Page 38 - Electromagnetics Handbook

2.2 The well-posed nature of the postulate
It is important to investigate whether Maxwell’s equations, along with the point form
of the continuity equation, suffice as a useful theory of electromagnetics. Certainly we
must agree that a theory is “useful” as long as it is defined as such by the scientists and
engineers who employ it. In practice a theory is considered useful if it predicts accurately
the behavior of nature under given circumstances, and even a theory that often fails may
be useful if it is the best available. We choose here to take a more narrow view and
investigate whether the theory is “well-posed.”
A mathematical model for a physical problem is said to be well-posed,or correctly set,
if three conditions hold:

1. the model has at least one solution (existence);
2. the model has at most one solution (uniqueness);
3. the solution is continuously dependent on the data supplied.

The importance of the first condition is obvious: if the electromagnetic model has no
solution, it will be of little use to scientists and engineers. The importance of the second
condition is equally obvious: if we apply two different solution methods to the same
model and get two different answers, the model will not be very helpful in analysis or
design work. The third point is more subtle; it is often extended in a practical sense to
the following statement:

3 . Small changes in the data supplied produce equally small changes in the solution.
That is, the solution is not sensitive to errors in the data. To make sense of this we
must decide which quantity is specified (the independent quantity) and which remains
to be calculated (the dependent quantity). Commonly the source field (charge) is taken
as the independent quantity, and the mediating (electromagnetic) field is computed from
it; in such cases it can be shown that Maxwell’s equations are well-posed. Taking the
electromagnetic field to be the independent quantity, we can produce situations in which
the computed quantity (charge or current) changes wildly with small changes in the
specified fields. These situations (called inverse problems) are of great importance in
remote sensing, where the field is measured and the properties of the object probed are
thereby deduced.
At this point we shall concentrate on the “forward” problem of specifying the source
field (charge) and computing the mediating field (the electromagnetic field). In this case
we may question whether the first of the three conditions (existence) holds. We have
twelve unknown quantities (the scalar components of the four vector fields), but only
eight equations to describe them (from the scalar components of the two fundamental
Maxwell equations and the two scalar auxiliary equations). With fewer equations than
unknowns we cannot be sure that a solution exists, and we refer to Maxwell’s equations
as being indefinite. To overcome this problem we must specify more information in
the form of constitutive relations among the field quantities E, B, D, H, and J. When
these are properly formulated, the number of unknowns and the number of equations
are equal and Maxwell’s equations are in definite form. If we provide more equations
than unknowns, the solution may be non-unique. When we model the electromagnetic
properties of materials we must supply precisely the right amount of information in the
constitutive relations, or our postulate will not be well-posed.

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