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On the Absorption/Emission of Energy by Electric Charge

The electric field energy of a uniform spherical shell, of charge q and radius R, is

Eelec =  q2/8pe0R.    (1)

The electromagnetic mass of such a shell is

melecmag = q2/6pe0Rc2.    (2)

Thus the total energy is

Etotal = melecmagc2 = (4/3)Eelec.    (3)

Etotal is greater than Eelec! This inequality purportedly caused much confusion when it was first derived.

In this article it is shown that the inequality is expected, provided charge  always exists as a continuum in space

Now If a spherical shell of charge is modeled as a collection of infinitesimal point charges, initially distributed evenly  over a spherical surface of infinite radius, then the total work expended to bring such a set of infinitely dispersed charges "in from infinity" and distribute them over the spherical surface of finite R is indeed simply

W = q2/8pe0R.    (4)

However, in the case of such a shell we have not explained why Etotal is greater than Eelec by the amount q2/24pe0R

Let us suppose that our initial spherical shell is not a collection of point charges, but is rather one of continuous charge with an infinite radius R and an infinitesimal surface charge density s. What amount of work must be done to shrink this shell down to one with our final values of R and s?

We can imagine that this shrinking process occurs in an infinite number of tiny steps, and that each step consists of two sub-steps: (1) An infinitesimal patch of surface charge must be pushed closer to the sphere's center, and (2) the patch must be shrunk so as not to overlap other patches.

Now hypothetically a small amount of work must be expended for each sub-step. When all of the infinitesimal amounts of work expended in sub-step 1 are added, we expect the sum to equal Eelec. Evidently the work expended for all of the sub-step 2's is

Wshrink = Etotal - Eelec = q2/24pe0R.    (5)

But the question arises: "How, if not in the electric field, is Wshrink manifest?" The answer would appear to be that it is manifest as stresses in the spherical surface of charge. These stresses (called Poincare stresses) would be analogous (but oppositely directed) to the surface tensions in a bubble of soapy water. The important point is that no such stresses are created when the initial and final shells are collections of point charges. They are created only when the spherical shell is shrunk down to a finite radius, R.

Before concluding this discussion it may be instructive to consider cases where the total charge of our final spherical shell is still q, but the surface charge density s is not single-valued. For example, s in a final shell's upper hemisphere might be greater than that in the lower hemisphere. In such cases we expect Eelec, and Wshrink to be greater than in the single-valued s case. Evidently the absorption and emission of energy can result in changes in s, even though q and R remain the same.

With remarkable insight Maxwell envisioned charge as always being a continuum whose density might be a function of position and time. The inequality of Eelec and Etotal suggests that charge never exists as a finite point. Although e (the fundamental quantum of charge) may be small, the charge of any particle (including quarks) must be continuously "smeared out" in space and time.

Of course the above discussion also applies to charge distributions in 3-dimensional volumes (such as solid spheres of charge). In such cases the parameter of interest would be r, the volume density. Here again a charge can absorb/emit energy by varying its density at internal points, even though q and R remain unchanged in the process. The possibility  constitutes a significant addition to (for example) the Bohr model, where the absorption and emission of energy is accompanied solely by changes in particle orbits. The added or alternate idea, that such emissions/absorptions might be manifest as changes in internal densities of charge, with little significant change in orbital parameters, might be particularly relevant in nuclear physics.