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