- #1
Demon117
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Homework Statement
A rectangular hollow conductor,open at one end of [itex]z[/itex], has a voltage impressed at [itex]z=0[/itex]. On the [itex]z=0[/itex] plane, the voltage as a function of [itex]x[/itex] and [itex]y[/itex] is simple cosines such that [itex]V=0[/itex] at [itex]x=y=+/-L/2[/itex]. All other surfaces are grounded.
[itex]V(x,y)\approx f(cos(Ax),cos(By))[/itex]
Calculate the potential [itex]\Phi(x,y,z)[/itex] using an "adequate" number of terms. Plot [itex]\Phi(x,y)[/itex] for at least 5 values of [itex]z[/itex] (arbitrary units)
Homework Equations
After looking at the geometry and the statement of the problem these are the boundary conditions that I came up with:
(i) [itex]V=0[/itex] when [itex]y=+/-\frac{L}{2}[/itex]
(ii) [itex]V=0[/itex] when [itex]x=+/-\frac{L}{2}[/itex]
(iii)[itex]V=V_{0}(x,y)[/itex] at [itex]z=0[/itex]
This last one is an assumption I am making, but I am not quite sure it is valid for this configuration:
(iv)[itex]V\rightarrow\infty[/itex] as [itex]z\rightarrow\infty[/itex]
This is rectangular geometry so the cosines make sense, but I am not quite sure how to apply boundary conditions to get my coefficients of the expansion. Heck, I may be entirely wrong with my solution.
The Attempt at a Solution
The generalized double sum that I come up with is
[itex]\sum_{m,n} K_{m,n}e^{-2\pi z\sqrt(m^{2}+n^{2})/L}cos(2mx\pi/L)cos(2ny\pi/L)[/itex]
But this seems either overcomplicated or just entirely incorrect, because whenever I try to find the coefficients of expansion I get vanishing terms, which means that there is no solution. . . . I'm lost and could really use some clarification in this problem. Thanks in advance!