Plenary 3: J Lekner

1. Electrostatics of two conducting
spheres
John Lekner
MacDiarmid Institute and Victoria University,
Wellington, New Zealand

2. 2
Pablo Etchegoin, Donald Pettit and Paul Callaghan

3. 1.Enhancement of external electric field (in
gap between spheres)
2. Capacitance coefficients of two spheres
3. Longitudinal and transverse
polarizabilities of two spheres
4. Electrostatic force between charged
spheres

4. Eric Le Ru Matthias Meyer

7. 2
Eave a/s
, a b
E0 3 1 ln 4a
2 s
2
Eave a/s
, b a
E0 2 1 ln 2a
2 s

9. 2
Eave a/s
, a b
E0 3 1 ln 4a
2 s
Field enhancement (equal spheres) as function of separation/radius

11. Polarizability, longitudinal and transverse

13. 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 2
2a 4 2 4 a a
3
(3) n 
1.2020569
n 1

14. ( z) d ln ( z) / dz
(1) 0.5772
2
1 1 1
2 2 ln 2, 2 , 2 14 (3)
2
1
(1 z ) z (z 1, 2, )
n 1 n(n z)

17. Capacitance coefficients of two spheres
Qa 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

18. The factor of 4
E. Weber, Electromagnetic fields, Wiley,1950, Volume
1, 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.

20. 1
C aa ab sinhU [a sinh nU b sinh(n 1)U ]
n 0
1
Cbb 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

21. Caa 1 sinhU a beU c sin Uy(e 2 y 1) 1
ln 2(a b coshU ) sinhU dy
ab 2b cU a beU c 0
(a b coshU ) 2 c 2 cos2 Uy
Cbb 1 sinhU aeU b c sin Uy(e 2 y 1) 1
ln 2(a coshU b) sinhU dy
ab 2a cU aeU b c 0
(a coshU b) 2 c 2 cos2 Uy
Cab 1 sinhU eU 1 2 sin Uy(e 2 y 1) 1
ln U coshU sinhU dy
ab 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

22. C (V ,V ) Caa 2Cab Cbb
ab a b
2 (1) O( s )
a b a b a b
C (V ,V ) 2a ln 2 O(s) (a=b)
2
Caa Cbb Cab
C (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

23. 1 1
W 2 QaVa 2 QbVb , Qa CaaVa CabVb , Qb CabVa CbbVb
Two spheres with
specified charges :
Qa Cbb Qb Cab Qb Caa Qa Cab
Va 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 )

24. James Clerk Maxwell, 1831-1879. With
colour wheel (L), Katherine and Toby (R)

27. 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 2
W
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

28. Mutual energy of two spheres, b=2a, Qb=Qa/2

29. 2ab a b
(Qa Qb ) 2 ln 4Qa Qb 2
2Qa 2Qb2
a b ( a b) s a b a b
W 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 b
F 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

30. William Thomson,
Lord Kelvin
1824-1907

33. 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 

34. Energy of two charged spheres which had been in contact, as
function of their separation

35. William Thomson, Lord Kelvin
1824-1907
James Clerk Maxwell, 1831-1879

36. Generalization of Kelvin force factor, as function of b/(a+b)

37. Longitudinal and transverse polarizabilities, as functions of b/(a+b)

39. bispherical coordinates
i( z )
u iv ln
i( z )
sin v

cosh u cos v
sinh u
z 
cosh u cos v

41.  
a , za
sinh u a tanhu a
ua ub
s za zb (a b)  tanh tanh
2 2
2 2 2
c a b
coshU cosh(u a ub )
2ab

42. 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

43. 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
2
V (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)

44. 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 dy
n 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