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Re: Waves and Energy



On Sun, 23 Feb 1997, David Abineri wrote:

I wonder if I can get help on this question from a high school class.

If two electromagnetic waves interfere destructively, what happens to the
energy associated with the waves. I am guessing that on the large scale,
in a double slit experiment that additional energy shows up at the points
of constructive interference that compensates for the loss at the nodes.
But, what about looking as just the waves that are meeting at a node?

If I may go a bit beyond the question as posed, I would like to offer some
general observations, just because I frequently hear this presented wrong.

Case A: tricked by E&H
Suppose you have a plane EM wave reflecting off a plane mirror. So you
then have the identical wave coming back again, and hence set up a
standing wave pattern. This is commonly compared with a wave on a string,
but how good is the comparison? The energy of an EM wave in free space is
split 50-50 between the electric and magnetic parts. If the energy
density of the original wave is E, then the average energy density of the
combined wave should be 2E. Now lets look at a peak in the electric
field: The electric field is double what it would have been with only
one wave, so the energy is 4 times. But since the electric part only has
half the energy, the total electric energy density is 2E.... what about
the magnetic part? Take your right hand and point the thumb towards your
nose, your index finger up, and the other three fingers horizontal to your
left. The thumb represents the wave coming towards you, your index finger
the electric part, and the other three the magnetic part. Now rotate your
hand abour the index finger to point the thumb exactly away from you. The
magnetic field is opposite. In other words, where we have addition of the
electric parts, we have cancellation of the magnetic part - zero magnetic
energy density. If you work it out, you find that the energy density in
this standing wave pattern is the average expected value of 2E everywhere,
it simply keeps swapping between the electric and magnetic parts of the
wave.
This fact is used a lot in communications. In many applications there is
no direct line of sight between a radio transmitter and receiver. So when
you are talking on your cell phone, you are receiving many waves that have
taken many routes (multiple diffractions and reflections) to get to you.
As a result, you are in a standing wave pattern. Since cell phones only
use one field component the recieved energy can vary typically by 10 dB
over distances of the wavelength (i.e. approximately 1 foot), so the
technology has to handle wide energy fluctuations as you move around.
More sophisitcated systems can overcome much of the problem of local
"multipath" fading by employing "diversity" antennas which can sense more
than one field component. Unfortunately, the energy is split into 6
different components (3 E and 3 H), whereas a typical diversity antenna is
sensitive to 2 or 3 components.

Case B: real energy minima

This is the case for the double slit you cite. Since the combining waves
are travelling in the same direction, the E and H fields are simply scalar
multiples of each other and will either both fade or both add at any given
spot. So in this case the energy density of the peaks is 4 times what it
would have been from a single slit. But analyzing how the energy is
distributed over the interference pattern will show that it all adds up to
to right value, twice that of a single slit.
However, this result has led to another common misconception, though I
stand to be corrected on this one:
I have had more than one person approach me with the idea that we could
use interference to get electromagnetic energy into a closed container.
For example, blasting brain tumors. The idea is that we set the beams up
such that there is a maximum at the tumor and minima elsewhere. So we
blast the tumor, but nothing else. This misconception, that such a thing
is possible, comes from looking at interference patterns on the wall and
being told that we can split the energy up so that it only occurs at
particular spots. This is true enough, but what we see on a wall doesn't
say what happens in 3-D. This gives us no help blasting brain tumors
because each on of the maxima can be traced back to the slits. It is
rather as though each of the maxima came from its own source. You could
not find a surface in front of the wall that would enable you to put a
barrier that had energy inside but none on its surface. (In the event that
I'm wrong about this however, who wants to approach a medical funding
agency with me???) Incidently, there is a closely related misconception
about holograms, but I've blathered on long enough.

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| Doug Craigen |
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| If you think Physics is no laughing matter, think again .... |
| http://cyberspc.mb.ca/~dcc/phys/humor.html |
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