Please note:
1. The word “radiation” does not always mean the same thing, don’t be afraid of it.
2. There are two types of radiation: ionizing and non-ionizing. Ionizing radiation will knock electrons off your DNA, typically not causing any problems, but in some cases can cause a genetic change that leads to cancer or other ills. Non-ionizing radiation will simply make things warm. In some cases, non-ionizing radiation will cause short term problems, but it has to be REALLY strong. Sticking your head in front of an airport radar dish strong. (Apparently makes a “clicking” noise in the ear canal)
Old CRT monitors (not the new flat ones) produced a small amount of radiation, since they are essentially a particle accelerator that’s tossing electrons at the front of the screen. The monitors were so heavy because of all the lead in the glass to contain this.
Modern LCD monitors don’t do this. They have fluorescent or LED lights behind the glass to provide backlight. They are no more harmful than the fluorescent lights at the grocery store.
Radio signals are non-ionizing radiation. Cell phones, wireless internet, etc. all are harmless. There are a few preliminary studies showing problems from cell phones, but again these are devices held against your head. Your computer is at least several inches from you, and is fine.
The primary way in which a PC tower would radiate is from the resonators inside. The computer is probably 1–2GHz, with a bus speed around 500MHz. The wires inside the computer will resonate at these frequencies and produce very low level radio waves (radiation) at these frequencies. 500MHz signals will go right through you without stopping. 2GHz (2.4GHz in particular) will bounce off most solid objects, but again won’t cause any sort of genetic or other damage.
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As an aside: folks who claim wifi allergies GENERALLY fail a double blind test, meaning its psychosomatic symptoms. That is, they’ll react equally to a blinking box that looks like a wireless router vs. an actual wireless router.