4

4. The Total Energy of a Distribution of Charge.

A consequence of Maxwell’s equations is that the electric field inside a resting spherical shell of charge (say one with radius R) is zero. Outside, the field is the same as that of a point charge at the shell’s center. The total energy in the shell’s electric field is thus

. (4_1)

Before the advent of "E=mc²" it was often stated that q²/8pe_oR is the work that would need to be expended to assemble the spherical shell from infinitely dispersed charge. The basis for this assertion lies in the fact that the work expended to change the separation between two spherical shells of charge (for example) equals the change in their collective electric field energy. q²/8pe_oR was accordingly said to be the energy of the resting charge.

After "E=mc²" was established, a matter of much speculation and discussion was the fact that m_ElecMagc² is greater than U_Elec:

. (4_2)

According to "E=mc²" there is an energy surplus:

. (4_3)

The question is: why the surplus?

A useful analogy in understanding this celebrated inequality is an inflated rubber balloon. At initial radius R the compressed air inside the balloon exerts an outward force of magnitude dF on each infinitesimal surface area increment dA. In order to increase the balloon’s radius by an amount dR, each dF must act through a displacement dR. But does (dF)(dR) account for all the work that must be done on surface increment dA? In fact it does not, for the following reason: the area of the surface increment must not only be displaced outward; it must be increased incrementally. That is, work must also be done to stretch the increment of rubber in its plane. The total work expended on area increment dA is thus

. (4_4)

The same reasoning theoretically applies to decreasing the radius of a spherical shell of charge. In decreasing R, some implied agent must not only push each surface charge increment closer to the shell’s center, but it must compress each charge increment to a slightly greater surface charge density.

Now U_Elec (or q²/8pe_oR) is the work needed to push a set of point charges in from infinity to R. But in the case of a spherical shell, of continuous charge density s(R), work must also be expended to increase s from its initial value of zero (at infinite R).

This additional work is not manifest as a change in the electric field energy. But thanks to "E=mc²" we now know what it is. The work expended to raise s from zero to a final value of q/4pR² is just

. (4_5)

The total work, expended to "assemble" the spherical shell of charge from infinitely dispersed charge (or from a spherical shell of infinite R and zero s) is therefore

(4_6)

The explanation for this inequality of U_Elec and m_ElecMagc² has only recently been suggested (or at least re-suggested). As recently as 1964 no less an authority than the late R.P.Feynman stated that Maxwellian theory "ultimately falls on its face." (See The Feynman Lectures on Physics, Volume II, Sect. 28-1.)

The aforementioned discussion can of course be extended to other charge distributions (e.g. solid spheres of charge). In short, the total energy of any resting distribution of charge consists of two parts: (1) the energy in the distribution’s electrostatic field, and (2) the stress energy created when the distribution of nonzero charge density is assembled from zero charge density.

Historically, Poincare seemed to have anticipated the physical basis of the surplus energy, and his explanation was given the name "Poincare stresses." Like more palpable media (such as rubber membranes), electric charge evidently possesses elastic properties. Some implied agent must do work (positive or negative) to raise or lower the density of any given charge increment. This work is in addition to any work that must be expended to move the increment closer to (or farther away from) the other increments in the distribution.