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Electrostatics of two conductingspheresJohn Lekner MacDiarmid Institute and Victoria University,         Wellington, New Z...
2Pablo Etchegoin, Donald Pettit and Paul Callaghan
1.Enhancement of external electric field (in  gap between spheres)2. Capacitance coefficients of two spheres3. Longitudina...
Eric Le Ru   Matthias Meyer
2  Eave             a/s                         , a b  E0         3 1 ln 4a               2 s         2Eave           a/s ...
2                                     Eave            a/s                                                                ,...
Polarizability, longitudinal   and transverse
Polarizability (a=b)                                4      L                          / 36     s       3             2 (3)...
( z)      d ln ( z) / dz                    (1)       0.5772                               21                         1  ...
Capacitance coefficients of two spheresQa    CaaVa        CabVb ,              Qb      CabVa             CbbVb          C(...
The factor of 4E. Weber, Electromagnetic fields, Wiley,1950, Volume1, page 232 (a=b capacitance formula):C(Q, Q) 2(Caa Cab...
1C aa     ab sinhU         [a sinh nU b sinh(n 1)U ]                    n 0                                               ...
Caa   1    sinhU    a beU          c                                  sin Uy(e 2 y 1) 1                 ln                ...
C (V ,V ) Caa    2Cab Cbb              ab                   a                 b                 2 (1)                     ...
1          1W    2   QaVa   2   QbVb ,   Qa   CaaVa   CabVb ,    Qb   CabVa   CbbVb                        Two spheres wit...
James Clerk Maxwell, 1831-1879. Withcolour wheel (L), Katherine and Toby (R)
a 2 b a 2 b 2 (a b) a 2 b 2 (a 3 a 2 b 2ab2 b 3 )C aa     a                                                      O (c 8 ) ...
Mutual energy of two spheres, b=2a, Qb=Qa/2
2ab                                  a                 b               (Qa   Qb ) 2 ln               4Qa Qb          2    ...
William Thomson,Lord Kelvin1824-1907
2                                               Q 4 ln 2 1                                                   a      Kelvin...
Energy of two charged spheres which had been in contact, as                function of their separation
William Thomson, Lord Kelvin         1824-1907                               James Clerk Maxwell, 1831-1879
Generalization of Kelvin force factor, as function of b/(a+b)
Longitudinal and transverse polarizabilities, as functions of b/(a+b)
bispherical coordinates                           i( z    )         u       iv ln                           i( z    )   ...
                           a             ,        za          sinh u a               tanhu a                            ...
ab sinhU    (a b s)                                 1    s (2a s )(2b s )(2a 2b s )   2              2(a b s )           ...
1                                        n 1 u                        n 1 u                       2  (coshu cos v)        ...
1                                S0 (U )                                                    n   0 e( 2n       1)U         ...
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
11.30 k8 j lekner
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11.30 k8 j lekner

