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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.)