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Department of Semiconductor Systems Engineering SoYoung Kim
Engineering Electromagnetics- 1 Lecture 8: Electric Dipole
SoYoung Kim
Department of Semiconductor Systems Engineering
College of Information and Communication Engineering
Sungkyunkwan University
Department of Semiconductor Systems Engineering SoYoung Kim
Outline
Electric Potential Review
Electric Dipole
Energy
Energy in Field
Department of Semiconductor Systems Engineering SoYoung Kim
Electric Dipole: Potential
Electric dipole: two point charges with opposite signs seprated by a small distance
Dipole moment p:
Electric potential due to an electric dipole
+Q
-Q
d d
In terms of dipole moment,
Dipole located at origin Dipole located at r’
2 1
1 2 1 2
2
2 1 2 1
2 2
1 1
4 4
If : ~ cos , ~
cos cos
4 4
o o
o o
r rQ QV
r r r r
r d r r d r r r
Q d pV
r r
32
( ') or ( )
4 4 '
r
o o
V Vr
p a p r rr
r r
p dQ
Department of Semiconductor Systems Engineering SoYoung Kim
Electric Dipole: Electric Field
From the definition of electric potential
Comparison with monopole
For monopole
Electric field varies inversely as r2
Potential varies inversely as r
For dipole
Electric field varies inversely as r3
Potential varies inversely as r2
3 3
1 cos sin
2 4r r
o o
V V Q d Q dV
r r r r
E a a a a
3(2 cos sin )
4E a a
r
o
p
r
Department of Semiconductor Systems Engineering SoYoung Kim
Electric Flux Line and Equipotential Surface
Electric flux line Lines whose direction at any point is same as the direction
of the electric field at the point
Lines to which electric flux density D is tangential at every
point
Equipotential surface Surface on which the potential is the same
No work is done in moving a charge on the equipotential surface
2D version is called equipotential lines
Equipotential lines and electric flux lines are always normal
Department of Semiconductor Systems Engineering SoYoung Kim
Energy in Electrostatic Field: Point Charge
Assume you move charges Q1, Q2, Q3 to positions P1, P2, P3
If you move the charges in the order of Q1, Q2, Q3, total work done is:
If you move in reverse order:
By adding the two:
In general,
Assume Vij is potential at point Pi due to charge Qj in this analysis
[J]
1 2 3
2 21 3 31 320 ( )
EW W W W
Q V Q V V
3 2 1
2 23 1 12 130 ( )
EW W W W
Q V Q V V
1 12 13 2 21 23 3 31 32
1 1 2 2 3 3
1 1 2 2 3 3
2 ( ) ( ) ( )
1( )
2
E
E
W Q V V Q V V Q V V
Q V Q V Q V
W Q V Q V Q V
1
1
2
n
E k k
k
W Q V
Solve Ex. 4.14
Department of Semiconductor Systems Engineering SoYoung Kim
Energy in Electrostatic Field: Distributed Charge
For distributed charge
1 (line charge)
2
1 (surface charge)
2
1 (volum e charge)
2
1( )
2D D
E LL
E SS
E vv
E vv
W V dl
W V dS
W V dv
W V dv
Department of Semiconductor Systems Engineering SoYoung Kim
Energy and Energy Density in Electrostatic Field
Electrostatic energy
Electrostatic energy density
2
2 31 1 [J/m ] or
2 2 2
E
E o E Ev
o
dW Dw E W w dv
dv
D E
U sing vector identity ( ) ( )
1 1 ( ) ( )
2 2
Applying divergence theorem ,
1 1 ( ) ( )
2 2
First term on R H S w ill becom e zero
A A A A A A
D D
D S D
Ev v
Ev
V V V V V V
W V dv V dv
W V d V dv
2
1 1 ( ) ( )
2 2
1 1
2 2
D D E
D E
Ev v
E ov v
W V dv dv
W dv E dv
[J]