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Plenary 3: J Lekner

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11.30 k8 j lekner

  1. 1. Electrostatics of two conductingspheresJohn Lekner MacDiarmid Institute and Victoria University, Wellington, New Zealand
  2. 2. 2Pablo Etchegoin, Donald Pettit and Paul Callaghan
  3. 3. 1.Enhancement of external electric field (in gap between spheres)2. Capacitance coefficients of two spheres3. Longitudinal and transverse polarizabilities of two spheres4. Electrostatic force between charged spheres
  4. 4. Eric Le Ru Matthias Meyer
  5. 5. 2 Eave a/s , a b E0 3 1 ln 4a 2 s 2Eave a/s , b aE0 2 1 ln 2a 2 s
  6. 6. 2 Eave a/s , a b E0 3 1 ln 4a 2 sField enhancement (equal spheres) as function of separation/radius
  7. 7. Polarizability, longitudinal and transverse
  8. 8. Polarizability (a=b) 4 L / 36 s 3 2 (3) O 2a 1 4a a ln 2 s 2 T 3 1 3 s s 3 (3) (3) ln 2 O 22a 4 2 4 a a 3 (3) n  1.2020569 n 1
  9. 9. ( z) d ln ( z) / dz (1) 0.5772 21 1 12 2 ln 2, 2 , 2 14 (3) 2 1(1 z ) z (z 1, 2, ) n 1 n(n z)
  10. 10. Capacitance coefficients of two spheresQa CaaVa CabVb , Qb CabVa CbbVb C(V , V ) Caa 2Cab Cbb Q(C ab Cbb ) Q(C ab C aa ) Va 2 , Vb 2 C aa Cbb C ab C aa Cbb C ab 2 Q C aa Cbb C ab C (Q, Q) Va Vb C aa 2C ab Cbb
  11. 11. The factor of 4E. Weber, Electromagnetic fields, Wiley,1950, Volume1, page 232 (a=b capacitance formula):C(Q, Q) 2(Caa Cab ) (wrong) J. B. Keller, J. Appl. Phys. 34 (1963) 991-993. J. D. Love, J. Inst. Math. Applics. 24 (1979) 255-257. A. D. Rawlins, IMA J. Appl. Math. 34 (1985) 119-120.
  12. 12. 1C aa ab sinhU [a sinh nU b sinh(n 1)U ] n 0 1Cbb ab sinhU [b sinh nU a sinh(n 1)U ] n 0 ab 1 C ab sinhU [sinh nU ] c n 1 2 2 2 c a b coshU 2ab
  13. 13. Caa 1 sinhU a beU c sin Uy(e 2 y 1) 1 ln 2(a b coshU ) sinhU dyab 2b cU a beU c 0 (a b coshU ) 2 c 2 cos2 UyCbb 1 sinhU aeU b c sin Uy(e 2 y 1) 1 ln 2(a coshU b) sinhU dyab 2a cU aeU b c 0 (a coshU b) 2 c 2 cos2 UyCab 1 sinhU eU 1 2 sin Uy(e 2 y 1) 1 ln U coshU sinhU dyab 2c cU e 1 c 0 cosh2 U cos2 Uy 1 2 2 2 c a b 2(a b) s 2 U arccosh O( s 3 / 2 ) 2ab ab ab 1 2ab b Caa 2 ln O( s ) a b ( a b) s a b ab 1 2ab a Cbb 2 ln O( s) a b ( a b) s a b ab 1 2ab Cab 2 ln O( s ) a b ( a b) s
  14. 14. C (V ,V ) Caa 2Cab Cbb ab a b 2 (1) O( s ) a b a b a bC (V ,V ) 2a ln 2 O(s) (a=b) 2 Caa Cbb CabC (Q, Q) Caa 2Cab Cbb 2 a b ab 2ab a b a b 1 / 2 ln O( s ) a b ( a b) s a b 2 a b a b a 1 4a C (Q, Q) ln Os (a=b) 2 2 s
  15. 15. 1 1W 2 QaVa 2 QbVb , Qa CaaVa CabVb , Qb CabVa CbbVb Two spheres with specified charges : Qa Cbb Qb Cab Qb Caa Qa CabVa 2 , Vb 2 Caa Cbb Cab Caa Cbb Cab 2 2 Q Cbb 2Qa Qb Cab Q Caa a b W 2 2(Caa Cab Cab )
  16. 16. James Clerk Maxwell, 1831-1879. Withcolour wheel (L), Katherine and Toby (R)
  17. 17. a 2 b a 2 b 2 (a b) a 2 b 2 (a 3 a 2 b 2ab2 b 3 )C aa a O (c 8 ) c2 c4 c6 ab2 a 2 b 2 (a b) a 2 b 2 (a 3 2a 2 b 2ab2 b 3 )Cbb b O (c 8 ) c2 c4 c6 ab a 2 b 2 a 2 b 2 (a 2 ab b 2 ) a 2 b 2 (a 2 ab b 2 )C ab O (c 9 ) c c3 c5 c7 Qa Qb2 Qa Qb Qa b 3 Qb2 a 3 Qa b 5 Qb2 a 5 2 2 2W 2a 2b c 2c 4 2c 6 2Qa Qb a 3b 3 Qa b 7 Qb2 a 7 3Qa Qb a 3b 3 (a 2 b 2 ) 2 10 O (c ) c7 c8 c9
  18. 18. Mutual energy of two spheres, b=2a, Qb=Qa/2
  19. 19. 2ab a b (Qa Qb ) 2 ln 4Qa Qb 2 2Qa 2Qb2 a b ( a b) s a b a bW O( s ) ab a b 2ab 2 a b 2 ln 2 2 a b a b ( a b) s a b a b 2 a b Qa Qb a b a b a bF 2 O(1) abs a b 2ab a b 2 2 ln 2 2 a b a b ( a b) s a b a b
  20. 20. William Thomson,Lord Kelvin1824-1907
  21. 21. 2 Q 4 ln 2 1 a Kelvin, 1853 (a=b): F0 2 2 (2a) 6(ln 2) ~ ~ ~ Qa Qb General case (2011): F0 2 f0 (a b) 2 (1 ) (2 1) ( ) (1 ) 1 (1 3 3 )2 ( ) (1 )f0 6 2 (1 )2 ( ) (1 ) 4 ln 2 1 f0 1 2 0.61490 b /(a b) 6(ln 2) 2 6 (3) 1 f0 ( 0 or 1) 2 0.83208 
  22. 22. Energy of two charged spheres which had been in contact, as function of their separation
  23. 23. William Thomson, Lord Kelvin 1824-1907 James Clerk Maxwell, 1831-1879
  24. 24. Generalization of Kelvin force factor, as function of b/(a+b)
  25. 25. Longitudinal and transverse polarizabilities, as functions of b/(a+b)
  26. 26. bispherical coordinates i( z ) u iv ln i( z ) sin v  cosh u cos v sinh u z  cosh u cos v
  27. 27.   a , za sinh u a tanhu a ua ubs za zb (a b)  tanh tanh 2 2 2 2 2 c a bcoshU cosh(u a ub ) 2ab
  28. 28. ab sinhU (a b s) 1 s (2a s )(2b s )(2a 2b s ) 2 2(a b s ) 1 3 2abs 2 2 O s a b
  29. 29. 1 n 1 u n 1 u 2 (coshu cos v) An e 2 Bn e 2 Pn (cosv) n 0 (solves Laplace’s equation) 1 n 1 u 2 1 2 (coshu cos v) e 2 Pn (cosv) n 0 1 n 1 u z  2 (coshu cosv) 2 (2n 1)e 2 Pn (cosv) n 0 1 n 1 u n 1 u 2V (u, v) E0 z 2 (coshu cos v) An e 2 Bn e 2 Pn (cosv) n 0 (solution on and outside the two spheres)
  30. 30. 1 S0 (U ) n 0 e( 2n 1)U 1 Abel-Plana formula:n2 n2 1 1 Im f (n2 iy) f (n1 iy) f (n) d f( ) 2 f (n1 ) 2 f ( n2 ) 2 dyn n1 n1 0 e2 y 1 U ln(eU 1) 1 [e 2 y 1] 1 sin 2Uy S 0 (U ) 2 2eU dy 2U 2U eU 1 0 e 2eU cos2Uy 1 1 2 U S 0 (U ) ln O(U 2 ) 2U U 144

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