Brief comments on the charged bowl problem

Brief Comments on Methods for the charged bowl problem PhL 7.9.10 As a first method, I think this problem can be fully solved in toroidal coordinates. I think this because the surfaces of constant potential in that system are in fact toroids and spherical bowls! There is good web action on this subject, and I have the problem in Lebedev's book. So this is a first possible "method". A second method is to use inversion. Start with an offset charged metal disk + neutralizing constant potential and end up with a grounded metal spherical bowl with a point charge at an arbitrary point on the cap. This is of course a special case of the Green's function for the bowl. I did this problem (both Φ and σ), and went on to rotate the bowl and invert a second time to get the Green's function for an iris with point charge in the hole. (And I later found out how to do this iris problem more directly, and I also did the Green's function for the disk.) The main point here is that this entire effort produces only a special case of the bowl Green's function, which is a very different problem from that of the charged bowl (but see next paragraph). Then if you integrate the solution of the above problem over the entire cap (perhaps first doing a ring), you can obtain results for a grounded metal bowl in the presence of a uniformly charged cap. I have done this for σ but not for Φ. At first, this situation seems of little interest until you realize that you can superpose on it a uniformly charged sphere which cancels the cap charge and raises the bowl potential from 0, and you end up with the charged bowl situation! In this way I have derived expressions for the charge density on the inner and outer surface of the bowl (they differ by a constant). This is a famous result, and Kelvin did it in 1847 and even plotted the charge distribution for various bowl cases. Perhaps he just gave numbers, I forget. Smythe uses this same method embedded in a sequence of problems. A third method is the Sneddon approach using dual series equations, which involves assuming a reasonable Smythian form for the potential. Comment on the first method. Realizing that the "bowl" is a constant parameter surface in toroidal coordinates, I am motivated to find the potential of a charged bowl in these coordinates. Even in messy ellipsoidal coordinate, we know the potential of an ellipsoid whose surface has the constant value ξ = ξ1, to wit V(ξ1) = V0 F[sin-1(g/ ξ1), f/g] / F[sin-1(g/a), f/g] // restatement of MF above V(ξ1) = V0 F[sin-1(f/ ξ1), g/f] / F[sin-1(f/a), g/f] // this line is "the Kelvin solution" semis: ξ1= a ≥ b ≥ c focals: g2 ≡ a2 - c2 and f2 ≡ a2 - b2 g > f The way this simple result arises (ie, it is not some messy double sum) is that one conjectures a single term separated form as the product of three Lamé functions two of which are 1, and the third is the above (which is the lowest second-kind Lamé). This result has constant V=V0 on the ellipsoid ξ = ξ1 , has the right large distance decay, and solves Laplace, so it must be the one and only solution. As of 9.26.10, I have not carried out this solution method, but I still intend to.