Uploaded by MIT on 29.12.2010

Transcript:

This is problem six, which is 33-26.

Let's do a time check--

boy, I must have gone fast. I'm one minute and 15 seconds

ahead of time.

Now we have a standing wave--

a standing electromagnetic wave. We have a frequency

which is 600 megahertz, and 600 megahertz would give the

wavelength lambda c divided by F, which is half a meter.

A standing wave is very different from a traveling

wave, but in terms of the wavelength and as far as the

wavelength is concerned, it is still c divided by F. In terms

of the magnitude of the magnetic vector, it is still

the magnetic field of the E vector divided by c.

But there is a big difference: there is a very, very big

difference, if we only look at the electric vector for now.

If there were a standing wave between these two mirrors

which reflect travelling waves back and forth, then at one

point in time, I might see an electric field that is

pointing like so--

E vector, E vector, E vector, E vector, E

vector, and E vector.

You may say, that looks like a traveling wave. Well, yes, it

looks like one, but it's not moving--

it's standing still.

A little later in time-- a quarter period later in time,

if I assume that this is the maximum displacement, and that

this is the amplitude of E vector-- a little later in

time, the E vectors [? will ?] everywhere in space 0.

Half a period later in time, the E vectors will have

reversed, and so on.

These will oscillate out of phase, and these will

oscillate in phase.

We have points in space here, which we call nodes, where the

E vector is always 0.

In this particular case, since this separation is 75

centimeters, you would have these nods at 0, 25, 50, and

75 centimeters.

You can only have a standing wave if the separation of

these reflecting mirrors--

whereby these travelling waves move through each other, which

causes the standing wave pattern--

if that separation is a multiple of half a wavelenght,

which you see it is.

It is 3 times 75 centimeters, and this is 3 times 25

centimeters.

Magnetic fields and E fields relate very differently to

each other in standing waves than they do

in traveling waves.

In travelling waves, I mentioned to you

that they're in phase--

that's no longer the case.

In traveling waves, the maxima--

the locations of maxima--

are therefore always at the same locations of the electric

fields as they are of the magnetic field, and that's no

longer the case either.

It's pretty nasty, but the magnetic field and electric

fields are 90 degrees out of phase.

Where the electric field has its nodes at all moments in

time-- they never change--

it's exactly where the magnetic fields have their

anti-nodes, their maxima.

They would still be perpendicular to the direction

of E, but they have their maxima here, and they would

have their minima right here.

They have their nodes here, where the electric fields have

their anti-nodes.

It's very difficult--

not all difficult, but it's very different, and the only

way you can see that is using Maxwell's Equations

rigorously, or Young does it in a very nice way.

He has two traveling waves from opposite directions with

the same amplitude--

not standing--

traveling electromagnetic plane waves, and then he

interferes them with each other.

They have the same amplitudes in E and B, and then he shows

that if you adopt that picture, you indeed can

demonstrate that E and B are 90 degrees out of phase, and

that the locations of the maxima of E and B--

or I should say, the nodes of E coincide with the anti-nodes

of B, and the nodes of B coincide with the anti-nodes

of E. It's actually a very nice way of showing it-- all

of that is completely consistent

with Maxwell's Equations.

I want you to appreciate that in a standing wave, the mean

pointing vector equals zero, and there is

a reason for that.

It's because B and E are 90 degrees out of phase.

If you make E oscillate with cosine omega t, then the B

vector would oscillate 90 degrees out of phase, and so

it would be the sine omega t.

That's what it means, 90 degrees out of phase.

To me, pointing vector would be proportional to cosine

omega t, if we assume that that's associated an E vector.

It's also proportional to sine omega t, because of the phase

difference with the B vector.

The mean value between the cosine omega t and sine omega

t, averaged over one complete oscillation, equals 0.

In a standing wave, there is never

any net energy transport.

That doesn't mean that if you look on a short time scale--

short relative to one complete period of oscillation--

then there is energy transport.

It goes in one direction, it goes back, and it

sloshes back and forth.

If you take the mean over one full period of oscillation,

then the net energy transport equals 0 in a standing wave,

which is not the case for a traveling wave.