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I've seen a number of sites/videos online that describe a method for measuring the speed of light, using a microwave oven and a chocolate bar. For example, this video on youtube. The basic idea is to microwave the chocolate for a few seconds, and measure the wavelength as the distance between the resulting melty regions. Then, reading the microwave frequency (listed on the door or back of the machine), you solve $c=\lambda f$.

It's a very nice demonstration, but the problem is: the only reason the frequency $f$ can be found listed on the machine is because it was designed to produce that frequency. The microwave source (magnetron) is a resonant cavity, and by adjusting the dimensions, one can figure out the resulting frequency if the speed of light is known.

My question is, could this experiment actually be done to "discover" the speed of light, i.e. if we lived in a world where the value of $c$ was unknown, but somehow we knew how to generate microwaves? Or in determining $f$, would we have already necessarily discovered the value of $c$ by some other means?

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    $\begingroup$ I suppose the important question is, in this hypothetical world where we don't know $c$, are we able to generate microwaves with known frequency? Or is that what you're actually asking? (i.e. is there a way to generate EM waves at a known frequency without tuning their wavelength to match a measured distance?) $\endgroup$
    – David Z
    Jan 10, 2015 at 9:07
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    $\begingroup$ @DavidZ Yes I suppose that is essentially my question. If you could generate the waves w/ known $f$, you can do the experiment and measure $c$. $\endgroup$
    – chase
    Jan 10, 2015 at 9:52
  • $\begingroup$ Also, a good microwave oven is not supposed to create a simple node pattern like that, there is a blender of sorts in the output path of the magnetron that should mix around the angles and modes. I guess that is not 100% perfect though but it should complicate the analysis (if it had been useful - the magnetron is tuned by cavity size like you write). $\endgroup$
    – BjornW
    Jan 10, 2015 at 10:02
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    $\begingroup$ There are plenty of ways (e.g. 2-slit experiment) to determine the wavelength. So do that first, then do the cooking experiment :-) . That way, all you're depending on is the validity of $ c = \lambda*2\pi\omega$ $\endgroup$
    – Carl Witthoft
    Jan 10, 2015 at 13:23
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    $\begingroup$ Fundamentally, the frequency of the waves is governed by something mechancial in the microwave-- the simplest wave generators are basically LC circuits where simple considerations tell you that $\omega = \frac{1}{\sqrt{LC}}$ $\endgroup$
    – Zo the Relativist
    Jan 10, 2015 at 19:14

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I think the main confusion is here is the difference between a scientific measurement and a pedagogical demonstration. You're right that the microwave and chocolate method has to assume the frequency on the back of the microwave is actually the frequency of radiation inside the cavity. A proper measurement of the speed of light would include a measurement of the frequency. However, at the frequencies present in a microwave oven (around 2.5 GHz), it takes incredibly expensive equipment to precisely measure the frequency. Oscilloscopes that can do this run to tens of thousands of dollars; spectrum analyzers run several thousand dollars. This is much too expensive for a demonstration in a classroom.

The point of the chocolate experiment is to demonstrate the interplay between the frequency, wavelength, and speed of EM waves--to show that wavelength really does mean a physical length, that the resulting speed matches what we expect, and that microwaves are just another form of light. Additionally, melting chocolate in a microwave provides examples of the power of radiation and a means to measure with a ruler that which is normally invisible. Plus, you know, chocolate is awesome. When learning physics, at times it can be easy to forget that the numbers on the page refer to real things that can be measured. When I was teaching physics, I tried to do experiments that required the least sophisticated equipment, often just stopwatches and meter sticks, in order to make the subject as far from esoteric as possible.

If you want a real measurement of the speed of light using frequency and wavelength, one could set up an adjustable-length resonant RF cavity with a probe antenna inside hooked up to an oscilloscope. Feed radio waves into it and adjust the length until you have the shortest length that resonates as determined by the amplitude of the signal on the scope. Read the frequency off the scope and the wavelength from twice the resonant cavity length. I specify radio waves here because they have a lower frequency that's easier to measure.

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I prove that two worlds, in which there are different electromagnetic constants ($\epsilon_0$ and $\mu_0$), this experiment always results in identical results (in case of identical apparatus). As $c^2 = \frac{1}{\mu_0 \epsilon_0}$, the value of $c$ can not be determined in this way.

A typical microwave oven consists of some cavity, in which electromagnetic resonance is created. Suppose the created EM-field admits the following solution in the first world (interpreting the real part as the actual field):

$$\vec{E} = \vec{E}_0(\vec{X}) e^{\omega t}, \vec{B} = \vec{B}_0 e^{\omega t}$$

It is a well-known result from linear algebra, that changing the constants in field equations will yield a corresponding solution in the second world, with just E- and B-fields multiplied by some constants. As the shape of the fringes (minimums/maximums) will be the same, the experiment yields exactly the same result.

The result could have been obtained in a different way: using analogy between cavities and LC-circuits.

$$\lambda = c \frac{1}{f} \propto \frac{1}{\sqrt{\epsilon_0 \mu_0}} \sqrt{LC} \propto \frac{\sqrt{\epsilon_0 \mu_0}}{\sqrt{\epsilon_0 \mu_0}} \propto 1 $$

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    $\begingroup$ In this other world, the size of the microwave cavity and the distance between melted regions will be the same, but the frequency f listed on the back of the microwave would not be, assuming the engineers who determined it are competent. But I think this is chase's point--the experiment isn't really a full measurement of the speed of light because it's just relying on measurements by the engineers who wrote down the frequency for you to read. $\endgroup$
    – Hypnosifl
    Jan 10, 2015 at 19:46
  • $\begingroup$ @Hypnosifl: Indeed. But one of the questions were: "In determining $f$, would we have already necessarily discovered the value of c by some other means?" The answer to that is no. $\endgroup$
    – kristjan
    Jan 10, 2015 at 20:51
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    $\begingroup$ Are you understanding "some other means" to mean some other means than just measuring the microwave and the distance between melted spots on the chocolate bar? If so, then by answering "no", you're saying we can determine $f$ from that experiment alone, even if we don't already know the value of other constants of nature like $c$, $\epsilon_0$, and $\mu_0$? $\endgroup$
    – Hypnosifl
    Jan 10, 2015 at 21:27
  • $\begingroup$ @Hypnosifl: My fault, I somehow understood the question exactly the other way. According to my answer (not my previous reply to you), the answer is yes, the value of $c$ has to determined in some other way, as any EM-resonator (with frequency achieved by either a cavity or LC-circuit) would yield exactly the same fringe pattern in different worlds. $\endgroup$
    – kristjan
    Jan 10, 2015 at 21:39
  • $\begingroup$ Hmm I think he can know f without ever knowing c. E=hf. But you need to repeat the experiment involved in finding h or the whole thing is invalid. $\endgroup$
    – Joshua
    Sep 19, 2015 at 22:40
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I don't really see a difference between the microwave measurement and the definitions of the SI units. We have

  • Unit of time: second: The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.
  • Unit of length: meter: The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

These definitions mean that the technically correct way to measure a distance is to construct a time reference from a cesium clock, then to use that clock to measure the light propagation delay over the distance of interest.

In the microwave experiment, you're using the resonant frequency $f$ of the magnetron as a time reference, using the relation $c=\lambda f$ to predict the distance between antinodes of the standing waves in the cavity, and comparing the measured $\lambda$ to a length reference you have at hand.

The fact that the microwave cavity has been designed to have the same resonant frequency as the $LC$ circuit driving the magnetron is a red herring; if the two frequencies were mismatched the oven would simply be less efficient at transferring power from the magnetron to the cavity.

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