Page 19 - Electromagnetics Handbook

of the supporting medium on the fields and are dependent upon the physical state of
the medium. The state may include macroscopic effects, such as mechanical stress and
thermodynamic temperature, as well as the microscopic, quantum-mechanical properties
of matter.
The value of the field at any position and time in a bounded region V is then determined
uniquely by specifying the sources within V , the initial state of the fields within V , and
the value of the field or finitely many of its derivatives on the surface bounding V .If
the boundary surface also defines a surface of discontinuity between adjacent regions of
differing physical characteristics, or across discontinuous sources, then jump conditions
may be used to relate the fields on either side of the surface.
The variety of forms of field equations is restricted by many physical principles in-
cluding reference-frame invariance, conservation, causality, symmetry, and simplicity.
Causality prevents the field at time t = 0 from being influenced by events occurring at
subsequent times t > 0. Of course, we prefer that a field equation be mathematically
robust and well-posed to permit solutions that are unique and stable.
Many of these ideas are well illustrated by a consideration of electrostatics. We can
describe the electrostatic field through a mediating scalar field (x, y, z) known as the
electrostatic potential. The spatial distribution of the field is governed by Poisson’s
equation
2
2
2
∂ ∂ ∂ ρ
+ + =− , θ
∂x 2 ∂y 2 ∂z 2 0
where ρ = ρ(x, y, z) is the source charge density. No temporal derivatives appear, and the
spatial derivatives determine the spatial behavior of the field. The function ρ represents
the spatially-averaged distribution of charge that acts as the source term for the field .
Note that ρ incorporates no information about . To uniquely specify the field at any
point, we must still specify its behavior over a boundary surface. We could, for instance,
specify on five of the six faces of a cube and the normal derivative ∂ /∂n on the
remaining face. Finally, we cannot directly observe the static potential field, but we can
observe its interaction with a particle. We relate the static potential field theory to the
realm of mechanics via the electrostatic force F = qE acting on a particle of charge q.
In future chapters we shall present a classical field theory for macroscopic electromag-
netics. In that case the mediating field quantities are E, D, B, and H, and the source
field is the current density J.

1.3 The sources of the electromagnetic field
Electric charge is an intriguing natural entity. Human awareness of charge and its
effects dates back to at least 600 BC, when the Greek philosopher Thales of Miletus
observed that rubbing a piece of amber could enable the amber to attract bits of straw.
Although charging by friction is probably still the most common and familiar manifes-
tation of electric charge, systematic experimentation has revealed much more about the
behavior of charge and its role in the physical universe. There are two kinds of charge, to
which Benjamin Franklin assigned the respective names positive and negative. Franklin
observed that charges of opposite kind attract and charges of the same kind repel. He
also found that an increase in one kind of charge is accompanied by an increase in the

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