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1
LOGO
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Chapter 6
Electrostatic Boundary
value problems
Part 2
3
CAPACITANCE
(1) Parallel-plate capacitor
1
2 0
D , E
.
. . 1
. .
s
s x xs x
d
Q SC
Q
S
Q
S
Q Q QdV E dl dx dx
S S S
E dl E dSV QR C R
SI VE
V d
dS E dl
dRC
S
d
a aa
4
(2) coaxial cylindrical capacitor
2
0 0
for a ,
. (2 )
, 2
. ln( )2 2
1
2
ln( )
2
ln( )
ln( )
2
L
z
a
b
b
Q E dS E d dz E L
QE
L
Q Q bV E dl d
L L a
R
Q LC
bV
a
b
a
LC R
L
b
a
a
5
(3) Two concentric spherical conductors
2
2
2
4
1
for a ,
. (4 )
, 4
1 1.
4 4
1
4
1
4
1
1
1
1
r
a
b
r b
Q E dS E r
QE
r
Q QV E dl dr
a b
a
r a b
QC
V
RC bR
a b
a
6
A parallel-plate capacitor has plate area 200cm2 and plate separation 3 mm. The charge density is 1 µC/m2 with air as dielectric. Find (a)The capacitance of the capacitor(b) The voltage between the plates
4
3
6 3
(200 10 )( ) 59 pF
3 10
(b)
(1 10 )(3 10 )V= =338.83 V
o
s
s
o
Q
Sa C
d
Q
S
Qd dC V
S
S
V d
7
The parallel-plate capacitor of Figure is quarter-filled with mica (εr = 6). Find the capacitance of the capacitor.
4
1 3
4
2 3
4
3 3
1 23
1 2
10 10(6)( )
2( ) 0.265 pF2 10
( )2
10 10( )
2 4.427 pF2 10
( )2
10 10( )
2 2.2135 pF2 10
6.008 pF
o
o
o
eq
Sa C
d
SC
d
SC
d
C CC C
C C
1
23
2mm
8
The Concentric shells r = 20 cm and r = 30 cm are held at V = 0 and V = 50, respectively.If the space between them is filled with dielectric material (εr= 3.1, σ= 10-12 S/m), find: •V, E, and D at r=25cm•the charge densities on the shells•the capacitance of the system.•the leakage resistance.
2
Using Laplace's equation :
0 ,
( 20 ) 0 5 0 ......(1)0.2
10( 30 ) 0 50 ......(2)
0.3 3
solve(1)and(2) A=30 , B=150
30( ) 150
30( 25 ) 150 30 V
0.25
AV V B
r
AV r cm B A B
AV r cm B A B
V rr
V r cm
9
2
2
2
2 2
2
2
2
2
30 V/m
30E( 25 ) = 480 V/m
(0.25)
30 0.823422(3.1 )( ) nC/m
0.823422D( 25 ) =0.132 nC/m
(0.25)
0.823422( 20 ) 20.56 nC/m
(0.20)
r
r r
o r o r r
r r
s n r
E Vr
r cm
D Er r
r cm
r cm D D
(b) a
a a
(c)
a a
a a
(d) a
2
2
12
0.823422( 30 ) 9.137 nC/m
(0.30)
4 (3.1 )4 C 207pF
1 1 1 1
0.2 0.3
1 1 1 1
0.2 0.3 R 132.6 G4 4 (10 )
s n r
o
r cm D D
a b
a b
a
(e)
(f)
10
Determine the capacitance of a conducting sphere of radius 5 cm deeply immersed in seawater (εr = 80).
44 4 (80 )(0.05) 444
1 1o
Qb C a pF
V
a b
11
Figure represents the cross section of two spherical
capacitors Determine their capacitances, where
a = 1mm , b = 2mm ,c = 3mm
1
2
4 3.540.778pF ,
1 1 1 1
0.001 0.002
4 (2.5)40.1669pF
1 1 1 1
0.002 0.003
(0.1669)(0.778)0.53102 pF
(0.1669 0.778)
oab
obc
ab bc
ab bc
C
a b
C
b c
C CC
C C
Series
12
1
1,1
2,2
,1 ,2
4
1 1
2 (2.5)20.2086 pF
1 1 1 1
0.001 0.003
2 (3.5)20.292 pF
1 1 1 1
0.001 0.003
0.2086 0.292 =0.50068 pF
ac
oac
oac
ac ac ac
C
a c
C
a c
C
a c
C C C
Parallel
