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Electrostatics 2
UCF Boundary Conditions
tt EE 12
. gives This .0 as 0 volume
since 0 charge, volumefor or ;0 charge, noFor
12
volumeinside
volumeinside
nn DDhQQ
021 LELE tt0 dLE
20h
0Volume
QdSD2
0)( 12n EEa 1LMedia 1
Media 2
E1
E2
E1t
E2t
0h1
S
Media 1
Media 2
D1
D2
D1n
D2n volumeinside12 QSDSD nn
SQ S volumeinside charge, surfaceFor SSnn DD )(or 1212 DDan
(1)
(2)
na
na
UCF Boundary Conditions for Potential
2V21 VV 2
11111
222
VV
EDED 22 1Media 1
Media 2
S
S
nV
nV
nV
nV
22
11
11
22 )()(
. charge, surface no is thereIf 2211 nV
nV
S )( 12 DDan
(1)
(2)1V
Potential should be continuous across the boundary
na
UCF Perfect Electric Conductor (PEC)
0Ean 0tE
0E
Media D
nV
S Dan
SDan
(1)
(2)
na
PEC
or
0BAB A B AV V V E dLOn PEC body (including boundary),
PEC is equal potential.
UCF Capacitance
+Q
-Q
+
-V
VQC
UCF Energy Stored in Electrostatic Field
ED 21
ew
V ee dVwW
Energy density
Total energy
UCFPoissons and Laplaces Equations
02
2
V
V V
0 V
VV D
From electrostatic equations:
VDE
0
In simple media
V E
Or
V ED
For source free region, we have
VV Then
If the dielectric distribution is uniform
Laplaces Equation
Poissons Equation
UCFUniqueness Theorem
02
2
V
V V
na
VV D
For
nV
nV
If on the boundary, V or is given, the solution is unique.
Laplaces Equation
Poissons Equation
or
given
V given
UCFProof of Uniqueness Theorem (1)
)( )()()()()(
have we,let weif
formulamath From
0or 0
have wegiven, is or either since boundary, On the
0)( have we
Equations, sLaplace'or sPoisson'satisfy and solutions twoAssuming
221
22121
2212121
21
22
boundary on 21
boundary on 21
212
21
VV
VVVVVVVVVV
VV
nV
nVVV
nVV
VV
VV
Proof:
UCFProof of Uniqueness Theorem (2)
solutions. identical twogiving Therefore, given. is when
0boundary on the since zero be toevaluatedeasily becan constant Theconstant
or0)(
have Then we
condition)boundary (from 0 ))((
Theorem) sGauss' (from )()(
)()()(
have wee,over volumequation above thegIntegratin
21
21
21
21
S
2121
S2121
21212
21
VVV
VVVV
VV
dSnV
nVVV
VVVV
dVVVVVdVVV
dS
UCF Laplaces Equations
Rectangular (Cartesian) Coordinate
Polar (Cylindrical) Coordinate
Spherical Coordinate
UCF Example 1 (1)Find the potential distribution between two parallel plates. Assume the two plates are large enough so that
0 and 0
yV
xV
z2
z1
z
d
V1
V2
UCF Example 1 (2)
02 VFrom Laplaces equation
0 and 0
yV
xV
022
dzVd
022
2
2
2
2
zV
yV
xV
)( and )( 2211 VzVVzV
Solution:we have
in Cartesian coordinate.
Assuming the two plates are large enough so that
we have
The boundary conditions are:
(1)
(2)
UCF Example 1 (3)
zdVzV 2)( 12
1221
12
12)(zzzVzVz
zzVVzV
The general solution for (1) is
have we,0 and 0 If 11 Vz
22
11
VBAzVBAz
12
1221
12
12 and zzzVzVB
zzVVA
BAzzV )( (3)Inserting (2) into (3) results in
from which, we have
Therefore
linear distribution
This page can be done using MatLab dsolve.
UCF Example 1 (4)
zaED dV2
zaE dVV 2
Find capacitance between two parallel plates.
zdVzV 2)(
0 and 0
yV
xVz2=d
0
z
d0
V2
Then
)()(
2
2 top
22 top
dS
VQC
SdVSQ
dV
dV
S
S
zz aa
zaS
S
UCF Example 2
0 ,0
r
02
1 0
0sin
V)V()V(
ddV
dd
0cossin 22
ddV
dVd
To use dsolve, we need to change the equation to: a