Page 59 - Electromagnetics

The use of P and M in place of D and H is sometimes called an application of the principle
of Ampere and Lorentz [199].
Let us examine the ramification of using (2.113)–(2.116) as the basis for a postulate
of electromagnetics. These equations are similar to the Maxwell–Minkowski equations
used earlier; must we rebuild all the underpinning of a new postulate, or can we use
our original arguments based on the Minkowski form? For instance, how do we invoke
uniqueness if we no longer have the field H? What represents the flux of energy, formerly
found using E×H? And, importantly, are (2.113)–(2.114) form invariant under a Lorentz
transformation?
It turns out that the set of vector fields (E, B, P, M) is merely a linear mapping of
the set (E, D, B, H). As pointed out by Tai [193], any linear mapping of the four field
vectors from Minkowski’s form onto any other set of four field vectors will preserve the
covariance of Maxwell’s equations. Boffi chose to keep E and B intact and to introduce
only two new fields; he could have kept H and D instead, or used a mapping that
introduced four completely new fields (as did Chu). Many authors retain E and H.
This is somewhat more cumbersome since these vectors do not convert as a pair under
a Lorentz transformation. A discussion of the idea of field vector “pairing” appears in
§ 2.6.
The usefulness of the Boffi form lies in the specific mapping chosen. Comparison of
(2.113)–(2.116) to (2.1)–(2.4) quickly reveals that

P = D −
0 E, (2.117)
M = B/µ 0 − H. (2.118)

We see that P is the difference between D in a material and D in free space, while M is
the difference between H in free space and H in a material. In free space, P = M = 0.

Equivalent polarization and magnetization sources. The Boffi formulation pro-
vides a new way to regard E and B. Maxwell grouped (E, H) as a pair of “force vectors” to
be associated with line integrals (or curl operations in the point forms of his equations),
and (D, B) as a pair of “flux vectors” associated with surface integrals (or divergence
operations). That is, E is interpreted as belonging to the computation of “emf” as a line
integral, while B is interpreted as a density of magnetic “flux” passing through a surface.
Similarly, H yields the “mmf” about some closed path and D the electric flux through
a surface. The introduction of P and M allows us to also regard E as a flux vector and
B as a force vector — in essence, allowing the two fields E and B to take on the duties
that required four fields in Minkowski’s form. To see this, we rewrite the Maxwell–Boffi
equations as

∂B
∇× E =− ,
∂t
B ∂P ∂
0 E
∇× = J +∇ × M + + ,
µ 0 ∂t ∂t
∇· (
0 E) = (ρ −∇ · P),
∇· B = 0,
and compare them to the Maxwell–Minkowski equations for sources in free space:

∂B
∇× E =− ,
∂t

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