Page 36 - Electromagnetics

by (B.49). This requires that ∇· B be constant with time, say ∇· B(r, t) = C B (r).
The constant C B must be specified as part of the postulate of Maxwell’s theory, and
the choice we make is subject to experimental validation. We postulate that C B (r) = 0,
which leads us to (2.4). Note that if we can identify a time prior to which B(r, t) ≡ 0,
then C B (r) must vanish. For this reason, C B (r) = 0 and (2.4) are often called the “initial
conditions” for Faraday’s law [159]. Next we take the divergence of (2.2) to find that

∂
∇· (∇× H) =∇ · J + (∇· D).
∂t
Using (2.5) and (B.49), we obtain

∂
(ρ −∇ · D) = 0
∂t
and thus ρ −∇ · D must be some temporal constant C D (r). Again, we must postulate
the value of C D as part of the Maxwell theory. We choose C D (r) = 0 and thus obtain
Gauss’s law (2.3). If we can identify a time prior to which both D and ρ are everywhere
equal to zero, then C D (r) must vanish. Hence C D (r) = 0 and (2.3) may be regarded
as “initial conditions” for Ampere’s law. Combining the two sets of initial conditions,
we find that the curl equations imply the divergence equations as long as we can find a
time prior to which all of the fields E, D, B, H and the sources J and ρ are equal to zero
(since all the fields are related through the curl equations, and the charge and current are
related through the continuity equation). Conversely, the empirical evidence supporting
the two divergence equations implies that such a time should exist.
Throughout this book we shall refer to the two curl equations as the “fundamental”
Maxwell equations, and to the two divergence equations as the “auxiliary” equations.
The fundamental equations describe the relationships between the fields while, as we
have seen, the auxiliary equations provide a sort of initial condition. This does not
imply that the auxiliary equations are of lesser importance; indeed, they are required
to establish uniqueness of the fields, to derive the wave equations for the fields, and to
properly describe static fields.

Field vector terminology. Various terms are used for the field vectors, sometimes
harkening back to the descriptions used by Maxwell himself, and often based on the
physical nature of the fields. We are attracted to Sommerfeld’s separation of the fields
into entities of intensity (E, B) and entities of quantity (D, H). In this system E is called
the electric field strength, B the magnetic field strength, D the electric excitation, and H
the magnetic excitation [185]. Maxwell separated the fields into a set (E, H) of vectors
that appear within line integrals to give work-related quantities, and a set (B, D) of
vectors that appear within surface integrals to give flux-related quantities; we shall see
this clearly when considering the integral forms of Maxwell’s equations. By this system,
authors such as Jones [97] and Ramo, Whinnery, and Van Duzer [153] call E the electric
intensity, H the magnetic intensity, B the magnetic flux density, and D the electric flux
density.
Maxwell himself designated names for each of the vector quantities. In his classic
paper “A Dynamical Theory of the Electromagnetic Field,” [178] Maxwell referred to
the quantity we now designate E as the electromotive force, the quantity D as the elec-
tric displacement (with a time rate of change given by his now famous “displacement
current”), the quantity H as the magnetic force, and the quantity B as the magnetic

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