Some web pages[1-4] are describing a method to measure speed of light using a microwave oven. They claim that by removing the turntable, hot spots can be found in the microwaved material (food). And the distance between two neighboring spots corresponds to the half wavelength of the microwave, which can be measured to be around 6cm. Then they can calculate the speed of light by using a wave formula,
where c is the speed of light, λ is the wavelength, and f is the frequency of the microwave which can be obtained in the sticker on the back of the oven and which is usually 2.45GHz.
Eq. (1) is valid for a monochromatic plain wave. For other forms of electromagnetic field (E&M field), we have to be careful. Apparently, the size of the oven is at the same order (~10cm) of the "wavelength" of the microwave, so the oven cannot be approximated to be infinitely large and, hence, the E&M field cannot be simply approximated to be a monochromatic plain wave. Also obviously, the oven cannot be approximated to be a pair of infinitely large walls, so the E&M field cannot be approximated to be a 1D standing wave. Therefore, we have to solve the Maxwell equations to see what is actually going on in the oven. And then we can see the physical validity of this method.
I. Resonant cavity approximation
In this section we ignore the coupling of the cooking chamber and the microwave source. The chamber can be approximated as an ideal rectangular resonant cavity. In this case, the E&M field can be solved exactly[5,6]. The components of electric field are given by:
where ω is the angular frequency of the microwave, and kx, ky, and kz are given by:
Lx, Ly, and Lz are dimensions of the cavity. The amplitude constants E1, E2, and E3 in Eqs. (2)-(4) are constrained by:
which comes from the non-divergence of electric field in charge-free space. We can see that by choosing different combination of (m, n, p), we can have different modes of electric field. Angular frequency of each mode is given by:
For simplicity, we assume the food does not perturb the field. The average power density absorbed by the food is:
where Im[ε] is the imaginary part of the dielectric constant of the food. <E2> can be found by:
In order to clearly see the power density distribution, we take a 29cm*29cm*19cm oven as an example. Since the microwave frequency is around 2.45GHz and because of Eq. (7), probable combinations of (m, n, p) are: (2, 4, 1), (4, 2, 1), (2, 3, 2), (3, 2, 2). Note that oven of different size has different possible (m, n, p) combinations. Because x and y direction are symmetric, here we can only consider (2, 4, 1) and (2, 3, 2). We can plot the power density distribution for each mode, which is shown in Fig. 1 and Fig. 2.
Fig. 1 Absorbed power density distribution (dark indicates high density) at the plane of z=8cm for mode (2, 4, 1) in 3 cases:(a) E2=0; (b) E1=0; (c) E1=E2
Fig. 2 Absorbed power density distribution (dark indicates high density) at the plane of z=8cm for mode (2, 3, 2) in 3 cases: (a) E2=0; (b) E1=0; (c) E1=E2
We can see either from Eq. (9) or from Fig. 1 and Fig. 2, that for mode (2, 4, 1), the distance between two neighboring hot spots is 7.25cm or 14.5cm. For mode (2, 3, 2), the distance is 9.67cm or 14.5cm. None is around 6cm. When the modes are mixed up together, the distribution would be more complicated.
II. Further discussion about the wavelength
What happened to the 12.25cm wavelength? Originally, wavelength is defined as the distance between repeating units of a propagating wave. However, we don't have a propagating wave here. The concept of wavelength is not well-defined. Nonetheless, we can redefine wavelength by Eq. (1). Moreover, if we define the “components” of wavelength as:
then we can find the wavelength by:
which can be proved from Eq. (7), Eq. (1), and
The neighboring distances calculated in Sec. I are actually only the x and y “components” of half wavelength. This raised another possible method of measuring the speed of light, which will be discussed in Sec. V.
III. Perturbation to the power density distribution
It becomes very hard to analytically solve the system when perturbing factors are put into consideration, for example, the coupling of the cooking chamber and the wave source (a wave guide or an antenna), the food (especially that is non-uniform distributed) as a dielectric in the chamber, the mesh of the chamber door, etc. The perturbing effects can be obtained from experiments or numerical computations. Experiments[8,9] showed that the food can change the power density distribution a lot. The results are shown on Fig. 3 and Fig. 4.
