- #1
koopatroopa
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The problem: Find the potential between 2 concentric spheres, of radii a & b, where the potential of the spheres are held at Ca*P3(cos(t)) & Cb*P5(cos(t)), where P3 & P5 are the 3rd & 5th Legendre functions, t is theta & Ca & Cb are constants.
The general solution to Laplace with azimuthal symetry:
V = sum over l of (Al*(r^l)+Bl*r^(-(l+1)))*Pl(cos(t))
I can use the orthoganality of the Legendre functions to elimimate all but one of the series terms when I plug in the boundry conditions, giving:
V(a,t) = (A*a^3 + B*a^(-4))P3(cos(t)) = Ca*P3(cos(t))
V(b,t) = (C*b^5 + D*b^(-6))P5(cos(t)) = Ca*P5(cos(t))
The solution should be a combination of these, but I'm a bit stumped about how to find the constants when I can't use r=zero or r=infinity to zero one of them (as is done in single shell problems). I have a feeling this may become obvious after a good sleep, but I'd be grateful for a nudge in the right direction.
The general solution to Laplace with azimuthal symetry:
V = sum over l of (Al*(r^l)+Bl*r^(-(l+1)))*Pl(cos(t))
I can use the orthoganality of the Legendre functions to elimimate all but one of the series terms when I plug in the boundry conditions, giving:
V(a,t) = (A*a^3 + B*a^(-4))P3(cos(t)) = Ca*P3(cos(t))
V(b,t) = (C*b^5 + D*b^(-6))P5(cos(t)) = Ca*P5(cos(t))
The solution should be a combination of these, but I'm a bit stumped about how to find the constants when I can't use r=zero or r=infinity to zero one of them (as is done in single shell problems). I have a feeling this may become obvious after a good sleep, but I'd be grateful for a nudge in the right direction.