EEE241:
Fundamentals of
Electromagnetics
Introductory Concepts, Vector
Fields and Coordinate Systems
Instructor: Dragica Vasileska
Outline
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Class Description
Introductory Concepts
Vector Fields
Coordinate Systems
Class Description
Prerequisites by Topic:
– University physics
– Complex numbers
– Partial differentiation
– Multiple Integrals
– Vector Analysis
– Fourier Series
Class Description
• Prerequisites: EEE 202; MAT 267, 274 (or
275), MAT 272; PHY 131, 132
• Computer Usage: Students are assumed to be
versed in the use MathCAD or MATLAB to
perform scientific computing such as numerical
calculations, plotting of functions and performing
integrations. Students will develop and visualize
solutions to moderately complicated field
problems using these tools.
• Textbook: Cheng, Field and Wave
Electromagnetics.
Class Description
• Grading:
Midterm #1
Midterm #2
Final
Homework
25%
25%
25%
25%
Class Description
Why Study Electromagnetics?
Examples of Electromagnetic
Applications
Examples of Electromagnetic
Applications, Cont’d
Examples of Electromagnetic
Applications, Cont’d
Examples of Electromagnetic
Applications, Cont’d
Examples of Electromagnetic
Applications, Cont’d
Research Areas of
Electromagnetics
•
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•
•
•
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Antenas
Microwaves
Computational Electromagnetics
Electromagnetic Scattering
Electromagnetic Propagation
Radars
Optics
etc …
Why is Electromagnetics
Difficult?
What is Electromagnetics?
What is a charge q?
Fundamental Laws of
Electromagnetics
Steps in Studying Electromagnetics
SI (International System) of
Units
Units Derived From the
Fundamental Units
Fundamental Electromagnetic Field
Quantities
Three Universal Constants
Fundamental Relationships
Scalar and Vector Fields
• A scalar field is a function that gives us
a single value of some variable for
every point in space.
• Examples: voltage, current, energy,
temperature
• A vector is a quantity which has both a
magnitude and a direction in space.
• Examples: velocity, momentum, acceleration
and force
Example of a Scalar Field
Scalar Fields
e.g. Temperature: Every location has
associated value (number with units)
26
Scalar Fields - Contours
• Colors represent surface temperature
• Contour lines show constant temperatures
27
Fields are 3D
• T = T(x,y,z)
• Hard to visualize
 Work in 2D
28
Vector Fields
Vector (magnitude, direction) at every point
in space
Example: Velocity vector field - jet stream
29
Vector Fields Explained
Examples of Vector Fields
Examples of Vector Fields
Examples of Vector Fields
VECTOR REPRESENTATION
3 PRIMARY COORDINATE SYSTEMS:
• RECTANGULAR
• CYLINDRICAL
• SPHERICAL
Choice is based on
symmetry of problem
Examples:
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL
Orthogonal Coordinate Systems: (coordinates mutually perpendicular)
Cartesian Coordinates
z
P(x,y,z)
Rectangular Coordinates
P (x,y,z)
y
x
z
z
P(r, θ, z)
Cylindrical Coordinates
P (r, Θ, z)
x
r
θ
y
z
Spherical Coordinates
θ r
P (r, Θ, Φ)
x
Φ
P(r, θ, Φ)
y
Page 108
-Parabolic Cylindrical Coordinates (u,v,z)
-Paraboloidal Coordinates (u, v, Φ)
-Elliptic Cylindrical Coordinates (u, v, z)
-Prolate Spheroidal Coordinates (ξ, η, φ)
-Oblate Spheroidal Coordinates (ξ, η, φ)
-Bipolar Coordinates (u,v,z)
-Toroidal Coordinates (u, v, Φ)
-Conical Coordinates (λ, μ, ν)
-Confocal Ellipsoidal Coordinate (λ, μ, ν)
-Confocal Paraboloidal Coordinate (λ, μ, ν)
Parabolic Cylindrical Coordinates
Paraboloidal Coordinates
Elliptic Cylindrical Coordinates
Prolate Spheroidal Coordinates
Oblate Spheroidal Coordinates
Bipolar Coordinates
Toroidal Coordinates
Conical Coordinates
Confocal Ellipsoidal Coordinate
Confocal Paraboloidal Coordinate
z
θ r
x
Φ
z
P(r, θ, Φ)
Cartesian Coordinates
P(x,y,z)
P(x,y,z)
y
x
y
Spherical Coordinates
P(r, θ, Φ)
z
Cylindrical Coordinates
P(r, θ, z)
z
P(r, θ, z)
x
θ
r
y
Coordinate Transformation
• Cartesian to Cylindrical
(x, y, z) to (r,θ,Φ)
(r,θ,Φ) to (x, y, z)
Coordinate Transformation
• Cartesian to Cylindrical
Vectoral Transformation
Coordinate Transformation
• Cartesian to Spherical
(x, y, z) to (r,θ,Φ)
(r,θ,Φ) to (x, y, z)
Coordinate Transformation
• Cartesian to Spherical
Vectoral Transformation
Vector Representation
z
z1
Z plane
Unit (Base) vectors
x plane
A unit vector aA along A is a vector
whose magnitude is unity
ẑ ŷ

