Section 4: Electromagnetic Field Propagation

There is a basic principle of antennae that is so unexpected (to the uninitiated student) that some people refuse to believe itʼs true the first time they hear it. The principle is called the Reciprocity Theorem and the consequences of this theorem are that (for the same input power level) if you can hear my transmission then I can hear yours. Where this becomes astounding is when I have a very elaborate, high gain, highly directive antenna thatʼs capable of beaming my 100 mW 802.11 signal across 20 miles. My antenna might be illegal, but if Iʼm trying to hack into your network, antenna legality is probably low on my list of concerns. You, on the other hand, have only the little nub antenna sticking out of the side of your notebook computer on your PCMCIA NIC. You have no special antenna, but you do have the same 100 mW power input that I do. The time-symmetric nature of Maxwellʼs wave equations form the basis for the fact that the same qualities of my antenna that allow it to beam my 100 mW signal over to you also allow it to work in reverse, picking up your signals from the air. Maxwellʼs equations are symmetric with respect to time.

Time Symmetry and the Reciprocity Theorem

To understand the significance of time symmetric physical properties, consider a pool table with a white ball near one end and the black ball in the center. The white pool ball is accelerated by the force of impact of the cue stick and travels towards the center of the pool table. In the center, the white ball strikes the black 8-ball in a straight, center-to-center impact. The inertia of the white ball is transferred to the black ball and it is now accelerated away from the white ball, in a straight line, leaving the white ball stationary at the point of impact. If you were to make a movie of the two balls striking and then played the movie backwards, it would show exactly the same thing except now it would be the black ball that starred in the opening scene of the movie. If the mass, velocity, and other characteristics of the Amazing Pool Ball Adventure movie were represented through mathematical equations the equations, would not be time dependent. Time could run forward or backward (negative time) and the results would be identical.

Consider the spatial volume of an expanding electromagnetic field. All of the fundamental metrics are measurable in the field. They are interacting in accordance with Maxwellʼs equations. The field propagates away from the antenna and the volume and its surface area get larger. As an electromagnetic field propagates it ultimately expands outward into the universe where, perhaps

thousands of years from now, some being in a distant galaxy will be reading your 802.11-transmitted email. The concept is, however, one of transmission volume. Now, from what volume of space can an antenna receive signals? It turns out that the shape (in 3-dimensional space) of the transmitted electromagnetic field also defines the shape of the volume from which the same antenna can receive signals in accordance with the rule called the Rayleigh-Helmholtz reciprocity theorem.

Baron Rayleigh of Essex, England, published a theory of light scattering in 1871 that was the first correct explanation for why the sky is blue. He earned the Nobel Prize in 1904 for the discovery of Argon gas. He donated the proceeds of his prize to the University of Cambridge where he became Chancellor in 1908. Rayleigh based some of his ideas on the work of Ferdinand Helmholtz who, after fighting in the Prussian army against Napoleon went on to invent the ophthalmoscope in 1851. Both men studied electromagnetic phenomena and both would develop theories that would become cornerstones of present day physics.

Copyright 2003 - Joseph Bardwell