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Assuming I shoot a beam of light to an electron, the electron will take the energy and reemit EMR at all direction randomly? If it will, is it happening at the same time?

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For a classical treatment see Thomson scattering. For a QM treatment see Compton scattering. – John Rennie Mar 6 '14 at 14:16
"The moving particle radiates most strongly in a direction perpendicular to its motion and that radiation will be polarized along the direction of its motion" so it means EMR aren't emitted in all direction at the same time if only one incident photon coming in – user40003 Mar 6 '14 at 14:32

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