Page 102 - Electromagnetics

Finally, using ˆ n 12 × (H 1 − H 2 ) = J s we arrive at (2.305).
The arguments above suggest an interesting way to look at the boundary conditions.
Once we identify S with the flow of electromagnetic energy, we may consider the condition
on normal S as a fundamental statement of the conservation of energy. This statement
implies continuity of tangential E in order to have an unambiguous interpretation for the
meaning of the term J s · E. Then, with continuity of tangential E established, we can
derive the condition on tangential H directly.

An alternative formulation of the conservation theorems. As we saw in the
paragraphs above, our derivation of the conservation theorems lacks strong motivation.
We manipulated Maxwell’s equations until we found expressions that resembled those
for mechanical momentum and energy, but in the process found that the validity of the
expressions is somewhat limiting. For instance, we needed to assume a linear, homoge-
neous, bianisotropic medium in order to identify the Maxwell stress tensor (2.288) and
the energy densities in Poynting’s theorem (2.299). In the end, we were reduced to pos-
tulating the meaning of the individual terms in the conservation theorems in order for
the whole to have meaning.
An alternative approach is popular in physics. It involves postulating a single La-
grangian density function for the electromagnetic field, and then applying the stationary
property of the action integral. The results are precisely the same conservation expres-
sions for linear momentum and energy as obtained from manipulating Maxwell’s equa-
tions (plus the equation for conservation of angular momentum), obtained with fewer
restrictions regarding the constitutive relations. This process also separates the stored
energy, Maxwell stress tensor, momentum density, and Poynting vector as natural com-
ponents of a tensor equation, allowing a better motivated interpretation of the meaning
of these components. Since this approach is also a powerful tool in mechanics, its ap-
plication is more strongly motivated than merely manipulating Maxwell’s equations. Of
course, some knowledge of the structure of the electromagnetic field is required to provide
an appropriate postulate of the Lagrangian density. Interested readers should consult
Kong [101], Jackson [91], Doughty [57], or Tolstoy [198].

2.10 The wave nature of the electromagnetic field
Throughout this chapter our goal has been a fundamental understanding of Maxwell’s
theory of electromagnetics. We have concentrated on developing and understanding the
equations relating the field quantities, but have done little to understand the nature of
the field itself. We would now like to investigate, in a very general way, the behavior
of the field. We shall not attempt to solve a vast array of esoteric problems, but shall
instead concentrate on a few illuminating examples.
The electromagnetic field can take on a wide variety of characteristics. Static fields
differ qualitatively from those which undergo rapid time variations. Time-varying fields
exhibit wave behavior and carry energy away from their sources. In the case of slow
time variation this wave nature may often be neglected in favor of the nearby coupling
of sources we know as the inductance effect, hence circuit theory may suffice to describe
the field-source interaction. In the case of extremely rapid oscillations, particle concepts
may be needed to describe the field.

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