Appendix A

The Solution To Zeno’s and Bardwell’s Paradoxes

The flaw in both Zenoʼs paradoxes and Bardwellʼs ERP Paradox relates to the fact that the mathematical solution proposed (an iterative series of calculations) requires that an infinite series of numbers be considered. If that were true then it would take Achilles an infinite amount of time to reach the turtle and power at an antenna would be infinite. The rules of calculus tell us, however, that the problem is not one of an infinite series but, rather, of a value that approaches a limit. The false argument proposes that there are an infinite series of time-points between two events. Then, to create the paradox, the stipulation is that an infinite series of points canʼt be considered in a finite amount of time. This is an invalid approach since the supposedly infinite series of calculations takes place on a bounded range (access point antenna to measurement point) that is finite in length. The question posed by the paradox is constructed within a finite set of boundaries. In the case of an 802.11 transmission, the field producing the Effective Radiated Power begins at the antenna (for the near field) or at the fuzzy boundary between the reactive near field and the far field (for far field measurements). At the starting point for ERP the power in the field is determined by the input power

presented across the impedance (resistance) of the antenna element, or elements. Since (simplistically stated), power equals voltage times amperage, and amperage equals voltage divided by resistance (Ohmʼs law) the input power to the antennaʼs radiating element can be calculated. After that, the near field power decreases proportionally to the cube of the distance and the far field power decreases proportionally to the square of the distance. The answer to Bardwellʼs ERP Paradox is, succinctly stated “The initial power output of an antenna is calculated based on the input power and the distance from the antenna to any measurement point is finite. The fact that this finite distance contains an infinite number of mathematical ʻhalvesʼis of no consequence to Maxwellʼs equations or to the basic equations for field strength based on point charges.”

The infinite series that underlies both Zenoʼs and Bardwellʼs paradoxes are applicable only in the realm of classical physics which, of course, is where the 802.11 WLAN engineer will spend their time. Interestingly, there are actually aspects of Bardwellʼs Paradox that do remain unsolved even by contemporary Quantum Electrodynamics (QED). The quantum physicist would be able to explain that there are issues relating to the field strength diverging that remain a mystery even today!

Copyright 2003 - Joseph Bardwell