Finite Cylinder in an electric field

  • Thread starter TOKAMAK
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Hi, I was just wondering if anyone could might know where maybe the following situation has been worked out: A finite cylinder of length L sticks out of a flat, horizontal plane. The plane and the finite cylinder are both grounded, and are placed into a previously uniform electric field, E = E0 (in the z-direction)

Anyway, I attempted the problem in the usual way, seperation of variables. Basically, I got as far as V ~ (J-zero(kr))*(sinh(kz))
(where the integral should be over all k>0). I thought maybe I should first concentrate on one section at a time, like get V below z=L, then make sure this expression matches V from above z=L, but that doesn't really get me anywhere, since at most V from above would be a linear combination of sinh and cosh, which is just e^kz. I can't get any useful coefficients (the f(k)'s, they are continuous of course) from boundary conditions, at least that I can see. Is this problem analytically possible? If not, I'll probably just try to use the greens' function for a spherical boss on a grounded plane; I just wanted to see for myself how the the E-field deforms around a grounded object. I've seen the drawings in text-books, and I know what should happen, but I kinda wanted to look at a simple object and see the exact expression for the E-field or the equipotentials
 

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