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Visible light has higher frequency than microwaves and hence higher energy. Then why doesn't it heat up objects like microwave does?

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First note: it is the individual photons of visible light which have higher energy than those of microwaves; but a beam of microwaves could easily have so large a photon flux (photons per second) that the energy it delivers per second is larger than for a beam of light from a torch (flashlight).

Any light that isn't reflected by, or transmitted through, a sample of matter is absorbed by it and does heat it up. A powerful enough beam of light would heat the matter strongly; for example a laser beam used for welding. [Ordinary lights don't produce nearly such powerful beams because we don't need them to see by – such is the sensitivity of our eyes.]

For heating purposes, the big difference between light and microwaves is depth of penetration. A large depth of penetration is equivalent to weak absorption and a small depth of penetration corresponds to strong absorption.

For many foods, microwaves penetrate a few centimetres and therefore heat up quite a volume of the food at the same time, whereas a powerful beam of light wouldn't penetrate far and would heat only a thin surface layer (rather like a toaster, though this gives out mainly near-infrared). [Heat would gradually travel deeper, by conduction, but this is quite a slow process.]

This leads us to ask: what determines how strongly e-m radiation is absorbed by matter? This is a simple question, but the answer isn't simple. Absorption is all about how charged particles, such as electrons or polarised molecules, in the material can move under the oscillating electric field of the waves (or equivalently when the particles encounter photons in the e-m radiation). This in turn depends on how the atoms are arranged and bonded together – we're in the realm of chemical physics. A basic treatment predicts strong absorption when the frequency of the radiation is near a natural frequency of oscillation of certain particles in the material, and there can be several such frequencies. As I said – not simple!

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    $\begingroup$ For water, in particular, rotational modes for water molecules correspond to microwave frequencies, while vibrational modes correspond to infrared frequencies. $\endgroup$ – probably_someone Jan 22 '18 at 15:12
  • $\begingroup$ Indeed, though it's a common misconception that the frequency of the microwaves (about 2.4 GHz) is chosen to be $equal\ to$ the natural frequency of libration of the water molecules. [If that were the case the absorption would be so strong that a microwave oven would act like a toaster, with the food absorbing almost all the energy in a thin surface layer.] $\endgroup$ – Philip Wood Jan 22 '18 at 15:31

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