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Microwaves are electromagnetic waves, they have wavelengths ranging from mm to cm scales. The electromagnetic spectrum includes radio, infra-red, visible light and ultra-violet light waves. The waves are energy and are made up of photons (discrete packets of energy).

The wavelengths of microwaves are easily absorbed by water molecules (just like how light is absorbed by a black object).

As microwave energy is shone through food, the water molecules take the microwave (light) energy and vibrate. This increased molecular movement shows itself to us on the large scale as heat energy.

The microwaves only penetrate a few centimeters into the food before they are all absorbed by water molecules. The heating to the center of food occurs via conduction (as well as convection for fluids such as soups and sauces).

Thermal conduction is where molecules vibrate and bang in to nearby molecules - thus causing them to vibrate- and so on in a Mexican wave of heat transfer.

Molecules in fluids (liquids and gases) can move more freely than in solids- so instead of just vibrating they can also move about in all directions. As hot fluids are less dense than cold fluids (because the particles move faster and push each other further apart), so they tend to rise and cause the whole fluid to mix quite well. This convection occurs as well as conduction in liquids.
It heats up the food from the heat that surrounds in side the space during the minutes or seconds the food is in the microwave.
From what I have learnt in Physics, Microwaves heat up the water molecules within a substance and that is what heats it up. We we're told that it would heat up the food from the inside out. But that is only from a GCSE level student so I doubt someone else will be able to give a more detailed and sophisticated answer in time :-)
Radiation.

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