Page 72 - Electromagnetics Handbook

2.6 The nature of the four field quantities

Since the very inception of Maxwell’s theory, its students have been distressed by the
fact that while there are four electromagnetic fields (E, D, B, H), there are only two funda-
mental equations (the curl equations) to describe their interrelationship. The relegation
of additional required information to constitutive equations that vary widely between
classes of materials seems to lessen the elegance of the theory. While some may find
elegant the separation of equations into a set expressing the basic wave nature of electro-
magnetism and a set describing how the fields interact with materials, the history of the
discipline is one of categorizing and pairing fields as “fundamental” and “supplemental”
in hopes of reducing the model to two equations in two unknowns.
Lorentz led the way in this area. With his electrical theory of matter, all material ef-
fects could be interpreted in terms of atomic charge and current immersed in free space.
We have seen how the Maxwell–Boffi equations seem to eliminate the need for D and H,
and indeed for simple media where there is a linear relation between the remaining “fun-
damental” fields and the induced polarization and magnetization, it appears that only
E and B are required. However, for more complicated materials that display nonlinear
and bianisotropic effects we are only able to supplant D and H with two other fields P
and M, along with (possibly complicated) constitutive relations relating them to E and
B.
Even those authors who do not wish to eliminate two of the fields tend to categorize
the fields into pairs based on physical arguments, implying that one or the other pair
is in some way “more fundamental.” Maxwell himself separated the fields into the pair
(E, H) that appears within line integrals to give work and the pair (B, D) that appears
within surface integrals to give flux. In what other ways might we pair the four vectors?
Most prevalent is the splitting of the fields into electric and magnetic pairs: (E, D) and
(B, H). In Poynting’s theorem E · D describes one component of stored energy (called
“electric energy”) and B · H describes another component (called “magnetic energy”).
These pairs also occur in Maxwell’s stress tensor. In statics, the fields decouple into
electric and magnetic sets. But biisotropic and bianisotropic materials demonstrate how
separation into electric and magnetic effects can become problematic.
In the study of electromagnetic waves, the ratio of E to H appears to be an important
quantity, called the “intrinsic impedance.” The pair (E, H) also determines the Poynting
flux of power, and is required to establish the uniqueness of the electromagnetic field.
In addition, constitutive relations for simple materials usually express (D, B) in terms
of (E, H). Models for these materials are often conceived by viewing the fields (E, H)
as interacting with the atomic structure in such a way as to produce secondary effects
describable by (D, B). These considerations, along with Maxwell’s categorization into
a pair of work vectors and a pair of flux vectors, lead many authors to formulate elec-
tromagnetics with E and H as the “fundamental” quantities. But the pair (B, D) gives
rise to electromagnetic momentum and is also perpendicular to the direction of wave
propagation in an anisotropic material; in these senses, we might argue that these fields
must be equally “fundamental.”
Perhaps the best motivation for grouping fields comes from relativistic considerations.
We have found that (E, B) transform together under a Lorentz transformation, as do
(D, H). In each of these pairs we have one polar vector (E or D) and one axial vector (B
or H). A polar vector retains its meaning under a change in handedness of the coordinate
system, while an axial vector does not. The Lorentz force involves one polar vector (E)

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