Page 60 - Electromagnetics

B ∂
0 E
∇× = J + ,
µ 0 ∂t
∇· (
0 E) = ρ,
∇· B = 0.
The forms are preserved if we identify ∂P/∂t and ∇× M as new types of current density,
and ∇· P as a new type of charge density. We define
∂P
J P = (2.119)
∂t
as an equivalent polarization current density, and
J M =∇ × M

as an equivalent magnetization current density (sometimes called the equivalent Amperian
currents of magnetized matter [199]). We define
ρ P =−∇ · P

as an equivalent polarization charge density (sometimes called the Poisson–Kelvin equiv-
alent charge distribution [199]). Then the Maxwell–Boffi equations become simply
∂B
∇× E =− , (2.120)
∂t
B ∂
0 E
∇× = (J + J M + J P ) + , (2.121)
µ 0 ∂t
∇· (
0 E) = (ρ + ρ P ), (2.122)
∇· B = 0. (2.123)

Here is the new view. A material can be viewed as composed of charged particles of
matter immersed in free space. When these charges are properly considered as “equiv-
alent” polarization and magnetization charges, all field effects (describable through flux
and force vectors) can be handled by the two fields E and B. Whereas in Minkowski’s
form D diverges from ρ, in Boffi’s form E diverges from a total charge density consisting
of ρ and ρ P . Whereas in the Minkowski form H curls around J, in the Boffi form B curls
around the total current density consisting of J, J M , and J P .
This view was pioneered by Lorentz, who by 1892considered matter as consisting of
bulk molecules in a vacuum that would respond to an applied electromagnetic field [130].
The resulting motion of the charged particles of matter then became another source
term for the “fundamental” fields E and B. Using this reasoning he was able to reduce
the fundamental Maxwell equations to two equations in two unknowns, demonstrating a
simplicity appealing to many (including Einstein). Of course, to apply this concept we
must be able to describe how the charged particles respond to an applied field. Simple
microscopic models of the constituents of matter are generally used: some combination
of electric and magnetic dipoles, or of loops of electric and magnetic current.
The Boffi equations are mathematically appealing since they now specify both the curl
and divergence of the two field quantities E and B. By the Helmholtz theorem we know
that a field vector is uniquely specified when both its curl and divergence are given. But
this assumes that the equivalent sources produced by P and M are true source fields in
the same sense as J. We have precluded this by insisting in Chapter 1 that the source
field must be independent of the mediating field it sources. If we view P and M as

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