Maxwell's Equations (ME) consist of two inhomogeneous partial
differential equations and two homogeneous partial differential
equations. At this point you should be familiar at *least* with
the ``static'' versions of these equations by name and function:

(10.1) | |||

(10.2) | |||

(10.3) | |||

(10.4) |

in SI units, where and .

The astute reader will immediately notice two things. One is that these
equations are not *all*, strictly speaking, static - Faraday's law
contains a time derivative, and Ampere's law involves moving charges in
the form of a current. The second is that they are *almost*
symmetric. There is a divergence equation and a curl equation for each
kind of field. The inhomogenous equations (which are connected to *sources* in the form of electric charge) involve the electric
displacement and magnetic field, where the homogeneous equations suggest
that there is no magnetic charge and consequently no screening of the
magnetic induction or electric field due to magnetic charge. One
asymmetry is therefore the presence/existence of electric charge in
contrast with the absence/nonexistence of magnetic charge.

The other asymmetry is that Faraday's law connects the curl of the
field to the time derivative of the
field, but its apparent
partner, Ampere's Law, does *not* connect the curl of
to the
time deriviative of
as one might expect from symmetry alone.

If one examines Ampere's law in its integral form, however:

(10.5) |

one quickly concludes that the current through the open surface bounded by the closed curve is

Consider a closed curve
that bounds *two distinct* open surfaces
and
that together form a *closed* surface
. Now consider a current (density) ``through'' the curve
,
moving from left to right. Suppose that *some* of this current
accumulates inside the volume
bounded by
.
The law of *charge conservation* states that the flux of the
current density *out* of the closed surface
is equal to the
rate that the total charge inside *decreases*. Expressed as an
integral:
&conint#oint;_S · dA = -t&int#int;_V/S&rho#rho; dV

With this in mind, examine the figure above. If we rearrange the
integrals on the left and right so that the normal
points *in* to the volume (so we can compute the current through the surface
moving from left to right) we can easily see that charge
conservation tells us that the current in through
minus the
current out through
must equal the rate at which the total charge
inside this volume increases. If we express this as integrals:
&int#int;_S_1 ·_1 dA - &int#int;_S_2 ·_2 dA & = &
Qt

& = & t&int#int;_V/S &rho#rho; dV
In this expression and figure, note well that
and
point
through the loop in the *same* sense (e.g. left to right) and note
that the volume integral is over the volume
bounded by the *closed* surface formed by
and
together.

Using Gauss's Law for the electric field, we can easily connect this
volume integral of the charge to the flux of the electric field
integrated over these two surfaces with *outward* directed normals:
&int#int;_V/S &rho#rho; dV & = & &epsi#epsilon;&conint#oint;_S · dA

& = & -&epsi#epsilon;&int#int;_S_1 · dA + &epsi#epsilon;&int#int;_S_2 · dA

Combining these two expressions, we get:
&int#int;_S_1 ·_1 dA & - & &int#int;_S_2 ·_2 dA =

& & t{-&epsi#epsilon;&int#int;_S_1 ·_1 dA +
&epsi#epsilon;&int#int;_S_2 ·_2 dA}
&int#int;_S_1 ·_1 dA & + &
t&epsi#epsilon;&int#int;_S_1 ·_1 dA =

& & &int#int;_S_2 ·_2 dA +
t&int#int;_S_2 &epsi#epsilon;·_2 dA
&int#int;_S_1 {+ &epsi#epsilon;t}·_1 dA
= &int#int;_S_2 {+ &epsi#epsilon;t}·_2 dA
From this we see that the flux of the ``current density'' inside the
brackets is *invariant* as we choose different surfaces bounded by
the closed curve
.

In the original formulation of Ampere's Law we can clearly get a
different answer on the right for the current ``through'' the closed
curve depending on which surface we choose. This is clearly impossible.
We therefore modify Ampere's Law to use the *invariant* current
density:
_inv = + &epsi#epsilon;t
where the flux of the second term is called the *Maxwell
displacement current* (MDC). Ampere's Law becomes:
&conint#oint;_C ·d& = & &mu#mu;&int#int;_S/C _inv · dA

& = & &mu#mu;&int#int;_S/C {+ &epsi#epsilon;t}·
dA
or
&conint#oint;_C ·d= &int#int;_S/C {+ t}·
dA
in terms of the magnetic field
and electric displacement
.
The origin of the term ``displacement current'' is obviously clear in
this formulation.

Using vector calculus on our old form of Ampere's Law allows us to
arrive at this same conclusion much more simply. If we take the
divergence of Ampere's Law we get:
·(×
) = 0 = ·
If we apply the divergence theorem to the law of charge conservation
expressed as a flux integral above, we get its differential form:
·- &rho#rho;t = 0
and conclude that in general we can *not* conclude that the
divergence of
vanishes in general as this expression requires, as
there is no guarantee that
vanishes everywhere in
space. It only vanishes for ``steady state currents'' on a background
of uniform charge density, justifying our calling this form of Ampere's
law a magnetostatic version.

If we substitute in
(Gauss's Law) for
, we
can see that it *is* true that:
·(×
) = 0 = ·{
+
t}
as an identity. A sufficient (but *not necessary!*) condition for
this to be true is:
×
=
+ t
or
×
- t =
.
This expression is identical to the magnetostatic form in the cases
where
is constant in time but respects charge conservation when
the associated (displacement) field is changing.

We can now write the *complete* set of Maxwell's equations,
including the Maxwell displacement current discovered by requiring
formal invariance of the current and using *charge conservation* to
deduce its form. Keep the latter in mind; it should not be surprising
to us later when the law of charge conservation pops *out* of
Maxwell's equations when we investigate their formal properties we can
see that we deliberately encoded it into Ampere's Law as the MDC.

Anyway, here they are. Learn them. They need to be second nature as we
will spend a considerable amount of time using them repeatedly in many,
many contexts as we investigate electromagnetic radiation.

(10.6) | |||

(10.7) | |||

(10.8) | |||

(10.9) |

(where I introduce and obvious and permanent abbreviations for each equation by name as used throughout the rest of this text).

Aren't they pretty! The no-monopoles asymmetry is still present, but we
now have *two symmetric* dynamic equations coupling the electric and
magnetic fields and are ready to start studying *electrodynamics*
instead of *electrostatics*.

Note well that the two *inhomogeneous* equations use the in-media
forms of the electric and magnetic field. These forms are already
coarse-grain averaged over the microscopic distribution of point charges
that make up bulk matter. In a truly microscopic description, where we
consider only bare charges wandering around in free space, we should use
the free space versions:

(10.10) | |||

(10.11) | |||

0 | (10.12) | ||

0 | (10.13) |

It is time to make these equations jump through some hoops.