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Boundary conditions for a metal cylinder placed in electric field of parallel electro

  1. Jan 15, 2010 #1
    1. The problem statement, all variables and given/known data


    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


    2. Relevant equations

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

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

    3. The attempt at a solution

    I am not sure about its boundary condition:

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

    and

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


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

    Thanks very much!

    Best Regards.

    Hector
     
  2. jcsd
  3. Jan 25, 2010 #2
    Re: Boundary conditions for a metal cylinder placed in electric field of parallel ele

    chould anyone help me please? Really thanks!
     
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