 A
a 
A
x̂
x1
Ax
y1
Ay
y
x
Unit vector properties
xˆ  xˆ  yˆ  yˆ  zˆ  zˆ  1
xˆ  yˆ  yˆ  zˆ  zˆ  xˆ  0
xˆ  yˆ  zˆ
yˆ  zˆ  xˆ
zˆ  xˆ  yˆ
Page 109
Vector Representation
z
Vector
 representation
z1
A  xˆAx  yˆAy  zˆAz
Z plane
Magnitude of A
x plane
   
A  A  A   Ax2  Ay2  Az2
Az
ẑ ŷ
x̂
Position vector A
xˆx1  yˆy1  zˆz1

A( x1 , y1 , z1 )
x1
Ax
y1
Ay
y
x
Page 109
Cartesian Coordinates
z
Dot product:
 
A  B  Ax Bx  Ay By  Az Bz
Az

B
Cross product:
Ax
xˆ
 
A  B  Ax
yˆ
zˆ
Ay
Az
Bx
By
Bz

A
Ay
y
x
Back
Page 108
Multiplication of vectors
• Two different interactions (what’s the
difference?)
– Scalar or dot product :
A  B | A || B | cos  B  A
• the calculation giving the work done by a force during a
displacement
• work and hence energy are scalar quantities which arise
from the multiplication of two vectors
• if A·B = 0
– The vector A is zero
– The vector B is zero
–  = 90°
A

B
– Vector or cross product :
A  B | A || B | sin  n
• n is the unit vector along the normal to the plane
containing A and B and its positive direction is
determined as the right-hand screw rule
A B  B  A
• the magnitude of the vector product of A and B is
equal to the area of the parallelogram formed by A
and B
• if there is a force F acting at a point P with position
vector r relative to an origin O, the moment of a force
F about O is defined by :
L  rF
• if A x B = 0
– The vector A is zero
– The vector B is zero
–  = 0°
A

B
Commutative law :
A B  B  A
A B  B  A
Distribution law :
A  (B  C)  A  B  A  C
A  (B  C)  A  B  A  C
Associative law :
A  BC  D  ( A  B)(C  D)
A  BC  ( A  B)C
A  B  C  ( A  B)  C
A  ( B  C )  ( A  B)  C
Unit vector relationships
• It is frequently useful to resolve vectors into components
along the axial directions in terms of the unit vectors i, j,
and k.
i  j  j  k  k i  0
A  Ax i  Ay j  Az k
i i  j  j  k  k  1
B  Bx i  B y j  Bz k
ii  j  j  k  k  0
A  B  Ax Bx  Ay B y  Az Bz
i j  k
i
j
k
jk  i
A  B  Ax
Ay
Az
k i  j
Bx
By
Bz
A B  C
Scalar triple product
The magnitude of A B  C is the volume of the parallelepiped with edges parallel to
A, B, and C.
A B
C
B
A
A  B  C  A  B  C  B  C  A  B  C  A  C  A  B  [ A, B, C ]
Vector triple product
A B  C
The vector A B is perpendicular to the plane of A and B. When the further vector
product with C is taken, the resulting vector must be perpendicular to A B and
hence in the plane of A and B :
( A  B)  C  mA  nB
where m and n are scalar constants to be determined.
C  ( A  B)  C  mC  A  nC  B  0
m  C  B
n  C  A
( A B)  C   (C  B) A  (C  A) B
Since this equation is valid
for any vectors A, B, and C
Let A = i, B = C = j:
  1
( A  B)  C  ( A  C ) B  ( B  C ) A
A  ( B  C )  ( A  C ) B  ( A  B )C
A B
C
B
A
VECTOR REPRESENTATION: UNIT VECTORS
Rectangular Coordinate System
z
Unit Vector
Representation
for Rectangular
Coordinate
System
âz
â x
â y
y
x
The Unit Vectors imply :
â x
Points in the direction of increasing x
â y
Points in the direction of increasing y
âz
Points in the direction of increasing z
VECTOR REPRESENTATION: UNIT VECTORS
Cylindrical Coordinate System
z
r
âz
P
âf
z
x
f
âr
y
The Unit Vectors imply :
âr
Points in the direction of increasing r
âf
Points in the direction of increasing j
âz
Points in the direction of increasing z
Cylindrical Coordinates
( ρ, Φ, z)
ρ radial distance in x-y plane
0r 
Φ azimuth angle measured from the positive
x-axis
0    2
Z
A1
  z  
Vector representation


ˆ A  zˆAz
A  aˆ A  ˆA  
Base
Vectors
Magnitude of A
   
A  A  A   A2  A2  Az2
Base vector properties
ˆ  zˆ,
ˆ  
Position vector A
ˆ1  zˆz1
Back
ˆ  zˆ  ˆ ,

