Of course, the field that radiates outward is, itself, a moving electromagnetic field. There is, at each point in this field, an electric component, always at 900 to the magnetic component. As the magnetic field expands it has a rotational characteristic. The direction of rotation can be determined using, what engineers have termed, the “Right Hand Rule.” If you imagine holding the conductor in your right hand, youʼll see that your fingers curl around the conductor in the direction that the field rotates when your thumb points in the direction the current is traveling. In the figure above you would grasp the wire from the top, with your thumb pointing to the left. Your fingers would curl around in the direction shown by the arrow, pointing in the direction that the magnetic field is rotating.

As the magnetic field expands the power radiated into space from the conductor varies inversely with the square of the distance. This is called the Inverse Square Law. According to the Inverse Square Law, if you were to measure the power of an antenna at a distance of, say, 10 feet and you found it to be 80 mW then at a distance of 20 feet (distance increased by a factor of 2) you would find the power to be 20 mW (power reduced by a factor of 4). When youʼre twice as far away the power is reduced to 1/4 of its previous value. If you were 3 times further away the power would be reduced to 1/9 of its previous value. Power varies inversely as the square of the distance.

The “Inverse Square Law” is fine and good, but it gives rise to a strange conclusion unless itʼs properly understood. Itʼs important to gain a working knowledge of the physical properties that make a magnetic field propagate outward from an 802.11 transmitter and induce current in a receiving antenna. When wireless LAN designers, engineers, or occasionally sales people try to draw valid conclusions about how equipment will interoperate and communicate in a particular environment they are faced with a major stumbling block unless they have the right understanding of how RF signals really behave. An engineer who only has partial knowledge of the physics underlying 802.11 communication can end up very confused. Even the Inverse Square Law, simple enough at first consideration, has a paradox hiding within it. The “Inverse Square Law” could be likened to the famous Greek math puzzles called “Zenoʼs Paradoxes.”

Zeno’s Paradoxes

Zeno of Elea was a Greek philosopher who lived around 500 BCE. He proposed four paradoxes that remained unsolved by mathematicians until Cantorʼs theory of infinite sets was put forth in the second half of the 19th century. Zenoʼs paradoxes describe situations not unlike the movement of

force in an electromagnetic field, which relate discrete calculations on individual points to continuous assessment of all of the points together. The distance from an antenna to a measuring point is finite; itʼs some precise number of meters. There are, mathematically speaking, an infinite number of points embodied in that distance. Since the Inverse Square Law relates, ultimately, to point-particles in an electromagnetic field, Zenoʼs paradoxes are apropos.

Zenoʼs first paradox asks for mathematical proof that something can get from Point A to Point B. Itʼs a paradox because we all know that you can walk from your house to your car, or that a signal can go from an access point to a client computer in a wireless LAN. Nonetheless, the mathematics underlying the proof is not immediately obvious and, in fact, it took hundreds of years before anybody had a clue how to prove it.

To understand Zenoʼs first paradox, think about a runner who wants to run 100 meters, in a finite time. To reach a 100-meter mark the runner must first reach the 50-meter mark and, to reach that, the runner must first get to the 25-meter mark. Of course, to get to the 25-meter mark the runner must first get to the 12.5-meter mark. The requirement that the runner must first reach the halfway

Copyright 2003 - Joseph Bardwell

point before they can reach the next milestone continues infinitely. The runner, therefore, can never start the race since, to reach even the most infinitesimal distance from the starting point they would first have to reach the point halfway there. The runner would have to reach an infinite number of ʻmidpointsʼin a finite time. Infinite points in finite time are impossible, so the runner can never arrive at the finish line. In general, anyone (or any electromagnetic field) wanting to travel from Point A to Point B must meet these requirements, and so motion becomes impossible, and what we perceive as motion is merely a strange illusion.

Zenoʼs second paradox also suggested a race, but this time between Achilles (a fast athlete) and a turtle. Achilles can run 5 meters per second but the turtle can only run 1 meter per second. The race will be over a 100 meter track. To be fair, Achilles gives the turtle a 5-meter head start. At the sound of the starting gun the turtle is 10 meters ahead of Achilles. After 1-second, Achilles has reached the spot where the turtle started. The turtle, during that one second, has run 1 meter further. Achilles continues running and reaches the 6-meter mark (where the turtle was when Achilles was at the 5- meter mark). Now, however, the turtle has moved away from the 6-meter mark and is, still, ahead of Achilles. This continues and whenever Achilles reaches the spot where the turtle has just been (a fraction of a second earlier) the turtle has again covered a little bit of distance beyond that spot, and is still winning the race. The paradox evolves since it would appear that no matter how hard Achilles tries he can never reach the turtle and the turtle ultimately wins the race.

Bardwell’s ERP Paradox

Now, consider “Bardwellʼs ERP Paradox” (a thought experiment created by the author), that sounds strikingly similar to the problem posed in Zenoʼs Paradoxes. This, of course, is not to be confused with the famous EPR paradox from 1935, when Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics.

Bardwell’s ERP Paradox

A measurement is made in the near field at some distance, D, from an antenna. It is determined that the measured power at that distance is P. By the inverse-square law (in the near field) it can be calculated that at distance D/2 the power would be 4P. Make D/2 a new point of measurement and repeat the experiment. You can see that for each D there exists a point D/2 that lies halfway to the antenna. At each successive selection of the measurement point the power increases by a factor of 4. As you approach the antenna itself the power will become infinite.

The paradox will be exposed in the discussions that follow so donʼt spend too much time struggling with it. If you would like a test to see whether or not someone is really “in the know” about all things electromagnetic, throw Bardwellʼs Paradox at them and see how they respond. The “answer” to Bardwellʼs Paradox (and Zenoʼs) is given in Appendix A and will be best appreciated after reading the remainder of the text.

How, then, shall we appreciate the nuances of electromagnetic field propagation and behavior? If we could see (with our eyes) what the RF field propagating from an 802.11 access point looked like then we would be better prepared to study it. Unfortunately, we can only use tools like spectrum analyzers and 802.11 protocol analyzers to gain indirect insight into the electromagnetic vibrations in the air around us.

Copyright 2003 - Joseph Bardwell