Page 126 - Electromagnetics

Chapter 3

The static electromagnetic field

3.1 Static fields and steady currents

Perhaps the most carefully studied area of electromagnetics is that in which the fields
are time-invariant. This area, known generally as statics, offers (1)the most direct op-
portunities for solution of the governing equations, and (2)the clearest physical pictures
of the electromagnetic field. We therefore devote the present chapter to a treatment
of static fields. We begin to seek and examine specific solutions to the field equations;
however, our selection of examples is shaped by a search for insight into the behavior of
the field itself, rather than a desire to catalog the solutions of numerous statics problems.
We note at the outset that a static field is physically sensible only as a limiting case
of a time-varying field as the latter approaches a time-invariant equilibrium, and then
only in local regions. The static field equations we shall study thus represent an idealized
model of the physical fields.
If we examine the Maxwell–Minkowski equations (2.1)–(2.4) and set the time deriva-
tives to zero, we obtain the static field Maxwell equations

∇× E(r) = 0, (3.1)
∇· D(r) = ρ(r), (3.2)
∇× H(r) = J(r), (3.3)
∇· B(r) = 0. (3.4)

We note that if the fields are to be everywhere time-invariant, then the sources J and
ρ must also be everywhere time-invariant. Under this condition the dynamic coupling
between the fields described by Maxwell’s equations disappears; any connection between
E, D, B, and H imposed by the time-varying nature of the field is gone. For static fields
we also require that any dynamic coupling between fields in the constitutive relations
vanish. In this static field limit we cannot derive the divergence equations from the curl
equations, since we can no longer use the initial condition argument that the fields were
identically zero prior to some time.
The static field equations are useful for approximating many physical situations in
which the fields rapidly settle to a local, macroscopically-static state. This may occur
so rapidly and so completely that, in a practical sense, the static equations describe the
fields within our ability to measure and to compute. Such is the case when a capacitor
is rapidly charged using a battery in series with a resistor; for example, a 1 pF capacitor
charging through a 1 resistor reaches 99.99% of its total charge static limit within
10 ps.

© 2001 by CRC Press LLC