Maxwell’s equations
In this chapter we come back to the complete set of the four
Maxwell equations that we took as our starting point in
Chapter 1. Until now, we have been studying Maxwell’s
equations in bits and pieces; it is time to add one final piece, and to
put them all together. We will then have the complete and correct story
for electromagnetic fields that may be changing with time in any way.
Anything said in this chapter that contradicts something said earlier is
true and what was said earlier is false—because what was said earlier
applied to such special situations as, for instance, steady currents or
fixed charges. Although we have been very careful to point out the
restrictions whenever we wrote an equation, it is easy to forget all of
the qualifications and to learn too well the wrong equations. Now we are
ready to give the whole truth, with no qualifications (or almost none). The complete Maxwell equations are written in Table 18–1,
in words as well as in mathematical symbols. The fact that the words
are equivalent to the equations should by this time be familiar—you
should be able to translate back and forth from one form to the other. The first equation—that the divergence of $\FLPE$ is the charge
density over $\epsO$—is true in general. In dynamic as well as in
static fields, Gauss’ law is always valid. The flux of $\FLPE$ through
any closed surface is proportional to the charge inside. The third
equation is the corresponding general law for magnetic fields. Since
there are no magnetic charges, the flux of $\FLPB$ through any closed
surface is always zero. The second equation, that the curl of $\FLPE$
is $-\ddpl{\FLPB}{t}$, is Faraday’s law and was discussed in
the last two chapters. It also is generally true. The last equation has
something new. We have seen before only the part of it which holds for
steady currents. In that case we said that the curl of $\FLPB$ is
$\FLPj/\epsO c^2$, but the correct general equation has a new part that
was discovered by Maxwell.
 Until Maxwell’s work,
the known laws of electricity and magnetism were
those we have studied in Chapters 3
through 17. In particular, the equation for the magnetic
field of steady currents was known only as
\begin{equation}
\label{Eq:II:18:1}
\FLPcurl{\FLPB}=\frac{\FLPj}{\epsO c^2}.
\end{equation}

Maxwell began by
considering these known laws and expressing them as
differential equations, as we have done here. (Although the
$\FLPnabla$ notation was not yet invented, it is mainly due to
Maxwell
that the importance of the combinations of derivatives, which we today
call the curl and the divergence, first became apparent.) He then
noticed that there was something strange about Eq. (18.1).
If one takes the divergence of this equation, the left-hand side will be
zero, because the divergence of a curl is always zero. So this equation
requires that the divergence of $\FLPj$ also be zero. But if the
divergence of $\FLPj$ is zero, then the total flux of current out of any
closed surface is also zero. The flux of current from a closed surface is the decrease of the
charge inside the surface. This certainly cannot in general be zero
because we know that the charges can be moved from one place to
another. The equation
\begin{equation}
\label{Eq:II:18:2}
\FLPdiv{\FLPj}=-\ddp{\rho}{t}
\end{equation}

has, in fact, been almost our definition of $\FLPj$. This equation
expresses the very fundamental law that electric charge is
conserved—any flow of charge must come from some supply.
Maxwell
appreciated this difficulty and proposed that it could be avoided by
adding the term $\ddpl{\FLPE}{t}$ to the right-hand side of
Eq. (18.1); he then got the fourth equation in
Table 18–1:
\begin{equation*}
\text{IV.}\quad c^2\FLPcurl{\FLPB}=\frac{\FLPj}{\epsO}+\ddp{\FLPE}{t}.
\end{equation*}
 It was not yet customary in
Maxwell’s time to think in terms of
abstract fields. Maxwell
discussed his ideas in terms of a model in
which the vacuum was like an elastic solid. He also tried to explain
the meaning of his new equation in terms of the mechanical
model. There was much reluctance to accept his theory, first because
of the model, and second because there was at first no experimental
justification. Today, we understand better that what counts are the
equations themselves and not the model used to get them. We may only
question whether the equations are true or false. This is answered by
doing experiments, and untold numbers of experiments have confirmed
Maxwell’s equations. If we take away the scaffolding he used to build
it, we find that Maxwell’s
beautiful edifice stands on its own. He
brought together all of the laws of electricity and magnetism and made
one complete and beautiful theory. Let us show that the extra term is just what is required to straighten
out the difficulty Maxwell
discovered. Taking the divergence of his
equation (IV in Table 18–1), we must have that the
divergence of the right-hand side is zero:
\begin{equation}
\label{Eq:II:18:3}
\FLPdiv{\frac{\FLPj}{\epsO}}+\FLPdiv{\ddp{\FLPE}{t}}=0.
\end{equation}

In the second term, the order of the derivatives with respect to
coordinates and time can be reversed, so the equation can be rewritten
as
\begin{equation}
\label{Eq:II:18:4}
\FLPdiv{\FLPj}+\epsO\,\ddp{}{t}\,\FLPdiv{\FLPE}=0.
\end{equation}

But the first of Maxwell’s equations says that the divergence of $\FLPE$
is $\rho/\epsO$. Inserting this equality in
Eq. (18.4), we get back Eq. (18.2), which we
know is true. Conversely, if we accept Maxwell’s equations—and we do
because no one has ever found an experiment that disagrees with
them—we must conclude that charge is always conserved. The laws of physics have no answer to the question: “What happens if
a charge is suddenly created at this point—what electromagnetic
effects are produced?” No answer can be given because our equations
say it doesn’t happen. If it were to happen, we would need new
laws, but we cannot say what they would be. We have not had the chance
to observe how a world without charge conservation behaves. According
to our equations, if you suddenly place a charge at some point, you
had to carry it there from somewhere else. In that case, we can say
what would happen. When we added a new term to the equation for the curl of $\FLPE$, we
found that a whole new class of phenomena was described. We shall see
that Maxwell’s
little addition to the equation for $\FLPcurl{\FLPB}$
also has far-reaching consequences. We can touch on only a few of them
in this chapter.