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To understand this, you need to know a little about how microwave ovens work, and for that we'll need to make a brief detour through physics and chemistry.

Temperature can be thought of, basically, as a measure of the average kinetic energy of molecules. Faster movement of the molecules means more kinetic energy and therefore a higher temperature.

Microwave ovens, however, don't directly make the molecules move faster. Instead, what happens is that molecules have certain specific frequencies of radiation that they absorb, which makes them vibrate and/or rotate in particular ways. Each molecule has a different "fingerprint" of radiation it absorbs, and analytical chemists can use this to study the absorption spectrum of a molecule in a particular frequency range and determine what the molecule is (in practice, it's more a matter of ruling out certain possibilities ... "This molecule can't have a -CH2- group, because it's missing the characteristic asymmetric stretch absorption frequency of that group").

In a microwave oven, we don't care so much about identifying molecules. What we're trying to do is get them to vibrate and/or rotate so fast they bump into their neighbor molecules and give them, effectively, a good hard "kick" to get them moving faster.

Microwave ovens have their klystron (the thing that emits the microwaves) tuned to emit in a frequency range that water absorbs strongly in. This works great for meats and veggies and soups and things like that, which contain high percentages of water. It works less well for certain other things, but it's safe to assume that most things you're going to eat have at least some water in them.

The frequency range also has some other absorption bands. Proteins, for example (something else that's probably in a lot of foods) absorb in the same region.

And here we get to bowls. Most bowls are made of materials that do not have strong absoprtiopn bands in the particular frequency range that microwave ovens use, so they heat up mostly because they're touching the stuff inside that is absorbing the microwaves (and getting "kicked" occasionally by those molecules).

Some kinds of plastics, though, are made from something that's chemically similar to proteins ... specifically, they contain amide groups, which is one part of the protein that absorbs in that frequency range. An example of such a plastic would be melamine (remember a few years ago when melamine was found to be a contaminant in certain foods from China? It was being added because it's similar enough to protein to "fool" the tests, so low-protein milk could be passed off as higher quality high-protein milk). Bowls made from melamine can get quite hot indeed in a microwave, because they're absorbing the radiation directly, not just being heated by the stuff inside.

And this is basically the reason. The bowl that's getting hot most likely has an absorption band in the microwave oven's frequency range, and the bowl that isn't getting as hot probably doesn't.

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