Re: [Phys-L] Laplace equation problem

["Prof John P. Ertel" <jpe@usna.edu>]

But Carl,
     Just as we verified with Shivok several Math colleagues, a perfectly proper name for a "rectangular parallelepiped" is cylinder because it rises uniformly from its base with constant cross-section.  Perhaps you could teach our Majors a little extra fact by calling it a "cylinder with square cross-section" or maybe  just "square cylinder".

ERTEL SENDS!
+===========================+

On Apr 2, 2013, at 19:54, Carl Mungan <mungan@usna.edu> wrote:

> 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
mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/
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