The study of field propagation and electromagnetic theory hinges on three basic concepts. The simplest is that of the existence of an electrical charge with either a positive or negative value. Quantum physicists and theoreticians can explore even this seemingly simple concept throughout an entire career lifetime. We will not be exploring it at that depth! The second principle is called Coulombʼs law and it stipulates that the electrostatic force between two charged particles is proportional to the product of their charges (q1 multiplied by q2 where q is the force of the charge) and is inversely proportional to the square of the distance between the two charges. The force acts along a line between the two charges. The third basic concept is that charges always influence each other. That is, if a force is exerted on a charge by a second charge (force F1) then it follows that a force is also exerted on the second charge by the first one (F2). This principle shows that the total resultant force between the two charges is the vector sum of F1 and F2.

An 802.11 transmitter radiates an electromagnetic field into the space around it. Aspects of the environment affect the field. There is an alternating electric current (positive and negative charge) being impressed on the metal radiating element in the transmitting antenna. The expanding electromagnetic field is an undulating energy field with both electric and magnetic characteristics. At the moment current is first applied to the antenna the electromagnetic field expands outwards at the speed of light. While the signal is being applied to the antenna the field energy ebbs and flows in a wavelike manner. These waves may be considered in their entirety by applying equations involving advanced calculus, specifically, the functions that allow an infinite set of values to be integrated over a surface area or in a 3-dimensional volume. The challenge is to gain a general knowledge of how all of these theoretical aspects of electromagnetism work and, in many cases, the only way to share that appreciation is through mathematical expression. Every attempt has been made to minimize any frightening aspects of mathematics or calculus that may creep into the discussions to follow.

At points that are very close (less than 1-foot for 802.11 networks) to an antenna the electromotive force (the “pushing” force of the field) exerts itself to some degree horizontally (across the x-axis), to some degree vertically (up and down on the y-axis), and to some degree to the front and back (along the z-axis). Electromotive force (represented by the symbol “E”) is, then, a vector quantity, having both magnitude and direction. The voltage in each direction is represented, by convention, using the symbols Ex, Ey, and Ez for the magnitude in each direction, x, y, and z. E, therefore, is the vector sum of Ex, Ey, and Ez. The variable “E” is only one of a number of metrics that are part of the study of electromagnetism. By gaining a general appreciation for a more complete set of fundamental metrics the character of the electromagnetic field can be more fully understood.

Scalar and Vector Measurement Metrics

When a child learns to read they preface their efforts by learning the alphabet. Itʼs only when they comprehend the individual letters that they can recognize organized groups of letters as words, then group those into sentences, and ultimately appreciate great works of literature. The study of electromagnetic theory, too, begins by learning about a basic group of symbols and what each of these symbols means. The symbols are variables that make up mathematical equations which describe the behavior of electromagnetic fields. The mathematical languages that are used include

algebra, trigonometry, and calculus. Calculus has sub-dialects that apply to different situations. PhD candidates in mathematics, physics, and engineering may be fluent in these mathematical languages. They, then, can read the exotic equations that describe electromagnetism and catch a glimmer of its mysterious character. We, however, will have to be content to struggle through only the elementary steps of mathematical representation. The goal will be to grasp the application of enough algebra, trigonometry, and calculus to get an appreciation for the complexities and beauty of electromagnetic field theory.

Copyright 2003 - Joseph Bardwell

Nature manifests itself with a multitude of measurable characteristics. We can eat too many calories (a unit of heat) and then we gain weight (a unit of gravitational attraction). Our refrigerator keeps the milk at 40OF (units of temperature) through the conversion of watts of electricity (units of power) into torque (units of rotational force) and heat (units of molecular motion) in the compressor motor that is at the heart of the cooling system. The total number of measurement units in the whole of the sciences is sufficiently vast to exceed any personʼs capacity for remembering them all. Like the letters of a childʼs alphabet, itʼs the variables that make up equations that serve as the first step in comprehending electromagnetism.

Measurements like weight, speed, charge, and temperature are called scalar quantities. Scalar quantities simply represent the magnitude something (i.e. gravitational attraction, distance per unit time, number of electrons, or excitation of molecules). When a quantity includes both a magnitude and a direction of action, itʼs called a vector quantity.

On the left is a diagram showing the x-axis, y-axis, and z-axis of a 3- dimensional set of coordinate axes. The line from the origin (O) to a point located in space (L) may be considered a representation of a vector. The line has a length and it has a direction both of which can be determined from the location of point L relative to the x-, y-, and z-axis. This point would be specified as L(x,y,z) with the displacement on the x-, y-, and z-axis provided.

Adding two vectors is straightforward. Consider a hiker who started at their camp and walked 3 miles East. Then they walked 2 miles North. The two legs of the hike are vectors because they both have magnitude and direction.

You can visualize how, on the map below, the two legs could be plotted and the hikerʼs position determined. This is the operation of vector addition on a flat plane (2 dimensions).

Figure 3.8 Hiking in the Las Trampas Wildlife Refuge

The vector sum for the two legs of the hikerʼs journey would be the dashed line in Figure 3.8 (above). The direction of the line could be calculated by the trigonometric Tangent function (tan) and the length of the line could then be determined using either the sine (SIN) or cosine (COS) function. These functions are explained in Appendix B.

Copyright 2003 - Joseph Bardwell