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EEGrad
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Homework Statement
Consider a shorted coaxial line with a resistive inner conductor with radius a, and a perfect outer conductor located at radius b. A DC voltage is applied at the input end. Find the electric and magnetic fields in the dielectric region (a<r<b), assuming the battery sets up a potential that varies as ln(r) on the input surface.
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Homework Equations
Not exactly sure if there is a better way, but I started by looking at Poisson's equation
[tex] \nabla^2 V = \frac{\rho}{\epsilon} [/tex]
The Attempt at a Solution
Since there is no charge in the dielectric, Poisson's equation becomes
[tex] \nabla^2 V(r,z) = 0 [/tex]
where V has rotational symmetry. In cylindrical coordinates,
[tex] \frac{1}{r} \frac{\partial }{\partial r} \left( r \frac{\partial V(r,z)}{\partial r} \right) + \frac{\partial^2 V(r,z)}{\partial z^2} =0 [/tex]
The boundary conditions look like:
[tex] V(r=a) = V_0 - \frac{V_0 z}{L}\\
V(r=b) = 0\\
V(z=L) = 0\\
[/tex]I'm not sure how to solve this differential equation. Can you use separation of variables to get:
[tex] V(r,z) = R(r)Z(z) \\
\frac{1}{R}\frac{1}{r} \frac{\partial}{\partial r}\left( r \frac{\partial R}{\partial r} \right) = - \frac{1}{Z} \frac{\partial^2 Z}{\partial z^2}\\
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Even if I do that though, I'm not quite sure how to solve the R portion. Any nudge in the right direction would be appreciated. Thanks!