Why the Total Energy of a Charge is Greater than its Electric Field Energy

Suggested background reading: The Feynman Lectures On Physics, V2, Sect 28.3.

Abstract. Discusses the reason why Poincare Stresses exist in a finite charge distribution that is assembled from infinitely dispersed charge. 

In this section we seek a solution to a famous "problem." According to Einstein, the energy equivalent of any mass is given by

. (1)

Specifically, the energy of a spherical shell of charge, q, of radius R, is



This is presumably the work that some constraining agent would have to do in order to assemble the spherical shell from infinitely dispersed charge. One of the elegant things about two point charges of like sign is that the work expended to bring them closer together equals the gain in electric field energy.

Now the electric field energy of the spherical shell of radius R is

, (3)

which is less than Etotal in Eq. 2. That is,

. (4)

We would like to see what the q2/24peoR term might be attributed to. The term evidently caused great confusion in the minds of Einstein, Lorentz, and other heirs to Maxwell's theory.

Let us begin by considering the reverse process, where a spherical shell of radius R and charge density si=q/4pR2 is allowed to expand to a spherical shell of radius R+dR and charge densitysf=q/4p(R+dR)2. We shall accomplish this expansion in two steps.

In Step 1 we divide the spherical surface up into an infinite number of infinitesimal square elements, each with infinitesimal charge dq, and then have the constraining agent allow each dq to move radially outward a distance dR. At the end of Step 1 we have a spherical "patchwork quilt" of dqs, with each dq separated from its neighbors by charge-free space. Fig. 1 illustrates a small part of the total spherical array at the conclusion of Step 1.

Figure 1

Close up at End of Step 1.

At the beginning of Step 1 the electric field at the inner surface of any dq is zero. And at any dqs outer surface it is q/4peoR2. Thus the radially outward electric force experienced by dq has magnitude



And of course the constraining agent keeps the spherical shell from flying apart by counteracting each such dF.

The total (negative) work done by the agent on a given dq in Step 1 is therefore

. (6)

Now the electric field energy of a uniform spherical shell of charge, q, with radius R+dR and charge density q/4p(R+dR)2, is

 . (7)

Thus (referring to Eq. 3) the difference between this shells electric field energy and the initial shells field energy is



And, subtracting this dEelec from dW1 in Eq. 6 produces



We thus find that the change in electric field energy, in going from the initial, uniform spherical shell to the "patchwork quilt" (partially depicted in Fig. 1) is the same as the electric field energy difference between uniform spherical shells of charge q and radii R+dR and R.

But we must still go from the "patchwork quilt" to a uniform spherical shell! In other words, the agent must allow each dq to expand until all of the charge-free space in Fig. 1 disappears. The agent must also do negative work during this second step, say dW2. But Step 2 entails no change in electric field energy. In other words,

. (10)

Or, repeating the process out to infinite R (and infinitesimal s),

. (11)

If we reverse the whole process and repeat it an infinite number of times, pushing an infinite set of dqs (each initially infinite in size and with infinitesimal charge density) in from infinity and shrinking each one as necessary, then the total work expended, pushing them inward, is just the gain in electric field energy:

. (12)

And as Eq. 1 suggests, the total work to shrink all the dqs (as required at the conclusion of each dR increment), so as to make up uniform spherical shells all along the way, is presumably

. (13)

This latter "shrinking" work never entails a change in the electric field energy.

Total work is thus



A useful (albeit reversed) analogy might be a spherical balloon. Each increment of surface area experiences a net force radially inward (owing to the surfaces curvature), and this inward force must be counteracted by the pressure (P) inside the balloon. In order to push each surface increment outward a distance dR, a force PdA must be applied, and work d2W1=P dA dR must be done. But each surface increment must also be expanded; d2W2 must be done to stretch the rubber. The total work is a combination of the work done to push all the surface increments outward, plus the energy expended to stretch all the surface increments.

It was perhaps with an analogy like this one in mind that Poincare suggested that the q2/24peoR term, in the case of a spherical shell of charge, is needed to increase stresses within the charge (when the radius is decreased). These stresses are accordingly called "Poincare stresses."  The important point is that a change in these stresses (Step 2 in this discussion) entails no change in Eelec, although it most assuredly entails a change in Etotal.