∫ A microwave oven arrived in the mail today…

Here’s what the left side of my kitchen looked like when I got up this morning:

And all of a sudden this afternoon, it looked like this:

Don’t worry–those pretzels were consulted before their territory was encroached upon like that. And yes, I worked much, much, MUCH harder than was called for to make those two pictures look as similar as possible. But the point is, I finally have a microwave, which is nice, because previously I had been reheating all my leftovers in a frying pan, which works okay for eggplant curry but didn’t go so well with my tortellini.

Anyway, as long as we’re talking about microwaves, I’d like to say a few things about why they’re cool. In the first place, there’s a pretty good story about how they were discovered, complete with exploding eggs, and unlike many other famous inventions which have acquired apocryphal tales of origin, this one seems to be true. Secondly, the science going on inside a microwave is pretty slick too. As a matter of fact, I have fond feelings for the scientific principles behind microwaves, since I can remember my dad explaining them to me in simple terms when I was still in elementary school but have also gotten to see how much complexity there is in the details while studying physics.

As dad explained to me (on a bike ride to Albertson’s, in all likelihood), the key to microwaves is that many molecules found commonly in food products (most notably water, but to a lesser extent also fats and other organic molecules) have what’s called an “electric dipole moment.”  That’s a fancy way of saying that the electric charge isn’t distributed evenly in the molecule, as is particularly evident in the case of water:

Electric charge distribution on a water molecule. Image from wikipedia.

When bonded like this, the hydrogen atoms are basically bare protons, so there’s a whole lot of positive charge hanging out on the back end of the molecule, and conversely, all the electrons will tend to collect up on the other side. The result is a lot like a bar magnet: it has two “poles”  with opposite sign; they’re just electrically charged instead of the north and south magnetic poles you might be used to.

However, they behave just like bar magnets in an important way: just like a magnetic field will make a bar magnet spin around until it’s lined up with the field (picture a compass), an object with an electric dipole moment will do the same thing if it’s exposed to an electromagnetic field. Thus, if you have a bunch of water molecules and you want them all to line up, you put them in a big, static, uniform electric field.

On the other hand, suppose you don’t want them all to line up, but maybe prefer that they do something else– like, say, spin around like crazy until they get really hot (if you don’t believe that spinning things get hot, make your dog chase his tale and see if he starts panting. Sort of.) If you wanted to do something like that, you could just keep changing the electric field around the molecule, and then, like the poor kid in the middle of a game of “keep away,”  the molecule has to constantly keep changing direction to try to keep up with the field changes. All this motion creates heat, and voila, your food is cooked, or at least warmed over. The only problem is, how are we going to create a changing electric field like this?

Well thankfully,  a “microwave” (the science thing not the oven) is nothing but an oscillating electric field*. In fact, it’s called a wave precisely because the strength of the electric field at any given point** keeps bobbing up and down like a boat on choppy water. So if you expose the food to some carefully controlled microwaves, you will see just the desired effect: as the field gets strong to the right of the molecule he will turn to the right:

Graph height indicates electric field strength

but now imagine “wiggling” the graph above just like making waves with a garden hose: suddenly the field will start to get weak there and stronger to the left, and eventually he will have to turn around:

Graph height indicated electric field strength

Repeat several billion times per second and you’ve got yourself a bag of popcorn.

So that’s the basic idea, but like I said, part of the reason the science of microwave ovens is special to me is because even though I was taught that basic principle a long time ago, it wasn’t until junior year of college that I could do the math to show exactly how and why it would work. So for that reason, I can’t resist throwing in one more tidbit that depends on some slightly more technical details. It turns out, if you solve Maxwell’s equations in order to figure out just what’s going on with the electric fields inside a microwave oven (you can do this yourself, if you know separation of variables and understand how to represent a perfect rectangular resonator cavity in terms of boundary conditions) then you’ll find it obey’s the following set of equations.

