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Re: Speed of Light With a Microwave (long reply!)



I don't have the article in front of me, so I don't recall their analysis,
but the eigenmodes of a rectangular box are definitely oriented along the
coordinate axes. The eigenmodes are of the form

p_{lmn} = cos(k_{x,l} x) cos(k_{y,m} y) cos(k_{z,n} z)

where k_x, k_y, and k_z are the wavenumbers in the x, y, and z directions
and l, m, and n are integers starting from 0.

The wave equation then gives the dispersion relation

k_{x,l}^2 + k_{y,m}^2 + k_{z,n}^2 = k^2 = omega^2 / c^2

where omega is the frequency and c is the speed of light for EM waves (I
had to say that; I'm an acoustician). The boundary conditions imply that

k_{x,l} = l pi/L_x; k_{y,m} = m pi/L_y; k_{z,n} = n pi/L_z

where L_x, L_y, and L_z are the dimensions of the enclosure, so the
frequency of the microwave has to be chosen according to

omega / c = pi * sqrt(l^2 / L_x^2 + m^2 / L_y^2 + n^2 / L_z^2)

or there won't be a solution. One could now form a table of (omega/c)
values for integer triples (l,m,n). I would imagine that the frequency of
the microwave oven was chosen to excite a particular eigenmode. The
distances between the spots in the x, y, and z directions give you the
wavenumbers in each direction, which gives you l, m, and n so that you know
which omega/c to use. Then the frequency of the microwave oven gets you c.

That's where I get confused, however. The measurement listed in the
article was only in the x and y directions. What happens in the z
direction? It's certainly possible that the frequency was chosen such that
n = 0, but I wouldn't have guessed that ahead of time.

Of course, since I know c and can get omega from the back of the oven, I
could work out what l, m, and n are and see for myself. But that, as noted
in the message, uses the answer directly. As a check, though, the l, m,
and n numbers would tell me what I should have measured for the wavenumbers.

Judging from the above, you can see I'm not primarily an experimentalist.
:) Am I missing something?

Ken Kousen
Research Scientist (someday to be HS Physics Teacher)
United Technologies Research Center
East Hartford, CT
kousen@utrc.utc.com
http://www.concentric.net/~kousen/


At 07:52 AM 5/16/97 -0800, you wrote:
Cherie,

I noticed the same thing and briefly contemplated writing a letter to TPT;
I'm sure someone has. The thing I also noticed is that you can read the
ruler in the picture and it clearly shows a distance that is about half of
that used in the article to get the speed of light. (Weren't we just
talking about the sinister effects of "knowing the right answer"?)
However, I think the situation is further complicated by the fact that one
should not necessarily expect the standing wave pattern to be oriented
horizontally. If it isn't, then the distances between melted spots on the
surface of the marshmallow will not bear any simple relationship at all to
the wavelength.

TPT often suffers from its loose policy of peer review. I think there are
some benefits to keeping the publication process simple, especially given
the low levels of editorial support that AAPT is likely providing.
Nevertheless, it is a shame when things like this make it to press and,
possibly, into the curricular bag of tricks of credulous readers.

John

On Fri, 16 May 1997, Cherie Lehman wrote:

In the most recent _Physics Teacher_, there appeared a one-page article
which described how one could use a pan of marshmallows in a microwave oven
to determine the speed of light. The method involves measuring the
distances between hotspots or more-cooked places on the top of the layer of
marshmallows. It is asserted that this distance is the wavelength of the
microwaves. I am having trouble understanding why this is so. It seems
reasonable that a standing wave may be set up inside the microwave. It also
seems reasonable that the hot spots would correspond to the anti-nodal
points. What I don't get is why the distance between the anti-nodes should
be a wavelength. One-half wavelength makes more sense to me as the distance
between consecutive antinodes. Help me, someone, please.

-----------------------------------------------------------------
A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
Professor of Physics mailto:mallinckrodt@csupomona.edu
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Cal Poly Pomona fax:909-869-5090
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Kenneth A. Kousen
Research Scientist
United Technologies Research Center
kousen@concentric.net
http://www.concentric.net/~kousen/