Far reaching as they are, the laws summarized by Maxwell's equations
are directly applicable to the description of only one of many
physical subsystems of scientific and engineering interest. Like
those before it, this chapter has been concerned with the
electromagnetic subsystem. However, by casting the electromagnetic
laws into statements of power flow, we have come to recognize how the
electromagnetic subsystem couples to the thermodynamic subsystem
through the power dissipation density and to the mechanical subsystem
through forces and force densities of electromagnetic origin.

The basis for a self-consistent macroscopic description of any
continuum subsystem is a power flow statement having the forms
identified in Sec. 11.1. Describing the energy and power flow in and
into a volume *V* enclosed by a surface *S*, the integral
conservation of energy statement takes the form
(11.1.1).

The differential form of the conservation of energy
statement is implied by the above.

Poynting's theorem, the subject of Sec. 11.2, is obtained
starting from the laws of Faraday and Ampère
to obtain an expression of the form of (2).
For materials that are Ohmic *(***J** = **E**) and that
are linearly polarizable and magnetizable (**D** = **E** and
**B** = **H**), the * power flux density* **S** (or *
Poynting's vector*), * energy density* *W*, and * power
dissipation density* *P*_{d} were shown in Sec. 11.3 to be

Of course, taking the free space limit where and assume
their free space values and * = 0* gives the free space
conservation statement discussed in Sec. 11.2.

In Sec. 11.3, we found that in EQS systems, an alternative to
Poynting's vector is (11.3.24).

This expression is of practical importance, because it can be evaluated
without determining **H**, which is generally not of interest in EQS
systems.

An important application of the integral form of the energy
conservation statement is to lumped parameter systems. In these
cases, the surface *S* of (1) encloses a system that is connected to
the outside world through terminals. It is then convenient to
describe the power flow in terms of the terminal variables. It was
shown in Sec. 11.3 (11.3.29), that the net power into the system
represented by the left-hand side of (1) becomes

provided that the magnetic induction and the electric displacement
current through the surface *S* are negligible.

This set the stage for the application of the integral form of the
energy conservation theorem to lumped parameter systems.

In Sec. 11.4, attention focused on the energy storage term, the
first terms on the right in (1) and (2). The energy density concept
was broadened to include materials having constitutive laws relating
the flux densities to the field intensities that were single valued
and collinear. With *E, D, H*, and *B* representing the field
magnitudes, the energy density was found to be the sum of electric and
magnetic energy densities.

Integrated over the volume *V* of a system, this function leads to the
total energy *w*.

For quasistatic lumped parameter systems, the total electric or
magnetic energy is often conveniently found following a different
route. First the terminal relations are determined and then the
total energy is found by adding up the increments of energy put into
the system as it is energized. In the case of an *n* terminal pair EQS
system, where the relation between terminal voltage *v*_{i} and
associated charge *q*_{i} is *v*_{i} (q_{1}, q_{2}, q_{n} ), the
increment of energy is *v*_{i} dq_{i}, and the total electric energy is
(11.4.9).

The line integration in an *n*-dimensional space representing the *n*
independent *q*_{i}'s was illustrated by Example 11.4.2.

Similarly, for an *n* terminal pair MQS system where the current *i*_{i}
is related to the flux linkage _{i} by *i*_{i} = i_{i} (_{1},
_{2}, _{n}), the total energy is (11.4.12).

Note the analogy between these expressions for the total energy
of EQS and MQS lumped parameter systems and the electric
and magnetic energy densities, respectively, of (8). The transition
from the field picture afforded by the energy densities to the lumped
parameter characterization is made by *E v, D
q* and by *H i, B *.

Especially in using the energy to evaluate forces of electrical
origin, we found it convenient to define coenergy density functions.

It followed that these functions were natural when it was desirable to
use *E* and *H* as the independent variables rather than *D* and *B*.

The total coenergy functions for lumped parameter EQS and MQS
systems could be found either by integrating these densities over the
volume or by again viewing the system in terms of its terminal
variables. With the total coenergy functions defined by

it followed that the coenergy functions could be determined from the
terminal relations by again carrying out line integrations, but this
time with the voltages and currents as the independent variables.
For EQS systems,

while for MQS systems,

Again, note the analogy to the respective terms in (12).

The remaining sections of the chapter developed some of the
possible implications of the "dissipation" term in the energy
conservation statement, the last terms in (1) and (2). In Sec. 11.5,
coupling to a thermal subsystem was discussed. In this section, the
disparity between the power input and the rate of increase of the
energy stored was accounted for by heating. In addition to Ohmic
heating, caused by collisions between the migrating carriers and the
neutral media, we considered losses associated with the dynamic
polarization and magnetization of materials.

In Secs. 11.6-11.9, we considered coupling to a mechanical
subsystem as a second mechanism by which energy could be extracted
from (or put into) the electromagnetic subsystem.
With the displacement of an object denoted by , we used an energy
conservation postulate to infer the total electric or magnetic force
acting on the object from the energy functions [(11.6.9), and its
magnetic analog]

or from the coenergy functions [(11.7.7) and the analogous
expression for electric systems].

In Sec. 11.8, where the Lorentz force on a particle was
generalized to account for electric and magnetic dipole moments, one
objective was a microscopic picture that would lend physical insight
into the forces on polarized and magnetized materials. The Lorentz
force was generalized to include the force on stationary electric
and magnetic dipoles, respectively.

The total macroscopic forces resulting from microscopic
forces had already been encountered in the previous two sections. The
force density describes the interaction between a volume element of the
electromagnetic subsystem and a mechanical continuum. The force
density inferred by averaging over the forces identified in Sec. 11.8
as acting on microscopic particles was

A more rigorous approach to finding the force density could be
based on a generalization of the energy method introduced in Secs.
11.6 and 11.7. As background for further pursuit of this subject, we
have illustrated the importance of including the mechanical continuum
with which the force density acts. Before there can be a meaningful
answer to the question, "Which force density is correct?" the other
force densities acting on the material must be specified. As an
illustration, we found that very different electric or magnetic force
densities would result in the same deformations of an incompressible
material and in the same net force on an object surrounded by free
space^{[1,2]}.