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If the atmosphere is filled with electromagnetic waves all oscillating at different wavelengths and speeds how is it that they don’t all interfere with each other? For example turning on your light seems to have no effect on the sound coming from your radio.

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Sometimes the light does interfere with the radio - especially with AM radios. It's a question of the frequency spectrum of the waves. The waves in the atmosphere indeed all mix up and form a gigantic linear superposition - that's why we have frequency selective bandpass filters to tune into communications channels. The idea is that interference is generally spread over wide bands, whereas communication channels concentrate their power into narrow frequency bands, so that they can overwhelm the background noise. This approach works even better with digital communications, as then one can use sophisticated error correcting codes so that corruption of any signal by noise can be reversed and you never know that it has happened. Your idea is also why some bands are not used - they're simply too noisy. Switched circuits and other appliances these days have to comply with electromagnetic compatibility standards, so that this effect is minimized, but the effect that you are asking about is still there.

In the light of twistor59's comment: If you are thinking that the light itself should interfere with the radio, then it is true that both light and radio waves are part of the huge EM wave superposition that bathes us all, but radio receivers and electronic circuits in general can only respond to frequencies of a hundred gigahertz at the most. Electronics is hindered by "inertia" effects: oscillations in a circuit are actually a shuttling of electromagnetic energy back and forth between inductive, capacitive and hybrid energy "reservoirs" in the circuit. These reservoirs take time to fill and drain. One can engineer the system so that these reservoirs are small, so that the circuit can respond to high frequency signals, but this it is simply not practical to enable macroscopic, everyday life size circuits to respond to higher frequencies than 100GHz. The current pinnacle of this kind of technology is the Gigahertz Oscilloscope and this uses highly specialized microwave engineering. House table radios respond to frequencies of up to merely about 1GHz. Seeable light on the other hand is at a frequency of roughly 600THz, or four orders of magnitude faster than any electronic circuit can respond to and about six orders of magnitude faster than the fastest home radio can respond to. The "receivers" of light radiation indeed tend to be atomic or molecular systems, whose individual electrons absorb and emit optical photons.

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I think the OP's example of the light should be interpreted as "light is EM wave, radio is EM wave, why don't they interfere"? Your first sentence is more about the spectral splatter from the transient when you switch a light on (which indeed is the real significant effect, but I had the impression that they had something else in mind in the question). But I may be wrong! –  twistor59 Aug 26 '13 at 6:39
    
@twistor59 Actually I think you're right - I've changed my answer. –  WetSavannaAnimal aka Rod Vance Aug 26 '13 at 7:13
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