Page 33 - Electromagnetics Handbook

Chapter 2

Maxwell’s theory of electromagnetism

2.1 The postulate
In 1864, James Clerk Maxwell proposed one of the most successful theories in the
history of science. In a famous memoir to the Royal Society [125] he presented nine
equations summarizing all known laws on electricity and magnetism. This was more
than a mere cataloging of the laws of nature. By postulating the need for an additional
term to make the set of equations self-consistent, Maxwell was able to put forth what
is still considered a complete theory of macroscopic electromagnetism. The beauty of
Maxwell’s equations led Boltzmann to ask, “Was it a god who wrote these lines ... ?”
[185].
Since that time authors have struggled to find the best way to present Maxwell’s
theory. Although it is possible to study electromagnetics from an “empirical–inductive”
viewpoint (roughly following the historical order of development beginning with static
fields), it is only by postulating the complete theory that we can do justice to Maxwell’s
vision. His concept of the existence of an electromagnetic “field” (as introduced by
Faraday) is fundamental to this theory, and has become one of the most significant
principles of modern science.
We find controversy even over the best way to present Maxwell’s equations. Maxwell
worked at a time before vector notation was completely in place, and thus chose to
use scalar variables and equations to represent the fields. Certainly the true beauty
of Maxwell’s equations emerges when they are written in vector form, and the use of
tensors reduces the equations to their underlying physical simplicity. We shall use vector
notation in this book because of its wide acceptance by engineers, but we still must
decide whether it is more appropriate to present the vector equations in integral or point
form.
On one side of this debate, the brilliant mathematician David Hilbert felt that the
fundamental natural laws should be posited as axioms, each best described in terms
of integral equations [154]. This idea has been championed by Truesdell and Toupin
[199]. On the other side, we may quote from the great physicist Arnold Sommerfeld:
“The general development of Maxwell’s theory must proceed from its differential form;
for special problems the integral form may, however, be more advantageous” ([185], p.
23). Special relativity flows naturally from the point forms, with fields easily converted
between moving reference frames. For stationary media, it seems to us that the only
difference between the two approaches arises in how we handle discontinuities in sources
and materials. If we choose to use the point forms of Maxwell’s equations, then we must
also postulate the boundary conditions at surfaces of discontinuity. This is pointed out

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