Electrodynamics is the study of the entire electromagnetic field. We have learned four distinct differential (or integral) equations for the electric and magnetic fields: Gauss's Laws for Electricity and for Magnetism, Ampere's Law (with the Maxwell Displacement Current) and Faraday's Law. Collectively, these are known as:

(9.1) | |||

(9.2) | |||

(9.3) | |||

(9.4) |

These equations are formulated above in in SI units, where
and
. , recall, is
the *permittivity* of the medium, where is called the *permeability* of the medium. Either of them can in general vary with
e.g. position or with frequency, although we will initially consider
them to be constants. Indeed, we will often work with them in a *vacuum*, where
and
are the
permittivity and permeability of free space, respectfully.

They are related to the (considerably easier to remember) electric and
magnetic constants by:

(9.5) | |||

(9.6) |

so that

(9.7) |

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* (source-free) equations, and that it is the *inhomogeneous* forms that are medium-dependent. This is significant for
later, remember it. Note also that if magnetic monopoles were
discovered tomorrow, we would have to make *all four* equations
inhomogeneous, and incidentally completely symmetric.

For the moment, let us express the inhomogeneous MEs in terms of the electric field and the magnetic induction directly:

(9.8) | |||

(9.9) | |||

(9.10) | |||

(9.11) |

It is difficult to convey to you how important these four
equations^{9.1} 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 try very hard to *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:

(9.12) | |||

(9.13) | |||

(9.14) | |||

(9.15) |