If you do use a microwave (setting aside for the moment the issue of "whether" you should microwave at all, which is addressed in part here), are there practical benefits to microwaving at lower power settings? Or is full power at a short time better than half power at twice the time?
The only reason to cook lower power for a longer time is to maintain texture in your food. It's extremely easy to overcook things in an instant or dry up the moisture to the point that you're eating shoe leather. Eggs are a pretty good example of this. Really easy to overdo it in a microwave.
Microwaves act on the water in the food, they do not "irradiate" your food. There is no danger, unless you're using a microwave in which the shielding has been breached - in which case it would not function properly for very long anyway.
I would the the answer is "maybe" - it depends on what it is about microwaving you think is the most harmful.
With a lower power setting, the power is the same but it's not applied constantly - your microwave cycles between full power and off. So if you're primarily worried about superheating your foods, this will be better. If you're worried about the "radiation", then it's probably that's it's every bit as bad.
That said, I'll still use a microwave when I have to, but I greatly prefer not to. Somewhat for fears of negatively impact the healthfulness of the foods, but primarily due to concern about how negatively it affects the foods taste and texture.
Because I generally can't stand how food tastes/feels when it comes out of the microwave.
NO it's not directly better to use lower power settings.
Why: Most microwaves do not change power/intensity.
To reduce power they take breaks. It's that sound change you hear at lower powers, such as when defrosting.
What's important is the time of exposure. Sadly I can't find the study that I read regarding this. Will update if I find it.
But there are uses for lower powers, such as defrosting, and cooking certain items like eggs that respond/taste better when they get short breaks from the heat/radiation.
Lets look at how the different ways to cook food work:
A standard oven/fire/external heat source works by heating up the outside of the food and starts cooking that part. The heat then slowly gets conducted to the inside and cooks that part. You can think of it as cooking from the outside-in. This is why you don't double the temperature and cook for half the time. You burn the outside and don't leave enough time for the heat to reach the inside.
A microwave works by aiming microwaves (convenient name, huh?) which are about 1 foot long at the food. These waves are at the correct frequency to shake water molecule (really -OH libration, so fats and sugars also interact with the microwaves) and heat it up. The microwave goes through all of the food at once and shakes all of the water molecules at once, so the food heats up evenly. Now, the inside heats as fast as the outside, but the outside actually can conduct its heat to the surrounding air and it cools down. So you actually heat the inside faster than the outside. You can think of it as cooking from the inside-out.
I would think that as long as you cook the food long enough to get to the proper cooking temperature, that a long-low power is probably "more gentle" on your food than a short high-power. But remember, your microwave doesn't adjust the power of the waves, they are what they are (for mine is 1500W), it just adjusts the duty cycle (how long the waves are on). So for something big, say a hunk of chicken, it's probably better to go slow because you're giving time for the heat on the inside to equalize with the outside. For something thin, like bacon, it probably won't make any difference other than you having to wait longer for your bacon.
I like to poach eggs in the microwave, and lower/slower works better. For mine, about a minute at 80% does the trick; at the office, 1:20 or so at 70%. At higher power and shorter time they turn rubbery on the outside and stay raw in the middle.
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