. (5_1)

The motivation for this formula is that if we initially have 2 distributions of charge (say 2 spherical shells of radius R) separated by distance D, and if we alter the distance between them, then the gain or loss of field energy precisely equals the positive or negative work we expend to make the change. (Note the implicit assumption that the dimensions of the distributions do not change.)

Quite often it is erroneously supposed that the field energy in a single spherical shell equals the work that must be expended to assemble the shell from infinitely dispersed charge. But it is readily appreciated that more work than the energy in the final field must be expended, in order to assemble the shell.

This can best be appreciated by considering a small change from radius r to radius r-dr. Let us divide the initial surface into N increments. We do a certain amount of work to push the N increments a distance dr toward the center. But we end up with overlaps at the increment fringes! In order to re-establish spherical symmetry, we must compress each increment in its plane. Having done that, we find that the gain in field energy equals the work done to push the N increments inward a distance dr. The work done to compress the N increments is "hidden" in elastic stresses in the surface of the shell.

It is not difficult to show that the total field energy in a shell of charge q and radius R is

. (5_2)

. (5_3)

Thus the stress energy is theoretically

. (5_4)

For a time it was believed that the total work needed to assemble a shell from infinitely dispersed charge must just equal the work in the final field, which is to say the work expended to push the charge increments in without compressing them. Poincare was first to recognize that the stresses in any distribution of charge also constitute a form of electromagnetic energy.