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An electron at rest creates a static distribution of an electric field around it. Wherever the particle moves, it carries or drags the field with itself. Does this electric field contribute to the mass/inertia of the particle? A particle can have mass even if it does not have any electric charge. Does the presence of a field modifies its mass or inertia? Please do not give mechanical analogies.

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For bodies composed of several charged parts, yes, EM interaction between those parts leads to increase of effective inertial mass, proportional to EM energy of the body.

For electron, no such parts were ever found, so we do not know whether and how big part of its mass is due to this effect. Electron's mass could be in part electromagnetic, but it could also be completely non-electromagnetic.

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