1. Numerical Integration | Trapezoidal, Simpson's 1/3 an

1. Numerical Integration | Trapezoidal, Simpson's 1/3 an1. Numerical Integration | Trapezoidal, Simpson's 1/3 an1. Numerical Integration | Trapezoidal, Simpson's 1/3 an

1. 16.360 Lecture 19
Maxwell equations
,
v
D

,
t
B
E





,0B

,
t
D
JH





,ED

 ,HB

 : electrical permittivity; :magnetic permativity
v: electric charge density per unit volume; J: current density per unit area.
,
v
D

,0E

,0B

,JH


Electrostatics Magnetostatics
E: electric field intensity
D: electric flux intensity
H: magnetic field intensity
B: magnetic flux intensity

2. 16.360 Lecture 19
,
vD

,0E

Electrostatics
Volume charge density
,lim
0 dv
dq
v
q
v
v 





Surface charge density
,lim
0 ds
dq
s
q
s
s






Line charge density
,lim
0 dl
dq
l
q
l
l 





,dvQ
v
v


3. 16.360 Lecture 19
,tsuq
v



,su
t
q
I
v




 
Current density J
,uJ
v


,sdJI
s



4. 16.360 Lecture 19
,
4
2
R
q
RE



,'EqF


Coulomb’s law
,
4
)(
3
1
11
1
RR
RRq
E 






,
4
)(
3
2
22
2
RR
RRq
E 






],
)()(
[
4
1
3
2
22
3
1
11
21
RR
RRq
RR
RRq
EEE 











,
)(
4
1
1
3
1

 


N
i
i
i
RR
RRq
E 




5. 16.360 Lecture 19
Electric field due to a charge distribution
,
'4
'
ˆ
2
R
dq
REd



,
'
'
ˆ
4
1
'
2

v
v
R
dv
REdE



,
'
'
'
ˆ
4
1
2

s
s
R
ds
REdE



,
'
'
'
ˆ
4
1
2

l
l
R
dl
REdE




6. 16.360 Lecture 19
Gauss’s law
,
vD

,QdvdvD
v
v
v
 


,

sv
sdDdvD

,QsdD
s


Gauss’s law

7. 16.360 Lecture 19
Electrical scalar potential
,ldEdV


,ldEqldFdW
e


,qdVdW
,
2
1
2
1
1221 

P
P
P
P
ldEdVVVV

,00  
CC
ldEdV

,0)(  
Cs
ldEsdE


8. 16.360 Lecture 19
Electrical potential due to point charge
,'
'4
1
)(
'
'
dv
RR
RV
v
v


 
 

,
4
2
R
q
RE



,
4
ˆ
4
ˆ
2
R
q
dRR
R
q
RldEV
RR

 





R
ldEV

Electrical potential due to continuous distributions
,
4
)(
1RR
q
RV 




,'
'4
1
)(
'
'
ds
RR
RV
s
s


 
 

,'
'4
1
)(
'
'
dl
RR
RV
l
l


 
 


9. 16.360 Lecture 19
Electric field as a function of Electrical potential
Poison’s equation
,ldEdV

 ,ldVdV


,VE

,
v
D

,


v
E

,


v
V ,
2


v
V Poison’s equation
,0
2
V Laplace’s equation

10. 16.360 Lecture 19
Electrical properties of material
• conductor
• dielectric
• semiconductor

11. 16.360 Lecture 20
Conductors
Electron drift velocityEu
ee


Hole drift velocity Eu
hh


Conducting current
,)( EuuJJJ
hvhevehvhevehe

 
,
hvheve
 
,EJ


Point form of Ohm’s law

12. 16.360 Lecture 20
Resistance
,
1
2
21
lEldEVVV
x
x
x
 

General form
,AEsdEsdJI
x
AA
  

,
A
l
I
V
R


,
1
2
1
2










A
x
x
A
x
x
sdE
ldE
sdJ
ldE
I
V
R 





13. 16.360 Lecture 20
Joule’s law
,
hhee
lFlFW 
General form
,
)(
vEJ
vEuEuuFuF
t
l
F
t
l
F
t
W
P
hvhevehhee
h
h
e
e
















v
dvEJP ,


14. 16.360 Lecture 20
Dielectrics
Electrical field induced polarization

15. 16.360 Lecture 20
Dielectrics
,
0
PED


P: electric polarization field
For homogeneous material:
,
0
EP
e


,
000
EEEPED
e

 
),1(
0 e
 
),1(
0
er



 Relative permittivity:
Electric susceptibility
Dielectric breakdown

16. 16.360 Lecture 20
Electric boundary condition
;0][
12
0
lim
 

ldEldEldE
d
c
b
a
h
C

,
111 ntEEE


,
222 nt
EEE


,0
21  lElE
tt

,
21 tt
EE


the tangential component is continuous
across the boundary of two media.

17. 16.360 Lecture 20
Electric boundary condition
;][lim
0
ssdDsdDsdD
s
bottomtop
h
C
 



,
21 ssDsD
snn  

the normal component of D changes, the
amount of change is equal to the surface
Charge density.
,
21 snnDD 


18. 16.360 Lecture 20
Dielectric-Conductor boundary
,
1 snD

,0
21 
ttEE


19. 16.360 Lecture 20
Conductor-Conductor boundary
,
221121 snnnn EEDD  

,
21 ttEE


,
2
2
1
1

ttJJ

 ,
2
2
2
1
1
1 s
nn JJ




 

,)(
2
2
1
1
1 sn
J 







20. 16.360 Lecture 20
Capacitance
,

s
sdEQ


,
V
Q
C


l
ldEV

,


RC
,





l
s
ldE
sdE
C 


,
1
2
1
2










A
x
x
A
x
x
sdE
ldE
sdJ
ldE
I
V
R 





21. 16.360 Lecture 20
Electrostatic Potential Energy
,ldWldFdW
ee

,
2
1
EDW
e


,
eWF

Image Method
Any given charge above an infinite, perfect conducting plane is electrically
equivalent to the combination of the give charge and it’s image with conducting
plane removed.