Solving the Laplace Equation for a Capacitor Setup

[hectoryx]

Homework Statement

The capacitor is assumed to consist of two parallel circular disc electrodes of radius R. The electrodes are of infinite small thickness, placed a distance 2H apart, and are equally and oppositely charged to potentials +U and -U. A metal cylinder is placed near the two electrodes and the position relationship can be found in the following picture:

http://i1021.photobucket.com/albums/af335/hectoryx/professional/1.jpg

Homework Equations

To solve the potential distribution in this situation, the Laplace Equation in cylindrical coordinate system is:

\[{\nabla ^2}\phi = \frac{1}{r}\frac{{\partial \phi }}{{\partial r}} + \frac{{{\partial ^2}\phi }}{{\partial {r^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {z^2}}} = 0\]

The Attempt at a Solution

I am not sure about its boundary condition:

\[\left\{ {\begin{array}{*{20}{c}}<br /> {\phi = + {\rm{U}},\begin{array}{*{20}{c}}<br /> {} &amp; {{\rm{z}} = {\rm{H}}} \\<br /> \end{array},0 \le r \le {\rm{R }}} \\<br /> {\phi = - {\rm{U}},\begin{array}{*{20}{c}}<br /> {} &amp; {{\rm{z}} = - {\rm{H}}} \\<br /> \end{array},0 \le r \le {\rm{R}}} \\<br /> \end{array}} \right.\]

and

\[\begin{array}{l}<br /> \phi = {\phi _c},\begin{array}{*{20}{c}}<br /> {} &amp; {{\rm{H}} + {\rm{d}} \le {\rm{z}} \le {\rm{H}}} \\<br /> \end{array} + {\rm{d}} + {\rm{L}},0 \le r \le {\rm{R}} \\ <br /> \frac{{\partial \phi }}{{\partial z}} = {\sigma _1},\frac{{\partial \phi }}{{\partial r}} = 0,\begin{array}{*{20}{c}}<br /> {} &amp; {{\rm{z}} = {\rm{H}} + {\rm{d}}} \\<br /> \end{array},0 \le r \le {\rm{R}} \\ <br /> \frac{{\partial \phi }}{{\partial z}} = {\sigma _2},\frac{{\partial \phi }}{{\partial r}} = 0,\begin{array}{*{20}{c}}<br /> {} &amp; {{\rm{z}} = {\rm{H}} + {\rm{d}}} \\<br /> \end{array} + {\rm{L}},0 \le r \le {\rm{R}} \\ <br /> \frac{{\partial \phi }}{{\partial z}} = 0,\frac{{\partial \phi }}{{\partial r}} = {\sigma _3},\begin{array}{*{20}{c}}<br /> {} &amp; {{\rm{H}} + {\rm{d}} \le {\rm{z}} \le {\rm{H}}} \\<br /> \end{array} + {\rm{d}} + {\rm{L}},r = {\rm{R}} \\ <br /> \end{array}\]<br />


Could anyone give me some help and tell me that whether the boundary conditions above are right?

Thanks very much!

Best Regards.

Hector

 

[hectoryx]

chould anyone help me please? Really thanks!