Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Laplace Equation, Non-Concentric Spheres

  1. Dec 18, 2012 #1
    1. The problem statement, all variables and given/known data

    General solution for eccentric spheres, smaller sphere (radius, b) completely embedded within larger sphere of radius c. The centers of both spheres lie on z-axis, distance a, apart (note: c>b+a). Problem is symmetric, so consider θ=[0,∏], r=[0,c]. The inner sphere is grounded while the outer sphere is held at a potential f(c,θ).


    2. Relevant equations
    General solution (see Morse & Feshbach, 1953) is given in coordinates of larger sphere as:

    V(r,θ)=Ʃ1((r/a)^s×Ʃ2(A_n×(a/b)^n×[(-1)^(n-s)n!/s!(n-s)!])-(a/r)^(s+1)×Ʃ3(A_n×(b/a)^(n+1)×[s!/n!(s-n)!]))P_s(cosθ)

    V(c,θ)=f(c,θ), θ=[0,∏], r=[0,c]

    Limits of summations
    Ʃ1, s=0 to s=s_max (can use s_max=50)
    Ʃ2, n=s to n=n_max (can use n_max=50)
    Ʃ3, n=0 to s=s

    3. The attempt at a solution
    I have expanded the boundary function, f(c,θ) as a series of Legendre polynomials and computed the unknown coefficients A_n. For the special case when the offset (a) is zero,the solution behaves well (and converges nicely to prescribed boundary conditions, BCs). However, in cases where the offset is greater than zero, the solution diverges in the region where radii r<a. Any ideas on what might be going wrong?
     
    Last edited: Dec 18, 2012
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Laplace Equation, Non-Concentric Spheres
  1. Laplace equation (Replies: 3)

  2. Laplace equation (Replies: 2)

Loading...