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I know that recent advances in technology are doing amazing things with very low amounts of electricity, with RFID being one of the most familiar examples, but I'm not very familiar with the principles involved. It makes me wonder: is it possible (with current technology, not "is it theoretically possible?") to use ambient sources (including body heat) to generate enough power to light up an LED, all in a device small enough for a person to hold in their hand? If so, how small could the whole assembly be made?

I've actually got a very specific application in mind, but if it turns out we don't have the capability for it, that'll save a lot of time right there.

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It depends on how bright the LED needs to be. Using body heat is a bad idea as the temperature gradient is small (effectively zero if it's covered by clothing).

You could use vibrations to power your device for example. However I doubt it'd meet the power requirement.

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is it possible (with current technology, not "is it theoretically possible?") is it possible (with current technology, not "is it theoretically possible?")

Well we will start off with "is it theoretically possible?" since if it is not even theoretically possible any further discussion is a waste of time.

The maximum possible amount of work that can be done from a heat engine is given by the Carnot efficiency:

$$ \eta = 1-\frac{T_c}{T_h} $$ Assuming the hot end is at body temperature and the cold end is ice then it is possible to use something like a thermocouple to generate electricity. $$ \eta = 1-\frac{273}{37+273} < 12\% $$

However this device can only be $12\%$ efficient at absolute maximum. Which means that if you want to use a ten watt $(10 W)$ LED then you need at least one hundred watts $(100 W)$ of heat energy to be taken from the body. Which is approximately all of it.

Bottom line Carnot Efficiency is shit, and lights use up a decent amount of energy so don't use heat engines.

If you want to generate light I recommend you use a hand crank flash-light. This heat could be used for very small electronics though.

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