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Is it possible that mass of an electron is decreased in the electromagnetic field of the atom? Does it follow $E=mc^2$?

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The Lorentz-invariant mass of an electron does not change in an electromagnetic field. (Lorentz-invariant mass is the only kind of mass modern physics uses; “relativistic mass” is today considered obsolete, confusing, and pointless.) The mass of an electron is a fixed number, which you can think of as a constant of nature.

The mass of a hydrogen atom is slightly less than the sum of the masses of a proton and an electron, because their electrostatic potential energy is negative (and twice as negative as their kinetic energy is positive). Furthermore, this mass varies slightly depending on whether the atom is on its ground state or in an excited state. So the mass of a composite system depends on the dynamics of its constituents, including their electromagnetic (and other) interactions.

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