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Find potential between 2 conc. cyl. with grounded strip

  1. Nov 19, 2017 #1
    1. The problem statement, all variables and given/known data
    Two concentric cylinders with radii a & b (b>a) with an infinitely long grounded strip along the z-direction are given potentials [tex]\phi_1[/tex] and [tex]\phi_2[/tex].

    Find [tex]\Phi(r,\phi)[/tex] for a<r<b

    Boundary conditions:
    [tex]\Phi(r,2n\pi)=0[/tex]
    [tex]\Phi(a,\phi)=\phi_1[/tex]
    [tex]\Phi(b,\phi)=\phi_2
    [/tex]
    2. Relevant equations
    The laplace equation in polar coordinates (as the solution is independent of z)
    [tex]
    \Delta^2\Phi=0[/tex]
    Ansatz:
    [tex]\Phi=R(r)Q(\phi)[/tex],
    compute
    [tex]\frac{r^2}{R(r)Q(\phi)}\Delta^2(R(r)Q(\phi))=0[/tex]
    [tex]\frac{1}{r}\frac{d}{dr}(r\frac{dR}{dr})+\frac{1}{Q(\phi)}\frac{dQ(\phi}{d\phi^2}=0[/tex]

    Assume that the first term equals n^2, which means that the second one equals -n^2
    -> Solutions: (n=0):
    [tex]R(r)=a_0 +b_0ln(r);[/tex]
    [tex]Q(\phi)=A_0+B_0\phi[/tex]

    and (n>1):
    [tex]R(r)=a_nr^n +b_nr^{-n};[/tex]
    [tex]Q(\phi)=A_ncos(n\phi)+B_nsin(n\phi) [/tex]

    [tex]\Phi(r,\phi)=(a_0 +b_0ln(r))(A_0+B_0\phi)+\sum_{n=1}^\infty(a_nr^n +b_nr^{-n})(A_ncos(n\phi)+B_nsin(n\phi))[/tex]
    3. The attempt at a solution

    From the boundary conditions we see that the solution needs to be periodic in [tex]\phi=2n\pi; n=0,1,...[/tex] which leads to

    [tex]A_n=A_0=B_0=0[/tex]
    This gives the solution:
    [tex]\Phi(r,\phi)=\sum_{n=1}^\infty(a_nr^n +b_nr^{-n})sin(n\phi)[/tex]
    Where I have now put B_n coefficients into a_n and b_n.
    Now I need to satisfy
    [tex]\Phi(a,\phi)=\phi_1[/tex]
    [tex]\Phi(b,\phi)=\phi_2[/tex]

    [tex]\Phi(a,\phi)=\sum_{n=1}^\infty(a_na^n +b_na^{-n})sin(n\phi)=\phi_1[/tex]
    [tex]\Phi(a,\phi)=\sum_{n=1}^\infty(a_nb^n +b_nb^{-n})sin(n\phi)=\phi_2[/tex]

    The problem with this is that I would say that the solutions need to be symmetric for phi = [0,pi] and phi = [0,-pi] i.e. The solution should be mirrored in these two domains Phi(r,phi)=Phi(r,-phi), and thus it would be sufficient to solve the boundary conditions only in one of these domains. But only having sinus functions would lead to a discontinuity at phi=plusminus pi... which makes me question the legitimity of this solution.

    Furthermore I don't know how to satisfy the boundary conditions on r=a,b; I guess that if I assume phi_1>phi_2 then a_n=0, likewise, phi_1<phi_2 gives b_n=0 and phi_1=phi_2 gives R(r) = constant.

    But, to satisfy the finite constant potentials at r=a,b means that we can not have sin(n\phi) solutions i.e. n=0 Which also means that we can't satisfy the fact that the strip is grounded (Phi=0).

    So all in all I would assume that the ansatz is wrong... can anyone give a helping hand??
     
  2. jcsd
  3. Nov 19, 2017 #2
    Obviously, if I may comment, the strip is situated at a<r<b and thus not in contact with the two cylinders (which would otherwise mean that they would all be equipotential and grounded). I wonder if the solution is a combination of two cases, r=a,b and a<r<b? What if I made the strip such that it was between a+d<r<b-d and then let d->0 (but never reach d=0?). Or is the ansatz simply wrong?
     
  4. Nov 19, 2017 #3

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Consider whether or not the boundary conditions can be satisfied with ##n## not being an integer.
     
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