ECE3300 Lecture 28-4 Plane Waves

Uploaded by cfurse on 15.08.2009

Now let's take our wave equations and analyze how they are
actually going to be used. If we have a simple wave, let's
analyze a wave that has a planar wave front. We call this a plane
wave. It means that everything along this particular plane is
absolutely equal. The magnitude is equal and the phase is equal,
and as it propagates the magnitude is equal and the phase is equal
on this plane and finally the magnitude is equal and the phase is
equal on this plane. This wave is propagating in this positive
direction, and these planes would extend in infinite extense. In
this case, we could write a simple equation that would say that
the electric field as a function of, let's say, Z. Electric field
is a function of Z is going to have a positive going term. It's
going to be part of the electric field going this way. And just
in case we hit something, we might also have an electric field
going in the negative direction. We can write this the same way
we did when we were doing the transmission line equations where we
have E to the minus J gamma Z plus E0 minus E to the plus J gamma
Z. And it also can be written this way. E0 plus E to the minus
alpha Z, E to the J, E to the minus J beta Z plus E0 minus E to
the plus alpha Z, E to the plus J beta Z. So this gives us a
forward-traveling wave and a backward-traveling wave. Similar
equations can be written for the magnetic field. Now, how do we
go from electric field to magnetic field? This is how it's done.
Right-hand rule just like we had before. The direction of
propagation is going to be my thumb. I'm going to write D on
that. The electric field is my first finger. I'm going to write
E. And my third finger is the magnetic field H. Let's write
those. And the equation that we will have is that E cross H is
equal to the direction of propagation. So I can do it one of
several ways. The easiest way I think is to put my fingers in the
direction of the electric field and cross them towards the
magnetic field. So E cross H gives me direction of propagation.
So in this case E cross H would give me direction of Z
propagation. The other way that I can do it is holding my hand as
the three corners of a square. I have E cross H is equal to the
direction of propagation. So I just spread them out like this. E
cross H is the direction of propagation. So if I'm going to have
an electric field, I need to define its vector direction. Let's
do an example where I have the electric field in the X direction.
This is the Z direction. This is the Y direction. So here's X,
Y, Z. Let's figure out which direction the magnetic field is in.
So we're going to take the electric field in this direction. My
direction of propagation this way. The magnetic field would have
been in the Y direction. So I would have had an HY term. Here's
the other thing that's going to tell us about their relative
magnitudes. E cross H divided by this function we're going to
call eta. Eta is gamma divided by omega mu. And so we can say
that the magnetic field is equal to E divided by eta. And the
only difference, actually the magnitude is the electric field and
the magnitude is the electric field. So if I wanted to write this
for an X polarized electric field, I would end up with a Y
polarized magnetic field. HY plus of Z plus HY minus of Z.
That's going to be the same thing as E -- let's take the
magnitude, of EX plus of Z divided by eta, plus EX -- let's take
its magnitude, of Z -- this is the negative one. Also divided by