Page 91 - Electromagnetics Handbook

(which do not involve k E or k H ) stay constant between problems. We see, for example,
that we may compensate for a halving of the length scale k l by (a) a quadrupling of the
permeability µ, or (b) a simultaneous halving of the time scale k t and doubling of the
conductivity σ. A much less subtle special case is that for which σ = 0, k
=
0 , k µ = µ 0 ,
and
= µ = 1; we then have free space and must simply maintain

k l /k t = constant
so that the time and length scales stay proportional. In the sinusoidal steady state, for
instance, the frequency would be made to vary inversely with the length scale.

2.9.5 Conservation theorems

The misconception that Poynting’s theorem can be “derived” from Maxwell’s equations
is widespread and ingrained. We must, in fact, postulate the idea that the electromagnetic
field can be associated with an energy flux propagating at the speed of light. Since
the form of the postulate is patterned after the well-understood laws of mechanics, we
begin by developing the basic equations of momentum and energy balance in mechanical
systems. Then we shall see whether it is sensible to ascribe these principles to the
electromagnetic field.
Maxwell’s theory allows us to describe, using Maxwell’s equations, the behavior of
the electromagnetic fields within a (possibly) finite region V of space. The presence of
any sources or material objects outside V are made known through the specification of
tangential fields over the boundary of V , as required for uniqueness. Thus, the influence
of external effects can always be viewed as being transported across the boundary. This
is true of mechanical as well as electromagnetic effects. A charged material body can
be acted on by physical contact with another body, by gravitational forces, and by the
Lorentz force, each effect resulting in momentum exchange across the boundary of the
object. These effects must all be taken into consideration if we are to invoke momentum
conservation, resulting in a very complicated situation. This suggests that we try to
decompose the problem into simpler “systems” based on physical effects.

The system concept in the physical sciences. The idea of decomposing a com-
plicated system into simpler, self-contained systems is quite common in the physical
sciences. Penfield and Haus [145] invoke this concept by introducing an electromagnetic
system where the effects of the Lorentz force equation are considered to accompany a
mechanical system where effects of pressure, stress, and strain are considered, and a
thermodynamic system where the effects of heat exchange are considered. These systems
can all be interrelated in a variety of ways. For instance, as a material heats up it can
expand, and the resulting mechanical forces can alter the electrical properties of the
material. We will follow Penfield and Haus by considering separate electromagnetic and
mechanical subsystems; other systems may be added analogously.
If we separate the various systems by physical effect, we will need to know how to
“reassemble the information.” Two conservation theorems are very helpful in this re-
gard: conservation of energy, and conservation of momentum. Engineers often employ
these theorems to make tacit use of the system idea. For instance, when studying elec-
tromagnetic waves propagating in a waveguide, it is common practice to compute wave
attenuation by calculating the Poynting flux of power into the walls of the guide. The
power lost from the wave is said to “heat up the waveguide walls,” which indeed it does.
This is an admission that the electromagnetic system is not “closed”: it requires the

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