# Thread: 3d poisson's equation problem

1. ## 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.
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I tried solving using the separation of variables method and failed miserably. I got V(x,y,z)=0

2. Originally Posted by vganpat
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).