If you’re familiar with PDEs you’re rolling your eyes at me right now– that’s just the familiar solution to the wave equation on a geometry that causes all the components to vanish at the rectangular boundary. Or in plain words, it’s the equation for what the microwaves do when they’re forced to stay inside your microwave.

The equations themselves might not mean anything to you, but this picture I made in mathematica might:

This is a snapshot of what the strength of the electric field might look like at some point in time along a slice of constant height inside the microwave oven. One side of the water molecules feel a strong attraction to the places where the field is strongly positive (the high points on the bumps) and the other side is attracted to where the field is negative (the low points). As time goes on, these peaks and valleys will bob up and down trading places, shaking the water molecules all about. Except, as you can imagine, there are a few places spread throughout the field that never ever change.

These are called “nodes,” and they may be easier to visualize in just one dimension. In the pictures above, you’ll notice that the point in the center never moves:

And in two dimensions, you get a grid of nodes filling the space inside the microwave:

Nodes marked in black; other nodes exist but are not visible when viewed from this angle

Now, contrary to what I drew above, a water molecule is much smaller than these waves, and thus if they’re sitting right at a node, all the action will seem to be going on far away, from their perspective. So far away, in fact, that they probably won’t feel particularly motivated to go to the trouble of trying to line up with the field. So these molecules will get a lot less exercise than the others, but instead of getting obese, they just stay relatively cold.

Well, if you’ve ever tried to microwave a large casserole or something similar, you already knew this was happening, you just didn’t need Maxwell’s equations to tell you. Because of course, this kind of phenomenon is the source of those pesky “cold spots” that you get where the food doesn’t seem to have been microwaved much at all, and it’s the reason why we use those little mechanized turntables to rotate the food as we heat it, so that all the water molecules move in and out of the cold spots and the entire thing gets cooked.

Incidentally, if you’re not familiar with this effect (and have a plate you can spare) you can microwave a whole bunch of mini marshmallows without rotating the plate, and you’ll discover (if you stop to check every so often) that some of them remain relatively unmelted. If you measure the distance between the unmelted patches, you should find that it’s around 5 or 6 centimeters, because the wavelength of a microwave is about 10-12 centimeters, and there are two nodes per wavelength. Of course, this probably won’t be exact, as a real microwave deviates somewhat from the ideal resonator cavity I assumed it to be above. But the overall pattern should be similar.

*well, okay, it’s two, coupled oscillating fields, one electric and one magnetic

**except the wave nodes, of course


About Colin West
Colin West is a graduate student in quantum information theory, working at the Yang Institute for Theoretical Physics at Stony Brook University. Originally from Colorado (where he attended college), his interests outside of physics include politics, paper-folding, puzzles, playing-cards, and apparently, plosives.

2 Responses to ∫ A microwave oven arrived in the mail today…

  1. Paul West says:

    Thanks for crediting me with your initial understanding of how microwave ovens work. This is one instance where I do vaguely recall having had such a discussion! I wish the math made as much sense to me as it does to you, but I’m intrigued by the marshmallow experiment. Did you really do it? I’ll have to saw up some of our old dried up marshmallows into small pieces and see if I can confirm your prediction!

  2. Colin West says:

    Oh good! Microwaves and vacuum cleaners are the two most memorable “Dad, how does this work?” conversations I can recall, although I think soap is a close second. And yes, a couple of friends and I did it, about two years ago, and the result was pretty satisfying. It wasn’t quite as ideal as you might hope for, partly just because it’s a goopy mess that’s not very precise, and partly because variation in the marshmallows themselves also produces some hot and cold spots, but there did seem to be a “macro” pattern of regularly spaced cold spots on top of the random noise.

    Can you think of anything better than marshmallows that might make it easier to see the pattern? They’re good because they don’t conduct heat well between themselves, so the heat from the warmest regions doesn’t bleed into the cold spots. Furthermore they’re cheap, and further furthermore they taste great on graham crackers once the experiment is done. But there might be a better alternative. :-)

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