Peter,

you  say: "...we agree that the microwaves vibrate water molecules, so, if you're right (and I do believe you), where is the water in the electrical components that suffer as a result of being bombarded?"

Well actually our agreement is only qualified. Certainly microwaves of a suitable frequency (such as those in the typical microwave oven) do vibrate the water molecules, and various other suitably polar molecules, such as common fats and oils. However, it does not follow that nothing else can absorb microwave energy. In particular any kind of microwave aerial also can absorb microwaves; that is precisely how they work!

In particular a transmission aerial such as a magnetron can absorb microwaves that suitably match the very microwaves that they transmit in the first place. It is very much like the way that if you twang the string of a musical instrument, creating a sound in the air, then the echo of that sound striking the string again, will amplify its vibrations again. And of course, if the energy cannot escape it is prone to convert into other forms, in the end typically heat. Poor magnetron!

You also asked: "... are we really saying that there is so much leakage (billions of waves a second), that having nothing in the microwave doesn't make much difference?"

Actually, what we said was billions of bounces (reflections if you like) a second. And some wastage at every bounce. We never said anything about having nothing in the oven not making much difference. Suppose the oven is empty; then the photons bouncing around have a half life, say, 100 nanoseconds. After a few half lives, say 10 or so, if you cut the power, the microwave glow will have died away for most practical purposes. However, if we put a microwave-absorbent plate of food or jug of milk in the oven, it will absorb microwaves strongly, and the half life of a bouncing microwave might drop to say, 10 ns. These figures are the wildest thumb sucks, but you can see that your magnetron in the second example would have only about one 10th as much energy to absorb, which I am sure you can see could make quite a lot of difference. It might not sound like much, put that way, but think of it like this: you might feel very comfortable carrying a 50 kg load a few hundred metres, but that does not mean that you could carry a 500 kg load anywhere, right?

You protest: "When I think of the energy required to warm a cup of tea in a microwave, I find it difficult to believe that when nothing is there, the whole energy furnished by the electrical supply has just .....gone......" and very right too!

Of course it has not just... gone... Energy is conserved remember? For all everyday practical purposes anyway. Some of it leaks out of the oven and warms a grateful universe, including you if you are nearby. A lot more of it goes into heating up your much-abused, long-suffering magnetron.

No mystery there, or am I misunderstanding your point?

You presume that: "...there is leakage when something is in there too, so something, to my simplistic mind, does not add up."

You are right about the leakage of course, but on the contrary, everything does add up, quite mercilessly in fact. Some leaks out, some heats up the magnetron (not a great deal we hope!) and most (we devoutly hope!) heats up our food. And between that wastage and that usage of power we consume precisely as much energy as we had fed into the oven.

Simple, right? Yes, but I sense your (hopefully residual) discomfort. What happened to the energy absorbed by the food? Am I right?

On the assumption that I am (forgive me if I am not) let me point out three major fates for any energy absorbed by the food. (There are umpteen others, but generally minor.)

Firstly, some of the absorbed energy spins or distorts polar molecules (prominently fats, proteins and water) and generally sets them bouncing around in a state we call being hot. That energy represents the equivalent of microwaves that have been absorbed and converted. As microwaves they are gone. As energy they still are there in the form of heat.

Secondly, part of the microwaves and the heat they generated causes chemical reactions, such as hydrolysis of some foods or charring of others. This might cause more heat to be given off, but also absorbs those microwaves responsible for the effect. Some of the new molecules formed in the process store such energy as is not given off in the form of heat.

Thirdly, the hot food re-radiates heat in the form of photons are far higher energy and far higher frequency (meaning far smaller wavelength). The wavelengths in question are in the infrared. Now, your microwave oven is partly metal, which is opaque both to microwaves and to infrared. However, the front panel, usually set into the door, is partly of glass and partly a metal grid. The grid does not permit long wavelengths such as those of microwaves (a few centimetres long) to pass through. The wavelengths of the infrared are in comparison minute, typically more like a few micrometres. They pass through the grid as comfortably as any torch beam. Now, there is leakage!

The glass of the door is not very infrared-opaque, though it is not especially transparent either, and it is a major route for escape of energy.But as I said, the whole lot adds up pretty comfortably!

Cheers

Jon