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Archive for January, 2013

Mapping Maxwell’s Equations with Del Vector Diagrams

Wednesday, January 16th, 2013

I saw an excellent presentation by my master’s degree thesis adviser, Dr. Robert Nevels, yesterday.  In the presentation, Dr. Nevels explained how something called a Wilton diagram could be used to visualize Maxwell’s equations and choices of gauge in a simple vector diagram.  If you’re near a university and/or have access to IEEE journals, you can read the entire article behind the presentation here.  I get something new every time I see one of these diagrams.  I’ll get you started on one excerpted from the article and then leave you to it.

For those who don’t work with antennas or other endeavors that call them back to EM on a regular basis, a brief refresher might be of some help.  First, we have Maxwell’s equations:

Dr. Nevels works in frequency domain notation.  It’s not too hard to get used to, just replace all time derivatives with j times omega.  Then keep in mind, that dividing by a derivative is ‘OK’ in engineering if not in strictly formal mathematics.

In the diagram below, you can see that the del operator has been put on the veritcal axis.  The first equation above says that  the dot product of del and the magnetic field is 0.  Therefore, the magnetic field and the del operator are perpendicular to each other and you can see that the B field is pointing out of the page.  From the third equation we can see that del and E are not always perpendicular, however from the fourth equation, we can see that del cross E should lie in a the plane that B is perpendicular to, hence the mapping of E in the diagram.  Because del cross B should be perpendicular to both del and B, it lies on the x axis.  Finally, the second equation above, (Faraday’s law), and vector addition tell us where to place the current density J.  As you can see, a number of other relations can be read from the diagram, but I’ll leave those for you.

From KH Yang, RD Nevels, “Diagrams for several electromagnetic field gauges”, Antennas and Propagation Society International Symposium, 2009. APSURSI ’09. IEEE

 

The Higgs Connection: Part VII of the Holiday Serial

Tuesday, January 8th, 2013

Since we’re coasting into the end of the holiday break, I better start bringing the serial to an end as well.  When we left off last time, Bryce DeWitt had lent an air of respectability to the Gravity Research Foundations essay contest and in so doing, had brought himself to the awareness of Agnew Hunter Bahnson Jr.  I alluded to the fact that Bahnson contacted DeWitt, but not to the complete nature of his communication.  Bahnson was still very much interested in finding a way to defeat gravity.  He wanted to be the first man in outer space and that motivated his gravity research activities.  He agreed with DeWitt that fundamental research would be key.  To this end he proposed to Bryce and Cecile DeWitt that they should head up a gravity study institute at a university.  The institute would be funded, (at least at first), by money raised by Agnew and his colleagues.  Bryce was inclined to turn down the offer, but John Archibald Wheeler, intervened and encouraged him to take the money.  Hence, the Institute for Field Physics at the University of Chapel Hill was born.

I became aware of the institute through an article that Bryce DeWitt had written about superconductors in the 1960s.

I noticed at the end of the article that the work had been funded by the United States Air Force.  Hoping that the article was part of a concerted research effort in the area, I performed a Google search on the funding number and came across a number of other research articles including the Higgs article referenced in Part I of this serial.

The grant in question had actually been made to the institute headed by the DeWitts.  By the time Higgs was finishing up his work, the institute had become quite successful and even had funds for visiting scientists like Dr. Higgs.  So, there you have it.  The concentrated efforts of two industrialists who would have rather seen a fringe physics solution to gravity wound up providing the institute and circuitously, the funding that supported the later stages of Dr. Higgs’ famous research.  As an interesting aside, when I asked one of the physicists involved how one went about obtaining Air Force funding in the ’60s, their answer was very amusing: “Oh, I don’t know, I suppose it was arranged at a cocktail party.  That’s how we did things in those days.”

If you’re a sci-fi fan, yes, the Robert Forward mentioned just above the funding notice is ‘that’ Robert Forward.  In addition to writing science fiction novels, he was also a research physicist who at the time was employed by Hughes Research Laboratories.  I’ve been unable to find out why he was collaborating with Bryce DeWitt on the study in question.