Teardown: Maxwell's equations

Arthur J. Glazar -August 19, 2012

If Maxwell's Equations were a piece of hardware, most of us would be eager to see what's inside the box. But you won't find the literary equivalent  of a physical teardown because the subject is basically mathematical and as such, requires a certain amount of rigor to explain and understand. If you're willing to trade mathematical rigor for readability, this article may be the refresher you've been looking for.

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THE EQUATIONS

Maxwell’s Equations can be written in several different forms. Here is one particular form, written in vector notation1:

(I)        curl H  =  σE + εĖ

(II)       curl E  =  - μ

Equations (I) and (II) are Maxwell’s First and Second equations. The symbols E and H represent electric and magnetic fields, respectively. It is striking to observe that E and H are somehow equated; that is, E and H appear on both sides of the equal signs. That is the essence of Maxwell’s Equations; i.e., electric and magnetic fields are inseparable.

Most reference sources assert that there are four Maxwell equations2. We will return to that matter later, but for now only (I) & (II) above, are the subjects of our attention.

The equations are verbalized as follows:

(I) "The curl of H equals sigma times E, plus epsilon times E-dot".

(II) "The curl of E equals minus (or negative) mu times H-dot".

Of course, this doesn't explain anything; it just allows us to say the equations out loud or in our heads

CONSTANTS AND UNITS

Epsilon (ε), sigma (σ) and mu (μ) are physical constants of particular materials which should be familiar to most radio amateurs. Epsilon is the dielectric constant of the material, sigma is its conductivity and mu is its magnetic permeability. The numerical values of epsilon, sigma and mu vary among different materials and are of no particular concern here. However, knowing their units is important3. In the balance of this discussion, units will be presented within square brackets like this:  [ampere]. Units of measure will play an important part in our discussion. We will add, multiply and divide units (just as numbers are manipulated) in order to illustrate important relationships.

The units of dielectric constant (ε) are: [ampere] [second] / [volt] [meter].

The units of conductivity (σ) are: [ampere] / [volt] [meter].

The units of permeability (μ) are:  [volt] [second] / [ampere] [meter].

[Volts] and [amperes] are not fundamental units. A volt is defined as the amount of physical work (i.e., force times distance) done in the process of moving a quantity of electrical charge from one point to another. Therefore [volt] = [newton][meter] / [coulomb]. Similarly, an ampere is defined as the quantity of electric charge moving past a point in a given time. Therefore, [ampere] = [coulomb] / [sec]. A newton is the unit of force in the SI system. You can find further information on the SI system at Wikipedia on the web.

FIELDS AND VECTORS

The concept of a "field" is a means to describe certain physical behavior within a region of 3-dimensional space.

Electric Field

By definition, an electric field exists in a region if a force is experienced by a stationary electrical charge placed in the region4.

The direction of the electric field is defined as the direction of the force experienced by the charge. See figure 1.

Figure 1: The direction of the electric field is defined as the direction of the force experienced by the charge.

Electric field strength, E, is the ratio of the force F [newtons] divided by the charge q [coulombs]. That is,  E = F / q  [newtons]  /  [ coulomb].5

More commonly, E is expressed in  the equivalent units  [volts]  /  [ meter]. We've already noted that a [volt] is defined as a [newton] [meter] per [coulomb]. Therefore, electric field strength equals

[volts]             [newton] [meter] /  [coulomb]                [newton]

----------    =     --------------------------------------      =      -----------

[meter]                       [meter]                                        [coulomb]

Notice that [meter] in both the numerator and denominator of the center expression "cancel" each other, leaving just [newton] / [coulomb]. We will use this method of manipulating units again later.

Because the electric field has both a magnitude and a direction, it is termed a vector quantity, and that is indicated by printing its symbol, E, in bold typeface as in equations (I) and (II).