Lecture Thoughts

Waveguides are fun. I mean on the page, although I imagine that they are fun to play with as fully realized physical objects too. But I was talking about them in the context of teaching undergraduate electromagnetism, as I am doing on my class this semester. I tell the class after the second week of class or so that we’re essentially done, and can all head to the beach since by then we’ve completed the derivation of Maxwell’s equations, which describe fully all electromagnetic phenomena. The rest of the class is essentially a semester of picking various situations in which we deploy the equations and study particular solutions. Of course, they realize that there is reason to stay, since that’s really the heart of it – studying those various situations and appreciating the range of delights those equations can yield. Among the most fascinating and delightful of those, er, delights, is light. Electromagnetic waves in general, and we study them in a whole lot of situations, including nipping along unfettered in free space, in conducting materials (where it penetrates for a while before dying away), at interfaces between different kinds of materials (we learn why metals are shiny quite readily this way, for example), and then there is the business of forcing the wave to propagate down a pipe – a waveguide. It turns out to be one of the nicest and cleanest examples of a dispersive setup (frequency dependent speed of propagation) one can devise, and so that’s another nice reason to study it, and it is simply cool the way you can tune the pipe to pick the characteristics of the wave you want to propagate, analogously to tuning a musical instrument. So I hope it is an example that the students like as much as I do.

I must confess that the last lecture I did on the topic, which was on Monday, was not as satisfying as I felt it should be, maybe because Spring break interrupted the treatment of the topic… So although we’ve started a new topic, and are continuing it today, I might just go beyond the text (Griffith sort of leaves the reader a bit high and dry on the topic all of a sudden, ending rather abruptly) a bit more and point out a few things more about the fields in the waveguides that might help cement them in the minds of the students. Let’s see if I have a bit of time before the lecture to solve for all the fields and show their shapes explicitly in an example or two…. [Update: I developed some Maple plots of some of the field configurations for a “TE01 mode” moving along the z direction. E is blue and B is red. This is not a particularly illustrative plot since there’s too much going on visually on one plot, but I thought it was nice and decorative, so kept it…You can sort of see that the magnetic fields swirl around in loops in the x-z plane, making a sort of series cylindrical columns that lie along the y direction, while the electric fields thread those columns in bunches, running along the y direction…]

The other thing that’s fun to mention in this context is waves in cavities, as it gives one the chance to mention two great examples – microwave ovens and the earth-ionosphere system. We met the latter earlier and I mumbled at length about my time as a boy listening to shortwave radio signals from stations very far away, which the ionosphere helps you with. There’s a notable day vs. night effect that is in part traceable in the analysis we do in class, which is fun…

What I’ve never done is the experiment where you use a microwave oven to determine the speed of light by looking at the pattern of hotspots and cold spots (they tell you where the nodes of the electromagnetic standing waves are)… I keep mentioning it in class as a nice example, but one of these days I must go and actually do it myself and see how good a measurement one can get…

Anyway, I’m sort of babbling here…. Better prepare that class.

-cvj