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Plenary 3: J Lekner
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Electrostatics of two conducting spheres
John Lekner
MacDiarmid Institute and Victoria University,Wellington, New Zealand
2
Pablo Etchegoin, Donald Pettit and Paul Callaghan
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
Eric Le Ru Matthias Meyer
ab
sasa
E
Eave
,2
ln21
/
2
2
0
ba
sasa
E
Eave
,4
ln21
/
3
2
0
Field enhancement (equal spheres) as function of separation/radius
ba
sasa
E
Eave
,4
ln21
/
3
2
0
Polarizability, longitudinal and transverse
a
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s
aaL
4ln2
136/
)3(22
4
3
2
2
32ln)3(
4
3
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1)3(
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3
2 a
sO
a
s
aT
2020569.1)3(1
3
n
n
Polarizability (a=b)
dzzdz /)(ln)(
5772.0)1(
1
),2,1()(
1)1(
n
zznn
zz
)3(14,2
,2ln2 21
2
21
21
bbbaabbbabaaaa VCVCQVCVCQ ,
bbabaa CCCVVC 2),(
22
)(,)(
abbbaa
aaabb
abbbaa
bbaba
CCC
CCQV
CCC
CCQV
bbabaa
abbbaa
ba CCC
CCC
VV
QQQC
2
),(2
Capacitance coefficients of two spheres
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.
E. Weber, Electromagnetic fields, Wiley,1950, Volume 1, page 232 (a=b capacitance formula):
)(2),( abaa CCQQC (wrong)
The factor of 4
1
1
0
1
0
1
][sinhsinh
])1sinh(sinh[sinh
])1sinh(sinh[sinh
nab
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C
y
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22
12
0
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0
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coscosh
)1(sinsinhcosh
2
1
1ln
sinh
2
1
cos)cosh(
)1(sinsinh)cosh(2ln
sinh
2
1
cos)cosh(
)1(sinsinh)cosh(2ln
sinh
2
1
)()(2
2arccosh 2/32
1222
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)()(
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)()(
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)()(
2ln
21
21
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sOba
a
sba
ab
ba
abC
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b
sba
ab
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aa
)(2ln2),( sOaVVC
)()1(2
2),(
sOba
b
ba
a
ba
ab
CCCVVC bbabaa
)(2
)(
2ln2/1
2),(
2
2
sO
bab
baa
bab
baa
sba
ab
ba
ab
CCC
CCCQQC
bbabaa
abbbaa
(a=b)
(a=b) sOs
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4
ln2
1
2),(
Two spheres with specified charges :
22,
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abbbaa
abbbbaa CCC
CQCQV
CCC
CQCQV
)(2
22
22
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aababbabba
CCC
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bbbaabbbabaaaabbaa VCVCQVCVCQVQVQW ,,21
21
James Clerk Maxwell, 1831-1879. With colour wheel (L), Katherine and Toby (R)
)()()(
)()22()(
)()2()(
97
2222
5
2222
3
22
86
322322
4
22
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2
86
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4
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2
2
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c
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babbaaba
c
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c
baaC
ab
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)()(32
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109
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7
33
6
5252
4
323222
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c
aQbQ
c
baQQ
c
aQbQ
c
aQbQ
c
b
Q
a
QW
bababa
babababa
Mutual energy of two spheres, b=2a, Qb=Qa/2
)(
22)(
2ln2
224)(
2ln)(
2
222
sO
ba
b
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a
sba
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)1(
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O
ba
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b
ba
a
ba
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abs
baF
ba
William Thomson, Lord Kelvin1824-1907
22
2
0 )2(ln6
12ln4
)2(
a
QF aKelvin, 1853 (a=b):
020 )(
~~~
fba
QQF ba
)1()()1(6
)1()(2)331(1)1()()12()1(22
2
0
f
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12ln422
10
f
83208.01)3(6
)1or0(20
f
)/( bab
General case (2011):
Energy of two charged spheres which had been in contact, as function of their separation
William Thomson, Lord Kelvin1824-1907
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)
)(
)(ln
zi
ziivu
vu
uz
vu
v
coscosh
sinh
coscosh
sin
bispherical coordinates
2tanh
2tanh)( ba
ba
uubazzs
)cosh(2
cosh222
ba uuab
bacU
aa
a uz
ua
tanh,
sinh
2
32
1
2
1
2
)(2
)22)(2)(2(
)(
sinh
sOba
abs
sba
sbasbsas
sba
Uab
)(cos)cos(cosh
0
2
1
21
21
vPeBeAvu nn
un
n
un
n
0
2
1
)(cos)cos(cosh21 21
nn
unvPevu
0
2
1
)(cos)12()cos(cosh2 21
nn
unvPenvuz
)(cos)cos(cosh2),(
0
2
1
021
21
vPeBeAvuzEvuV nn
un
n
un
n
(solves Laplace’s equation)
(solution on and outside the two spheres)
1
)()(Im2)()()()(
212
0221
1212
1
2
1
y
n
n
n
nn e
iynfiynfdynfnffdnf
12cos2
2sin]1[2
12
)1ln()(
2
12
0
21
0
Uyee
Uyedye
eU
eUUS
UU
yU
U
U
0
)12(0 1
1)(
nUne
US
Abel-Plana formula:
)(144
2ln
2
1)( 2
0 UOU
UUUS
