• ### Unit 6: Maxwell's Equations

At this point in the course, we have developed the mathematical structure for and a general understanding of all of Maxwell's Equations. Now we want to sit back and summarize our findings by identifying what they are, what they mean, and how we can use them.

There are four Maxwell equations that describe all classical electromagnetism. Maxwell's equations take on a particularly simple form when describing the behavior of electric and magnetic fields in regions devoid of matter; that is, in a vacuum. (Note that for most purposes, air is close enough to being a vacuum that the presence of an atmosphere can be ignored.) These are Maxwell's free space equations.

There are four Maxwell free space equations. These include the two flux equations - the electric and magnetic forms of Gauss' law. These state that the electric or magnetic flux through a closed surface is proportional to the electric or magnetic charge enclosed within that surface. Note that in the magnetic case, there are no magnetic charges (also called magnetic monopoles), so that the magnetic flux through and closed surface is zero.

The other two free space Maxwell's equations are Faraday's Law of Induction and a modified version of Ampere's Circuital Law. Once again, these electric and magnetic equations have similar formalisms, thereby emphasizing the close relationship of the electric and magnetic fields. Faraday's Law of Induction states that the induced EMF in any closed circuit is proportional to the time rate of change of the magnetic flux through the circuit, while Ampere's Law states that the integrated magnetic field around a closed curve is proportional to the currents passing through a surface bounded by the curve. Maxwell's main contribution (beyond realizing that these four equations provided a complete theory of electromagnetism) was the discovery and description of the displacement current, which is a source of the magnetic field associated with the rate of change of the electric displacement field in a region.

Inside materials, Maxwell's Equations are modified by the electric permittivity and magnetic permeability of the materials, but they remain the basis for the classical model of electromagnetism. In this unit, we will concentrate on Maxwell's Equations as a single theory that unites the half-century of previous work on electromagnetism.

Completing this unit should take you approximately 8 hours.