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Mapping Maxwell’s Equations with Del Vector Diagrams

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


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