We begin with Maxwell's Equations (ME):

(9.1) | |||

(9.2) | |||

(9.3) | |||

(9.4) |

in SI units, where and . By this point, remembering these should be second nature, and you should really be able to freely go back and forth between these and their integral formulation, and derive/justify the Maxwell Displacement current in terms of charge conservation, etc. Note that there are two inhomogeneous (source-connected) equations and two homogeneous equations, and that the inhomogeneous forms are the ones that are medium-dependent. This is significant for later, remember it.

For the moment, let us express the inhomogeneous MEs in terms of just
and
, explicitly
showing the *permittivity* and the *permeability*
^{9.1}:

(9.5) | |||

(9.6) |

It is difficult to convey to you how important these four equations are
going to be to us over the course of the semester. Over the next few
months, then, we will make Maxwell's Equations dance, we will make them
sing, we will ``mutilate'' them (turn them into distinct coupled
equations for transverse and longitudinal field components, for example)
we will couple them, we will transform them into a manifestly covariant
form, we will solve them microscopically for a point-like charge in
general motion. We will (hopefully) *learn* them.

For the next two chapters we will primarily be interested in the
properties of the field in regions of space without charge (sources).
Initially, we'll focus on a vacuum, where there is no dispersion at all;
later we'll look a bit at dielectric media and dispersion. In a
source-free region, and
and we obtain **Maxwell's Equations in a Source Free Region of Space:**

(9.7) | |||

(9.8) | |||

(9.9) | |||

(9.10) |

where for the moment we ignore any possibility of dispersion (frequency dependence in or ).