Ve 230 Homework Set #1. Review of Vector Calculus.
Assign date: 2013/5/16
Due date: 2013/5/23.
1) Find the area of a circle or radisu a in the xy plane centered at the origin using:
a) rectangular coordinates x 2 + y 2 = a 2 Hint: a 2 x 2 dx =
1
x a 2 x 2
Open and closed
B
An open line path
A
B
A
fdl
B
A
E dl
A close line path ( Starts from and ends in the same point)
C
fdl
C
E dl
General surface integration path
E ds
E ds
s
A closed surface integration path
s
The circle indicates it is closed path or su
Dipoles, conductors and
polarization
Electric field lines and equipotential lines(surfaces)
E field direction is
tangential to the E field
line at any point
2 dimensional
3 dimensional
E dl 0
The differential
equation for E field lines
Equipotential line
E field is an irrotational field .why, so what?
The physical meaning of potential, the relation with E
E field lines & equpotential lines properties
The properties of conductors Q, E, V
Dielectricss Q, polarization(trying to cancel) dielectric constant (n
About the bound charges
Bound charges=polarization charges
Bound charges decrease external E field
Polarization
When an external Eo is applied
Bound charges establish Ep in the opposite direction
Electric field is reduced from Eo to E E0 E p
E Eo
Conduc
Capacitors
Capacitance and Capacitor
Capacitors
Q
C
U
A capacitor consists of two conductors separated by free space or
dielectric materials.
The two conductors may be of arbitrary shapes
Materials dielectrics such as glass, polymer, liquid crystal and ai
Dielectrics
E
Nonpolar molecules
Polar molecules
There is no free charge either on the surface of or inside a dielectric material
External E field polarizes a dielectric material, and aligns the molecules
There are polarized charges on the surface of a
Capacitors
Capacitance and Capacitor
Capacitors
Q
C
U
A capacitor consists of two conductors separated by free space or
dielectric materials.
The two conductors may be of arbitrary shapes
Materials dielectrics such as glass, polymer, liquid crystal and ai
Capacitors
Assume there is +Q on the inner conductor,
Q on the outer conductor
Apply Guasss law in mediathe Guasss surface
Being the cylindrical wall a r b
Then the potential:
Then the capacitance:
Q
E ar
2 rL
r a
r a
Q
Vab E dl ar
(ar dr )
r b
r b
2 rL
Solution of electrostatic
problems I
VE230 Yan Li
E 0
E V
D 0
E E E E V
Uniform media
V
V
V 2V 2V 2V
V (ax a y az ) (ax
ay
az
) 2 2 2
x
y
z
x
y
z
x
y
z
2
2
2
V
V
V
2
V V
2
2
x
y
z 2
V
2
Poissons equation
V
V
V
V V
2
2
x
y
z 2
2
2
2
2
2
Fundamental Postulates
B
D
magnetic flux density
electric flux density
The fundamental postulates of magnetostatics that specify the
divergence and the curl of B in free space are
Charges are sources
D
D 0 E 0
0
B 0
B 0 J
no magnetic flow sources, No m
Possion Laplace method of
image
VE230 Yan Li
E 0
E V
D 0
E E E E V
Uniform media
V
V
V 2V 2V 2V
V (ax a y az ) (ax
ay
az
) 2 2 2
x
y
z
x
y
z
x
y
z
2
2
2
V
V
V
2
V V
2
2
x
y
z 2
V
2
Poissons equation
V
V
V
V V
2
2
x
y
z 2
2
2
2
2
2
2
2
2
Gradient,Divergenceand
GausssLaw
Electromagnetics
YanLi
Gradient
A vector that represents both the magnitude and direction of
maximum space rate of increase of a scalar field
sclope
Gradient
Itpointsinthedirection inwhich
thegivenfunctionchangesmost
rapi
VE230 Electromagne1cs I
Yan Li
[email protected]
ee1-202 34207644
VE230 Electromagne1cs I
Yan Li
[email protected]
ee1-202 34207644
CREOL, University of Central Florida
University of
Rochester
Orlando
Unive
E potential
VE230
Yan Li
Static E field
For a point charge
E
q
Calculate
r r'
E
4 0 r r ' 3
vector
E
so
q
4 0
E
r r'
r r'
q
[
3
scalar
1
4 0 r r '
3
r r '
+
1
r r'
3
r r '
]
Static E field
E
q
[
0
1
4 0 r r '
3
r r '
+
1
r r'
rr' 0
03
(r r ') =0
5
Ve 230 Homework Set #2. Electrostatics in Vacuum
Assign date: 2013/5/23
Due date: 2013/5/30.
1) Two small conducting balls, each of mass m, are at the end of
insulating strings of length l joined at a point. Charges are placed
on the balls so that they ar
Scalars and Vectors
Scalar Fields are single valued functions space-time
of
(temperature, charge density, f ( r , t )
energy)
Vector Fields are multivalued functions of space-time
F(
that indicate a direction (velocity, current, force) r , t )
Vector Nota
Review for final
The names of E, D, P,(B,H,M). What are their relations? What are the definition of P and M.
What is the polarization process and the magnetization process?
what is electric dipole? What is magnetic dipole?
D E P relation with Q
Capacitanc
Review for final
The names of E, D, P,(B,H,M). What are their relations? What are the definition of P and M.
What is the polarization process and the magnetization process?
what is electric dipole? What is magnetic dipole?
D E P relation with Q
Capacitanc
Class 1 Vector Analysis I
VE230 Electromagne1cs I
Yan Li
[email protected]
EE1-202 34207644
1
Scalars and vectors
Scalar: only magnitude no direc8on
such as
Vector: both magnitude and direc8on
such as
!" !
Cartesian Coordinate System
z
z!
x!
y!
y
x
Coordinate system where all coordinate surfaces are planar is
called Cartesian. Note that in this system, the direction of
coordinate vectors is the same at every point in space
Cylindrical Coordinate Syste
Reviewofthepreviousclasses
Coulombslaw
Efield
Reviewofthepreviousclasses
Line charge
density
l
dq l dl '
Surface charge
density
s
dq s ds '
Volume charge
density
v
dq v dv'
Efieldduetochargedistributions
Example3
Example4
Vector Analysis
OUTLINE
Two Null Identities
Helmholtzs Theorem
Two Null Identities
( A) 0
( f ) 0
Can you prove these in Cartesian
Coordinate?
Null Identity I
( A) 0
Stokes theorem
For a closed surface, no path C
Null Identity I
For a closed surface,
Class 2 Electrosta-cs I
J
Electrosta-cs
Static electric fields due to Charges at rest
Electric force
t
Coulombs Law
In 1785 French physicist
Charles-Augus-n de Coulomb
By measuring the electric
forces, he stated the
Ifinthecenterispointcharge,then
divergenceat5ispositiveinfinity
Ifthecenterisavolumewithfinite
chargedensity,thedivergenceat5is
positive
review
Vectoranalysis
Gradient
Directionandmagnitudeforthemaximumslope
A Ay A
x
z
A
Divergence
x
y
z
Volumedensityoff
VE230 Electromagnetics I
Sung-Liang Chen ()
Summer 2017
1
About me
1999 2003 B.S., Electrical Engineering National Taiwan University
2003 2005 M.S., Electro-optical Engineering, National Taiwan University
2007.09 2011.12 Ph.D., Electrical Engineering, Uni