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The Charged Bowl in Toroidal Coordinates.Pdf The Charged Bowl in Toroidal Coordinates Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: January 5, 2016 Maple code is available upon request. Comments and errata are welcome. The material in this document is copyrighted by the author. The graphics look ratty in Windows Adobe PDF viewers when not scaled up, but look just fine in this excellent freeware viewer: http://www.tracker-software.com/pdf-xchange-products-comparison-chart . Overview and Summary.........................................................................................................................3 1. Bipolar and toroidal coordinates .......................................................................................................7 1.1 Bipolar coordinates .........................................................................................................................7 1.2 Toroidal coordinates .......................................................................................................................9 2. The charged bowl potential in toroidal coordinates ......................................................................16 2.1 Level curves, the charged ellipsoid problem and a dashed hope ..................................................16 2.2 Cartesian, spherical, toroidal and ellipsoidal atomic forms ..........................................................17 2.3 Smythian forms and one for the bowl...........................................................................................18 2.4 Solution to the charged bowl potential problem: the Mehler-Fock Transform............................19 2.5 Limiting cases of the potential......................................................................................................22 3. The charged double-bowl potential in toroidal coordinates..........................................................26 3.1 The general double bowl solution.................................................................................................26 3.2 Special case: a bowl with a flat lid................................................................................................29 3.3 Special case: a sessile drop (lens-shaped double bowl) ................................................................29 4. The charged bowl surface charge densities and capacitance ........................................................31 4.1 Bowl surface charge densities.......................................................................................................31 4.2 Comparison with known results, and the small-hole limit............................................................35 4.3 The full-sphere and flat-disk limits of σ .......................................................................................39 4.4 Bowl capacitance ..........................................................................................................................40 5. Plots of the toroidal coordinates u(ρ,z) and ξ(ρ,z)..........................................................................43 6. A selection of charged bowl potential plots....................................................................................49 7. Mehler integrals ................................................................................................................................52 7.1 List of useful Mehler integrals......................................................................................................52 7.2 Evaluation of the integral X appearing in (4.1.10)........................................................................53 7.3 Where to find Mehler integrals (some errata noted) .....................................................................55 7.4 Mehler Functions: hypergeometric forms and plots .....................................................................56 8. The bowl potential in elementary functions....................................................................................61 9. Using Maple to plot the bowl potential ...........................................................................................64 10. Potential and capacitance of a charged torus ...............................................................................67 10.1 The potential of a charged torus..................................................................................................67 10.2 How many terms should one keep in the potential series? .........................................................69 1 10.3 Using Maple to plot the torus potential.......................................................................................71 10.4 Capacitance of a torus.................................................................................................................74 10.5 Surface charge density on a torus ...............................................................................................77 10.6 Using Maple to plot the torus surface charge density.................................................................82 Appendix A. Maple code text for plotting the bowl and torus potentials .......................................86 Appendix B : Converting expressions from toroidal to cylindrical coordinates ............................89 B.1 The conversions............................................................................................................................89 B.2 Application: The Disk Potential..................................................................................................93 Appendix C: Kelvin's approach to the charged bowl problem........................................................96 Appendix D: Smythe's approach to the charged bowl problem ....................................................100 Appendix E: Dual equations and their connection to the charged bowl problem .......................105 Appendix F: The Abel Transform in various forms .......................................................................110 Appendix G: Solving the charged bowl problem using a double Abel transform .......................113 G.1 Find g1........................................................................................................................................113 G.2 Run a check on g1 ......................................................................................................................117 G.3 Find an ........................................................................................................................................117 G.4 Find Ψ and f2..............................................................................................................................119 Appendix H: Support.........................................................................................................................121 H.1 Solving for A(τ) and B(τ) in (2.4.10) and double-bowl support ................................................121 H.2 Computation of the integral in (2.4.7)........................................................................................127 H.3 Computation of the integral in (4.4.2)........................................................................................128 H.4 Computation of two integrals used in Appendix J and K ..........................................................129 H.5 Limits of toroidal functions (and combinations) as n→∞..........................................................133 H.6 Limits of toroidal functions (and combinations) as z→∞..........................................................135 H.7 Limits of toroidal functions as z→1...........................................................................................137 H.8 Alternate series for T(z) near z = 1 ............................................................................................139 Appendix I. Capacitance in the thin-wire and horn torus limits....................................................141 I.1 The torus thin wire limit R→0.....................................................................................................141 I.2 Warm-up exercises to prepare for the degenerate torus limit.....................................................145 I.3 Capacitance of a degenerate torus (horn torus) ...........................................................................150 Appendix J. The Fourier Cosine Series Transform and the Mehler-Fock Transform................154 J.1 Regular boundary value problems and associated transforms.....................................................154 J.2 The Fourier Cosine Series Transform .........................................................................................156 J.3 Application to f(θ) = (a-bcosθ)1/2 ...............................................................................................159 J.4 The Q2 and QQ' sums.................................................................................................................162 J.5 The Generalized Mehler-Fock Transform...................................................................................164 Appendix K: Integration of torus surface charge density..............................................................171 K.1 Warmup Exercises: Circumference and Area of a Torus..........................................................171 K.2 Integration of the toroidal surface charge density.....................................................................173 Appendix L: Some Mehler Integrals.................................................................................................176
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