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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
sO
s
aaL
4ln2
136/
)3(22
4
3
2
2
32ln)3(
4
3
2
1)3(
4
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
nbb
naa
nUUc
abC
UnanUbUabC
UnbnUaUabC
ab
bacU
2cosh
222
UyU
eUydyUU
ce
e
cU
U
cab
C
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eUydyUbUa
cbae
cbae
cU
U
aab
C
UycUba
eUydyUUba
cbea
cbea
cU
U
bab
C
y
U
Uab
y
U
Ubb
y
U
Uaa
22
12
0
222
12
0
222
12
0
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
sOab
sba
ab
bacU
)()(
2ln
)()(
2ln
)()(
2ln
21
21
21
sOsba
ab
ba
abC
sOba
a
sba
ab
ba
abC
sOba
b
sba
ab
ba
abC
ab
bb
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
aaQQC
4
ln2
1
2),(
Two spheres with specified charges :
22,
abbbaa
abaaabb
abbbaa
abbbbaa CCC
CQCQV
CCC
CQCQV
)(2
22
22
ababaa
aababbabba
CCC
CQCQQCQW
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
2
2
86
322322
4
22
2
2
cOc
bababa
c
bababa
c
ba
c
abC
cOc
babbaaba
c
baba
c
abbC
cOc
babbaaba
c
baba
c
baaC
ab
bb
aa
)()(32
2222
109
2233
8
7272
7
33
6
5252
4
323222
cOc
babaQQ
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
ba
a
sba
ab
ba
b
ba
a
ba
bQ
ba
aQQQ
sba
abQQ
ab
baW
bababa
)1(
22)(
2ln2
2
2
2
O
ba
b
ba
a
sba
ab
ba
b
ba
a
ba
bQ
ba
aQ
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
61490.0)2(ln6
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
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