### 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.**

Upon successful completion of this unit, you will be able to:

- state Maxwell's equations and identify each mathematical operator and physical quantity in the equations;
- state in words the information contained in each of Maxwell's equations;
- describe how Maxwell's equations resulted in the prediction of electromagnetic waves and the realization that light was an electromagnetic wave;
- list several examples of electromagnetic phenomena; and
- solve problems involving properties of electromagnetic waves.

### 6.1: Maxwell's Equations

Most of this chapter reviews Maxwell's time-independent equations but often from a different viewpoint that acts synergistically with our previously covered material. Take particular note of the definition on page 7 of the electric flux density vector

**D**= ε**E**, where**E**is the electric field vector and ε is the dielectric constant times the free space permittivity ε_{o}. Similarly, on page 12 the magnetic flux density vector**B**= μ**H**, where**H**is the magnetic field vector and μ is the magnetic permeability, sometimes described as the relative permeability times the permeability of free space. For this class, these equations that allow the use of Maxwell's Equations in a material are assumed to be scalar functions of position. The general case is that they are tensors, but the scalar approximation simplifies gaining an initial understanding of the behavior of electromagnetism systems. Work through the examples until you understand how to approach solving similar problems.This is an illustrative explanation of displacement current, which is arguably the linchpin of Maxwell's full theory of electromagnetism, as well as the single most confusing concept in that theory.

There is no electric current between the plates of a capacitor. However, Ampere's Circuital Law tells us that the integral of the magnetic field B around a closed loop C is proportional to the flux of the current density through a surface S attached to the loop. This is independent of the shape of S. In the demonstration, consider a closed loop positioned around one of the wires carrying electric current into the charging capacitor. If S is chosen so that the wire penetrates the surface, the flux of the current density through S is simply the electric current.

Now draw another surface S' so that it passes between the capacitor plates, thereby making no contact with the current carrying wire. There is no electric current between the capacitor plates, so it would appear that the flux of the current density through S' is zero. However, Ampere's Circuital Law tells us that the magnetic field integral around loop C is still the same non-zero value. We appear to meet a contradiction.

What the apparent contradiction is actually telling us is that Ampere's Circuital Law is incomplete. The electric field between the plates of a charging capacitor changes with time, so it would appear that a time-varying electric field must generate a magnetic field which is consistent with the current charging the capacitor. The way Maxwell chose to think about this by identifying a fictitious current between the capacitor plates called the displacement current such that the total displacement current flux between the plates was equal to the current charging the capacitor. Although there is no actual electrical current between the plates, we still refer to the source of magnetic field associated with a changing electric field as the displacement current.

### 6.2: Electromagnetic Waves

Watch this lecture, pausing to take notes, before moving on to the reading below. Please test your understanding of this lecture by attempting the problem #6 from this problem set. Check your solution here.

This chapter develops the time-varying version of Maxwell's Equations and uses them to examine not only the properties of EM radiation, but also why anything emits this radiation in the first place. Again, stick with the math until you can see the physics.

This demonstration schematically indicates the time-dependent electric and magnetic fields associated with an electromagnetic wave. Note that the electric and magnetic fields are mutually perpendicular to one another and to the path the wave follows.

As the electric field changes in time, a magnetic field is generated as described by Maxwell's generalization of Ampere's Circuital Law. As the magnetic field changes in time, an electric field is produced as described by Faraday's Law of Induction. Given this word picture, why are the electric field strength and the maximum magnetic field strength proportional at all times? Which direction does the wave velocity point, toward positive or negative x? (Hint: Look at Poynting's Theorem.)

Try this experiment using your microwave oven. Why do the melted spots appear in a regular pattern?

### 6.3: Energy and Intensity of Electromagnetic Waves

Watch this lecture, pausing to take notes, before moving on to the reading below. Please test your understanding of this lecture by attempting the five problems in this problem set. Check your solutions here.

### 6.4: Spectrum of Electromagnetic Radiation

Read chapter 24, "Electromagnetic Waves". Think about the accompanying conceptual questions to assess your understanding of the chapter.

### Unit 6 Assessment

Take this assessment to see how well you understood this unit.

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**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

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