Fig. 3 Absorbed power density distribution in an experiment. (a) The oven is empty. (b) A cup of water is put in the oven.
Fig. 4 Absorbed power density distribution in another experiment. (a) The oven is empty. (b) Something is loaded in the oven.
With wave source considered, one experiment and numerical simulation got the distribution shown in Fig. 5. The pattern is quite similar to those in Fig. 1 and Fig. 2. However, another simulation predicted that the distribution varies a lot when wave source is considered, as shown in Fig. 6. Fig. 7 is another experiment result, which is quite different from Fig. 1 and Fig. 2.
Fig. 5 Absorbed power density distribution with wave source considered. (a) Numerical simulation result. (b) Experimental result.
Fig. 6 Absorbed power density distribution with wave source considered in another numerical simulation. (a) 3D plot. (b) Contour plot.
Fig. 7 An actual absorbed power density distribution measured in an experiment.
Furthermore, it is possible that several field modes coexist in the oven, which can show pattern that is totally different from Fig. 1 and Fig. 2.
IV. Discussion about the speed-of-light-measurement experiment
With results from Sec. I and Sec. II, we can see the measurement of speed of light stated at the very beginning of this article does not make much physical sense. The concept of wavelength is not even well-defined. Although we can redefine wavelength by Eq. (1), as stated in Sec. II, the wavelength is not a physical existence. It is defined mathematically and it cannot be measured directly.
Why are people claiming that they can get 6cm apart hot spots? In my opinion, they were expecting 6cm apart hot spots when they did the experiment, so they did not analyze the overall hot spots distribution and, thus, they ignored other hot spots. Those 6cm apart hot spots might be caused by all sorts of perturbing factors mentioned in Sec. III. As far as I know, none can provide evident photos showing meaningful 6cm apart hot spots. Fig. 8, Fig. 9, and Fig. 10 are photos trying to show 6cm apart hot spots. However, these photos doesn't make much sense. Fig. 8 showed only 1 pair spots. Did they appear periodically over the testing material (egg white)? Could they appear again if the experiment was repeated? In Fig. 9, I cannot see any clear hot spots. Moreover, the testing material (chocolate chips) is non-uniform, which changes the power density distribution a lot, as mentioned in Sec. III. Fig. 10's experiment put the testing material in a line, but how did she know in which direction the microwave travels? Therefore, none of these photos can show evidence of meaningful 6cm apart hot spots.
Fig. 8 So called 6cm apart hot spots in egg white.
Fig. 9 So called 6cm apart hot spots in chocolate chips.
Fig. 10 So called 6cm apart hot spots in marshmallows.
V. Another possible way to measure the speed of light
Although the method of measuring speed of light mentioned above is a failure, discussion in Sec. II raised another possible method. First of all, we can use the wet thermal paper method to measure the horizontal and vertical plain (2D) power density distribution. We may get a pattern similar to those in Fig. 1 and Fig. 2. If not, the perturbation is too strong. This experiment cannot be carried out before the perturbation is removed. If fortunately the correct pattern appears, we can measure the distance of neighboring hot spots to get the “components” of half wavelength. Then we can calculate the newly defined wavelength by Eq. (11). Finally, we can find the speed of light by Eq. (1).
As discussed in Sec. III, due to the existence of all sorts of perturbation, we probably cannot get a pattern similar to those in Fig. 1 and Fig. 2. Therefore this new method may be impractical, although it is a plausible way.
E&M field in microwave oven is analyzed. It is not a simple plain wave or a 1D standing wave. Thus the method of measuring speed of light by measuring 6cm apart hot spots does not make sense. Instead, the method by measuring “components” of wavelength is more reasonable. However, it is valid only in the ideal resonant cavity approximation. In a real microwave oven, all sorts of perturbation can fail this method.
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