summary of disk green attempts

Summary of disk green attempts PhL 7.6.10 These attempts were made April 4-22, 2010. I am trying to find the Green's Function for a disk with the point charge in the plane of the disk. All these attempts use a 3-region Smythian Form which is probably just no good, and that is why I have so much trouble. Eventually I learn (elsewhere) to solve this problem by inversion and get a result in terms of elementary functions for the potential and the induced charge. Attempt #1. Here I attempt a 3-region Smythian form for the disk (radius a) with point charge in plane Green's problem (distance b from origin). I use cylindricals, so atoms include Bessel functions like Jm(kρ). Each region is a cylinder, and there are four sets of coefficients. At the point charge boundary I try my usual pillbox thing, but in the z direction I have expo decay so no orthog in z, and I end up stuck: (q/ε)(1/b) δ(z) = 2π/(2-δm,0) !Syntax Error, Idk k e-k|z|{ Bm(k) Jm'(kb) + Cm(k) H(1)m'(kb) – Dm(k) H(1)'m(kb) } I then state that the entire rest of the doc is scraps, so I skip it here and move on to the next attempt. Attempt #2. In order to avoid the above problem, I try again, this time with cos(kz) so z is then an oscillatory direction. This means the Bessel functions are I and K type. As before, we have 4 coefficients. I get down to the point analogous to the above, but now have (q/ε)(1/b) δ(z) = 2π/(2-δm,0) !Syntax Error, Idk k cos(kz){ Bm(k) Im'(kb) + Cm(k) Km'(kb) – Dm(k) K'm(kb) } I then digress on the Fourier Cosine Transform (and eventually added it to transforms.doc) and the above becomes (q/ε)(1/b) = 2π/(2-δm,0) (π/2) k { Bm(k) Im'(kb) + Cm(k) Km'(kb) – Dm(k) K'm(kb) } which is more reasonable. I come up with four conditions, this is just one of them. I seem to grind things down so there is only one set of coefficients cm(k). Somehow I am led here to something like the dual integral equation situation. I then learn for the first time about Neumann's Factor (or Number), and collect historical information on it (invented by Watson). I continue on, but am led nowhere good. I suspect that this triple-cylinder Smythian form is just no good. Attempt #3. Undaunted, I start over again using the cos(kz) basis and the same three cylindrical regions. This time I try harder to find my coefficients cm(k). I end up with a contradiction (again suggesting the is a bad form). Attempt #4. Now I am back to e-k|z| for z. This time I am trying to get some dual integral equations. In this and previous attempts I keep arriving at this point, heading toward this dual integral things, V1(0,ρ,φ) = 0 ρ ≤ a Dirichlet ∂zV2(0,ρ,φ) = 0 a < ρ ≤ b Neumann ∂zV3(0,ρ,φ) = 0 b < ρ < ∞ Neumann, except for the Green's charge After removing the φ dependence, I end up with something that looks like such things: !Syntax Error, Idk Am(k) Jm(kρ) = F(ρ,m,b) ρ ≤ a !Syntax Error, Idk k [ Bm(k) Jm(kρ) + Cm(k) H(1)m(kρ)] = 0 a < ρ ≤ b !Syntax Error, Idk k Dm(k) H(1)m(kρ) = 0 b < ρ < ∞ Of course since I have three regions, I guess I have triple integral equations (these are in fact a subject of literature! [ and are in Sneddon ]) I flail a while, then.. I start with new forms where I show the point charge explicitly, so the expansions are then only for the induced charge on the disk. But this too just peters out. Here is my closing comment: " Somehow, I am just plain approaching this problem wrong. I just don't have any clear view of how a solution is to be obtained. I wanted to try the "Smythian form" approach because that is what Jackson did for his charged disk problem, and I wanted to do this in cylindricals for the same reason. The Smythian forms might be OK, but I just don't know how to solve for the coefficients. There MUST be a better way to go about this problem. Somehow, this problem is harder than all the Smythe problems he and I have done to date. So I will refer to this series of Attempts (now 1,2,3,4) as my 3-region cylindrical-coordinates Smythian Form method. I tried pretty hard for several weeks at least to make this method fly, but I could not make it fly. " Since the time of writing these docs in April (it is now July), I have learned how to do this problem by inversion. And I know how to do it in oblates. I think my trouble in these "attempts" is that the three cylinder Smythian form is just plain wrong. Yes, it contains valid atoms, but that does not a valid form make. If I were trying it again, I would break out the point charge, and treat the induced charge on the disk as a single expansion that was Jackson-like for the charged disk. By forcing V = - q/R on the disk, and forcing ∂V/∂z = 0 in the rest of the z=0 plane, a set of dual integral equations would be quickly arrived at. This would be a one-region fit, not a 3 region fit. I basically do this for the iris in my iris folder, and it would be the same idea for the disk. Of course then I don't know how to solve the dual integral equations, so I get nowhere. But I think there have been modern advances in this subject. [ Sneddon shows how to do it all! ]