Let us begin by imagining a grand thought experiment where a resting spherical shell of charge q, with finite radius R’, is formed from a spherical shell of charge, q, with an infinite radius. According to classical electromagnetic theory, viewed from the final inertial rest frame K’, the work done to shrink the shell from the one of infinite radius to the final shell of radius, R’, is:
E’=q2/8πεoR’. (1)
Now in general
m=E/c2. (2)
Hence the so-called “electromagnetic mass” (say m’elecmag for short) can be defined to be
m’elecmag= q2/8πεoR’c2. (3)
The momentum of the final shell, at rest in K’, is of course:
px’=py’=pz’=0. (4)
We now switch to frame K, which moves in the -x’ direction of K’ at speed v. According to the Lorentz transformation of energy, from the perspective of frame K the shell will have the energy:
E=γ(E’+vpx’) (5)
=γE’. (1)
Hence we can say that, from the perspective of K,
melecmag=γq2/8πεoR’c2 (6)
=γm’elecmag.
Since
γ>1, (7)
it follows that
melecmag> m’elecmag. (8)
The terms “melecmag” and “m’elecmag” cannot therefore be assumed to signify the same value. Suggesting that they do is referred to as “overloading”, and in the present case doing so is clearly an error in logic.
The author is frankly astonished that Einstein, when considering the “grand thought experiment”, assumed that melecmag and m’elecmag ought to be equal. (The explanation might be that he reportedly believed, for a period of time, that mass is an invariant quantity.) In any case, the idea that melecmag and m’elecmag could serve as symbols for the same physical quantity led to what Feynman described as “great confusion” for Einstein and some of his contemporaries. It seems, from Feynman’s assertion that the confusion is due to a flaw in Maxwellian theory, that he agreed melecmag and m’elecmag ought to be equal.(2)
- Note that this changing of frames is not the same thing as allowing the spherical shell in K’ to expand to a sphere of radius R>R’.
- See The Feynman Lecrures on Physics, chapter 28.