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Re: [Phys-L] Laplace equation problem

I am planning to put the following problem on a test tomorrow and I'm somewhat surprised by the answer. So let me ask the list for insight:

A square waveguide extends from -infinity to infinity in the z direction. It has two grounded plates at y=-0.5 and at y=0.5. The plate at x=-0.5 has potential -sin(pi*y). The plate at x=0.5 has potential +sin(pi*y). What is the surface charge density on the plate at y=-0.5?

(Actually the first part of the problem asks one to find the potential everywhere inside the waveguide. I give them the hint to just write down the appropriate form to match the boundary conditions. For those a bit rusty on this kind of problem, this means a sinh in the x direction and a sine in the y direction, with each coordinate multiplied by pi. You can now find the overall scaling constant by fitting to the boundaries.)

Please see what you get for the preceding question and send me any comments so I can revise the problem if necessary before the test! Thanks, Carl

Okay, I found the difficulty. I seem to have chosen anomalous coordinates for the edges. If I instead choose to put the edges at x=1, x=-1, y=1, and y=-1 then I get a much more reasonable answer.

However... what if I *do* stick with my original choice? I then have the strange situation of having a tangential electric field along the surface of a metal plate! How can that possibly be? I'm still puzzled by that. -Carl

ps: It would also be better if I didn't call the two edges parallel to the y axis "plates" since they obviously cannot be metal because the potential is not constant for them. (You might also quibble with my calling the whole thing a "waveguide" when we're doing statics and not guiding waves. But I do that just because I don't know what else to call this thing. A "box" or a "slot" are obviously no good. I could call it a "cylinder" but then some of my students would expect the cross section to be circular. In class I told them I call it a "waveguide" just because the shape reminds me of a microwave guide.)
Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363