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It is important to note that condition of being at rest in the
system's CM is not enough to determine its rest energy.
If the parts of the system interact with one another, then the
distance between them is very important.
The field energy of e-p pair is much greater when the members of the
pair are widely separated than when brought to rest close together.
By bringing them close together while still keeping them at rest
(call this process "adiabatic contraction") we can make their net EM
field negligible.