__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

E_{elec} = q^{2}/8pe_{0}R.
(1)

The electromagnetic mass of such a shell is

m_{elecmag} = q^{2}/6pe_{0}Rc^{2}.
(2)

Thus the total energy is

E_{total} = m_{elecmag}c^{2
}= (4/3)E_{elec}. (3)

E_{total} is *greater*
than E_{elec}! 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 = q^{2}/8pe_{0}R.
(4)

However, in the case of such a
shell we have not explained why E_{total} is *greater*
than E_{elec} by the amount q^{2}/24pe_{0}R

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 E_{elec}. Evidently the work expended for all of the sub-step
2's is

W_{shrink} = E_{total}
- E_{elec} = q^{2}/24pe_{0}R.
(5)

But the question arises:
"How, if not in the electric field, is W_{shrink} 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 E_{elec}, and W_{shrink} 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 E_{elec} and E_{total}
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.