Page 99 - Electromagnetics

Interpretation of the energyand momentum conservation theorems. There
has been some controversy regarding Poynting’s theorem (and, equally, the momentum
conservation theorem). While there is no question that Poynting’s theorem is mathe-
matically correct, we may wonder whether we are justified in associating W em with W k
and S em with S k merely because of the similarities in their mathematical expressions.
Certainly there is some justification for associating W k , the kinetic energy of particles,
1
with W em , since we shall show that for static fields the term (D · E + B · H) represents
2
the energy required to assemble the charges and currents into a certain configuration.
However, the term S em is more problematic. In a mechanical system, S k represents the
flow of kinetic energy associated with moving particles — does that imply that S em rep-
resents the flow of electromagnetic energy? That is the position generally taken, and it is
widely supported by experimental evidence. However, the interpretation is not clear-cut.
If we associate S em with the flow of electromagnetic energy at a point in space, then
we must define what a flow of electromagnetic energy is. We naturally associate the
flow of kinetic energy with moving particles; with what do we associate the flow of
electromagnetic energy? Maxwell felt that electromagnetic energy must flow through
space as a result of the mechanical stresses and strains associated with an unobserved
substance called the “aether.” A more modern interpretation is that the electromagnetic
fields propagate as a wave through space at finite velocity; when those fields encounter a
charged particle a force is exerted, work is done, and energy is “transferred” from the field
to the particle. Hence the energy flow is associated with the “flow” of the electromagnetic
wave.
Unfortunately, it is uncertain whether E × H is the appropriate quantity to associate
with this flow, since only its divergence appears in Poynting’s theorem. We could add

any other term S that satisfies ∇·S = 0 to S em in (2.297), and the conservation theorem

would be unchanged. (Equivalently, we could add to (2.299) any term that integrates to
zero over S.) There is no such ambiguity in the mechanical case because kinetic energy
is rigorously defined. We are left, then, to postulate that E × H represents the density
of energy flow associated with an electromagnetic wave (based on the symmetry with
mechanics), and to look to experimental evidence as justification. In fact, experimental
evidence does point to the correctness of this hypothesis, and the quantity E×H is widely
and accurately used to compute the energy radiated by antennas, carried by waveguides,
etc.
Confusion also arises regarding the interpretation of W em . Since this term is so con-
veniently paired with the mechanical volume kinetic energy density in (2.300) it would
seem that we should interpret it as an electromagnetic energy density. As such, we can
think of this energy as “localized” in certain regions of space. This viewpoint has been
criticized [187, 145, 69] since the large-scale form of energy conservation for a space re-
gion only requires that the total energy in the region be specified, and the integrand
(energy density) giving this energy is not unique. It is also felt that energy should be
associated with a “configuration” of objects (such as charged particles) and not with an
arbitrary point in space. However, we retain the concept of localized energy because it
is convenient and produces results consistent with experiment.
The validity of extending the static field interpretation of

1
(D · E + B · H)
2
as the energy “stored” by a charge and a current arrangement to the time-varying case
has also been questioned. If we do extend this view to the time-varying case, Poynting’s
theorem suggests that every point in space somehow has an energy density associated

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