
omega times the complex epsilon. Complex epsilon, epsilon C, or
sometimes written as epsilon star, in some textbooks, is epsilon
naught times epsilon R minus J sigma over omega, where omega, of
course, is 2 pi times the frequency at hertz. This is also
written as something called epsilon prime, you can see right here,
this part is epsilon prime, minus J epsilon double prime. So this
is epsilon double prime. So these are the four Maxwell's
equations, and they're very important to us and they're what help
us figure out how ways propagate. If we had a charged-free
region, that's also a very important approximation. So if we have
a charge and current-free region, that means there's no charge and
there's no current in this region, then row V goes to 0 and the
conduction current goes to 0. So in that case we have that the
divergence of the electric flux density is equal to 0. Of course
the divergence of the magnetic flux density is equal to 0. The
curl, the cross-product of the electric field, is still minus DB
by DT. And we also have del cross H is equal to time derivative
of the electric flux density. Now, these two equations are
normally not very interesting for us to solve. They're
interesting equations, but they're not used in our solutions.
These two equations right here are what we're going to use in the
solutions of all of our waves. Because the magnetic field
generates the electric field, and because the electric field
generates the magnetic field, these two equations kind of go in a
circle. You can see how it's going from B to E and from E back to
B. That means that these two are coupled, first order
differential equations. They are differential in both space --
this is related to spatial D by DX, D by DY, D by DZ. And there's
also a time domain derivative. So we have coupled first order
differential equations. It means that we have to solve both of
these equations simultaneously at the same time in order to get
the correct solution.