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Finding potential from Laplace's eq'n

  1. Dec 6, 2005 #1
    We start with Laplace's eq'n in 2-d place polar co-ordinates and find all single valued separable solutions which gives us the general solution:

    V(r,theta) = A + Blnr + sum from 1 to infinity of {(Cn*sin(n*theta) + Dn * cos(n*theta))*(En*r^-n + Fn *r^n) }

    we also note that V is always finite

    we then have to find V in certain regions with certain B.C's

    we are given that V(a,theta)=Vo + Vo*cos(theta) and that V(b,theta)=2Vo*(sin^2(theta))
    and we need to show that for a<r<b:
    V(r,theta)=Vo{1 + a/(a^2 - b^2) * (r - (b^2)/r)*cos(theta)
    - b^2/(b^4 - a^4)*(r^2 - (a^4)/r^2)*cos(2*theta)}

    I can easily find the potential for r<a and r>b but am not sure what to do for middle section!! thank very much
  2. jcsd
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