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Math Help - 3d poisson's equation problem

  1. #1
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    3d poisson's equation problem

    Problem:
    Consider a 3d rectangular region with dimensions 0<=x<=w,0<=y<=h,0<=z<=L.
    Potential is 0 on all surfaces except for the surface on the z=0 plane, where
    v(x,y)=(w/2-|x-w/2|)(h/2-|y-h/2|).
    Calculate V(x,y,z) assuming no charges or dielectrics are present within the region.
    -------------------------------------------------------------------------
    I tried solving using the separation of variables method and failed miserably. I got V(x,y,z)=0
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  2. #2
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    Quote Originally Posted by vganpat View Post
    Problem:
    Consider a 3d rectangular region with dimensions 0<=x<=w,0<=y<=h,0<=z<=L.
    Potential is 0 on all surfaces except for the surface on the z=0 plane, where
    v(x,y)=(w/2-|x-w/2|)(h/2-|y-h/2|).
    Calculate V(x,y,z) assuming no charges or dielectrics are present within the region.
    -------------------------------------------------------------------------
    I tried solving using the separation of variables method and failed miserably. I got V(x,y,z)=0
    I would recommend defining U(x,y,z)= (w/2-|x-w/2|)(h/2-|y-h/2|)z and define [tex]W(x,y,z)= V(x,y,z)- U(x,y,z)[/itex] then W will be 0 on all surfaces. \nabla^2W= \nabla^2V- \nabla^2U= -\nabla^2U.
    You should be able to find \nabla^2U (treat -w/2< x< 0, 0< x< w/2, -h/2< y< 0, and 0< y< h/2 separately) and then apply "separation of variables" to solving for W(x,y,z).
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