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11.30 k8 j lekner

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Plenary 3: J Lekner

Plenary 3: J Lekner

Published in: Health & Medicine

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  • 1. Electrostatics of two conductingspheresJohn Lekner MacDiarmid Institute and Victoria University, Wellington, New Zealand
  • 2. 2Pablo Etchegoin, Donald Pettit and Paul Callaghan
  • 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. Eric Le Ru Matthias Meyer
  • 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. 2 Eave a/s , a b E0 3 1 ln 4a 2 sField enhancement (equal spheres) as function of separation/radius
  • 7. Polarizability, longitudinal and transverse
  • 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. ( 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. 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. 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. 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. 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. 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. 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. James Clerk Maxwell, 1831-1879. Withcolour wheel (L), Katherine and Toby (R)
  • 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. Mutual energy of two spheres, b=2a, Qb=Qa/2
  • 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. William Thomson,Lord Kelvin1824-1907
  • 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. Energy of two charged spheres which had been in contact, as function of their separation
  • 23. William Thomson, Lord Kelvin 1824-1907 James Clerk Maxwell, 1831-1879
  • 24. Generalization of Kelvin force factor, as function of b/(a+b)
  • 25. Longitudinal and transverse polarizabilities, as functions of b/(a+b)
  • 26. bispherical coordinates i( z ) u iv ln i( z ) sin v  cosh u cos v sinh u z  cosh u cos v
  • 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. 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. 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. 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