- #1
Illeism
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Homework Statement
I'm trying to derive Equation (1) from the paper: http://idv.sinica.edu.tw/jwang/EP101/Paul-Trap/Winter 91 ajp demo trapping dust.pdf
We are working with a cylindrically symmetric geometry along the z-axis.
[itex]r^2 = x^2 + y^2[/itex]
We have electrodes described by the hyperbolas:
[itex]z^2 = z_0^2 + r^2/2[/itex]
[itex]z^2 = r^2/2 - z_0^2[/itex]
The top and bottom electrode (described by [itex]z^2 = z_0^2 + r^2/2[/itex]) are held at a potential [itex]V_0[/itex] with respect to our ground ring (described by [itex]z^2 = r^2/2 - z_0^2[/itex]) held at potential [itex]0.[/itex]
We want to find the potential V(r,z) for the region inside the hyperbolas (the region containing the origin).
The solution is:
[itex]V(z,r) = V_0(\frac{1}{4z_0^2})(2z^2+r_0^2-r^2)[/itex]
Homework Equations
[itex]\nabla^2 V = 0[/itex]
(no charge inside volume)
The Attempt at a Solution
So, for boundary conditions we have:
[itex]V(z,r)=V_0[/itex] when [itex]z^2 = z_0^2 + r^2/2[/itex]
[itex]V(z,r)=0[/itex] when [itex]z^2 = r^2/2 - z_0^2[/itex]
I can't think of any appropriate image that would generate hyperbolic potentials, and following Jackson's section on boundary value problems (physics.bu.edu/~pankajm/LN/hankel.pdf) seems to be giving unnecessarily complex solutions, even though the final solution is quite simple.
At this point I'm at a loss. Is there a better way to approach this problem?