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Electromagnetic waves are caused by changing electric and magnetic fields, and these are caused by a charge possible oscillating like an antenna or a varying current etc.

My question is, with the sun, where is this source that causes the electric and magnetic fields to oscillate.

Everywhere I've read stated that it was due to the energy released from nuclear fusion, but when looking at the process of nuclear fusion there are no charges produced. How is this 'energy' supposedly producing the same effect as an electron oscillating?

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This is not technically very exact since the reasoning is classical but that energy you're talking about mostly turns into heat and when something is hot it's constituents; ionic cores and electrons, move around, collide, and oscillate violently and since these constituents are charged you essentially have a bunch of oscillating charges sending out light since that is what accelerating charges do.

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The Sun is made of plasma, which is a gas of bare ions and electrons. The energy released from nuclear fusion heats this plasma, and the moving charges emit electromagnetic waves.

Note that in other, colder radiating bodies (like a tungsten filament or a human body), the radiation is emitted by other means; in this case, electron transitions in atoms emit photons of various energy levels, and since each atom's energy level is affected by its environment, the discrete levels smooth out into what we call the blackbody curve.

Interestingly enough, despite the clear difference in mechanism at the molecular level, all of the systems I just described (including the Sun) follow the same approximate law: the blackbody curve! This is because the approximation in question is derived from statistical mechanics, which tends to ignore the exact mechanisms and motions at the microscopic level in favor of a macroscopic description.

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The visible, UV, and IR radiation which reaches the earth from the sun is due to blackbody radiation from the surface (photosphere and chromosphere) of the sun. You can see more about blackbody radiation here and in the answer by @annav here. Basically, temperature indicates molecular or atomic vibration, which means moving charges. Higher temperature means more movement which means more EM radiation. Anna's link gives a fuller picture.

The fusion process, deep in the core of the sun, transfers mass-energy into both thermal (aka, atomic kinetic energy) and photons (which are EM energy bundles emitted from nuclei when the nuclei re-arrange during decays and reactions). Fusion involves strong, weak, and electromagnetic interactions, so the emission of photons is not surprising. All of this energy takes a long time to transition to the surface due to the high densities of the inner layers of the sun, plus it gets spread out over a much larger volume, so the temperature at the surface (~6000 K) is much lower than the temperature in the core (~15 MK).

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