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Homework Help: PDE, 2D Laplace Equation, Sep. of Variables, Finding Potential

  1. Feb 19, 2012 #1
    1. The problem statement, all variables and given/known data
    A square rectangular pipe (sides of length a) runs parallel to the z-axis (from [itex]-\infty\rightarrow\infty[/itex]). The 4 sides are maintained with boundary conditions
    (i) V=0 at y=0 (bottom)
    (ii) V=0 at y=a (top)
    (iii) V=constant at x=a (right side)
    (iv) [tex]\frac{\partial V}{\partial x}=0[/tex] at x=0 (left side).
    Use separation of variables to find the potential V(x, y, z) inside the pipe.

    2. Relevant equations
    General solutions for 2D Laplace equation
    [tex] X(x)=Ae^{kx}+Be^{-kx}\qquad Y(y)=C\cos ky+D\sin ky[/tex] where A, B, C, and D are constants.

    3. The attempt at a solution
    I know since that C=0 in order to satisfy the boundary conditions V=0 for y=0 and y=a. I however do not know how to satisfy the boundary condition for the left side where dV/dx=0.
    This is what I've tried:
    [tex]\frac{d}{dx}[Ae^{kx}+Be^{-kx}]=kAe^{kx}-kBe^{-kx}[/tex]
    and set A=0 because [itex]kAe^{kx}[/itex] does not go to zero.
     
    Last edited by a moderator: Feb 19, 2012
  2. jcsd
  3. Feb 19, 2012 #2

    vela

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    Set the derivative equal to 0. This gives you a relationship between A and B.
     
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