Page 34 - Electromagnetics

clearly by Tai [192], who also notes that if the integral forms are used, then their validity
across regions of discontinuity should be stated as part of the postulate.
We have decided to use the point form in this text. In doing so we follow a long
history begun by Hertz in 1890 [85] when he wrote down Maxwell’s differential equations
as a set of axioms, recognizing the equations as the launching point for the theory of
electromagnetism. Also, by postulating Maxwell’s equations in point form we can take
full advantage of modern developments in the theory of partial differential equations; in
particular, the idea of a “well-posed” theory determines what sort of information must
be specified to make the postulate useful.
We must also decide which form of Maxwell’s differential equations to use as the basis
of our postulate. There are several competing forms, each differing on the manner in
which materials are considered. The oldest and most widely used form was suggested
by Minkowski in 1908 [130]. In the Minkowski form the differential equations contain
no mention of the materials supporting the fields; all information about material media
is relegated to the constitutive relationships. This places simplicity of the differential
equations above intuitive understanding of the behavior of fields in materials. We choose
the Maxwell–Minkowski form as the basis of our postulate, primarily for ease of ma-
nipulation. But we also recognize the value of other versions of Maxwell’s equations.
We shall present the basic ideas behind the Boffi form, which places some information
about materials into the differential equations (although constitutive relationships are
still required). Missing, however, is any information regarding the velocity of a moving
medium. By using the polarization and magnetization vectors P and M rather than the
fields D and H, it is sometimes easier to visualize the meaning of the field vectors and
to understand (or predict) the nature of the constitutive relations.
The Chu and Amperian forms of Maxwell’s equations have been promoted as useful
alternatives to the Minkowski and Boffi forms. These include explicit information about
the velocity of a moving material, and differ somewhat from the Boffi form in the physical
interpretation of the electric and magnetic properties of matter. Although each of these
models matter in terms of charged particles immersed in free space, magnetization in the
Boffi and Amperian forms arises from electric current loops, while the Chu form employs
magnetic dipoles. In all three forms polarization is modeled using electric dipoles. For a
detailed discussion of the Chu and Amperian forms, the reader should consult the work
of Kong [101], Tai [193], Penfield and Haus [145], or Fano, Chu and Adler [70].
Importantly, all of these various forms of Maxwell’s equations produce the same values
of the physical fields (at least external to the material where the fields are measurable).
We must include several other constituents, besides the field equations, to make the
postulate complete. To form a complete field theory we need a source field, a mediating
field, and a set of field differential equations. This allows us to mathematically describe
the relationship between effect (the mediating field) and cause (the source field). In
a well-posed postulate we must also include a set of constitutive relationships and a
specification of some field relationship over a bounding surface and at an initial time. If
the electromagnetic field is to have physical meaning, we must link it to some observable
quantity such as force. Finally, to allow the solution of problems involving mathematical
discontinuities we must specify certain boundary, or “jump,” conditions.

2.1.1 The Maxwell–Minkowski equations

In Maxwell’s macroscopic theory of electromagnetics, the source field consists of the
vector field J(r, t) (the current density) and the scalar field ρ(r, t) (the charge density).

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