ˆ
zˆ  ˆ  
Pages 109-112
Cylindrical Coordinates
Dot product:
 
A  B  Ar Br  Af Bf  Az Bz
B
A
Cross product:
 
A  B  Ar
fˆ
zˆ
Af
Az
Br
Bf
Bz
rˆ
Back
Pages 109-111
VECTOR REPRESENTATION: UNIT VECTORS
Spherical Coordinate System
âf
z
P

âr
r
x
f
â
y
The Unit Vectors imply :
âr
â
Points in the direction of increasing r
âf
Points in the direction of increasing j
Points in the direction of increasing 
Spherical Coordinates
(R, θ, Φ)
Vector representation

A  Rˆ AR  ˆA  fˆAf
Magnitude of A
   
A  A  A   AR2  A2  Af2
Position vector A
Rˆ R1
Base vector properties
Rˆ  ˆ  ˆ ,
ˆ  ˆ  Rˆ ,
ˆ  Rˆ  ˆ
Back
Pages 113-115
Spherical Coordinates
Dot product:
 
A  B  AR BR  A B  Af Bf
B
A
Cross product:
 
A  B  AR
ˆ
fˆ
A
Af
BR
B
Bf
Rˆ
Back
Pages 113-114
VECTOR REPRESENTATION: UNIT VECTORS
Summary
RECTANGULAR
Coordinate
Systems
aˆ
x
aˆ y aˆ z 
CYLINDRICAL
Coordinate
Systems
aˆ
r
aˆ f aˆ z 
SPHERICAL
Coordinate
Systems
aˆ
r
aˆ aˆf 
NOTE THE ORDER!
r,f, z
r, ,f
Note: We do not emphasize transformations between coordinate systems
METRIC COEFFICIENTS
1. Rectangular Coordinates:
Unit is in “meters”
When you move a small amount in x-direction, the distance is dx
In a similar fashion, you generate dy and dz
Cartesian Coordinates
Differential quantities:
Differential
 distance:
dl  xˆdx  yˆ dy  zˆdz
Differential surface:

ds x  xˆdydz

ds y  yˆ dxdz

ds z  zˆdxdy
Differential Volume:
dv  dxdydz
Page 109
Cylindrical Coordinates:
Differential Distances:
y
Distance = r df
df
r
x
( dr, rdf, dz )
Cylindrical Coordinates:
Differential Distances: ( dρ, rdf, dz )

dl  d  aˆ    df  aˆf  dz  aˆz
Differential Surfaces:

ds   df  dz  aˆ 

dsf  d  dz  aˆf

ds z  df  d  aˆ z
Differential Volume:
Spherical Coordinates:
Differential Distances:
y
Distance = r sin df
( dr, rd, r sin df )
z
df
P

r sin
x
r
x
f
y
Spherical Coordinates
dlR  dR
dl  Rd 
Differential quantities:
Length:

ˆ dl  
ˆ dl
dl  Rˆ dl R  


ˆ Rd   
ˆ R sin d
 Rˆ dR  
dl  R sin d
Area:

ds R  Rˆ dl dl  Rˆ R 2 sin dd

ˆ dl dl  
ˆ R sin dRd 
ds  
R 

ˆ dl R dl  
ˆ RdRd 
ds  
Volume:
dv  R 2 sin dRd d
Back
Pages 113-115
METRIC COEFFICIENTS
Representation of differential length dl in coordinate systems:
rectangular
cylindrical
spherical

dl  dx  aˆx  dy  aˆy  dz  aˆz

dl  dr  aˆr  r  df  aˆf  dz  aˆz

dl  dr  aˆr  rd  aˆ  r sin df  aˆf
Example
• For the object on the right
calculate:
• (a) The distance BC
• (b) The distance CD
• (c) The surface area ABCD
• (d) The surface area ABO
• (e) The surface area A OFD
• (f) The volume ABDCFO
AREA INTEGRALS
• integration over 2 “delta” distances
dy
dx
Example:
y
7 6
AREA =
6
  dy  dx
= 16
3 2
2
Note that: z = constant
3
7
x
In this course, area & surface integrals will be
on similar types of surfaces e.g. r =constant
or f = constant or  = constant et c….
SURFACE NORMAL
Representation of differential surface element:
Vector is NORMAL
to surface

ds  dx  dy  aˆ z
DIFFERENTIALS FOR INTEGRALS
Example of Line differentials

dl  dx  aˆ x
or

dl  dr  aˆ r
or

dl  rdf  aˆf
Example of Surface differentials

ds  dx  dy  aˆ z
Example of Volume differentials
or

ds  rdf  dz  aˆr
dv  dx  dy  dz
Cartesian to Cylindrical Transformation
Ar  Ax cos f  Ay sin f
Af   Ax sin f  Ay cos f
Az  Az
r   x2  y2
f  tan 1 ( y / x )
zz
rˆ  xˆ cos f  yˆ sin f
fˆ   xˆ sin f  yˆ cos f
zˆ  zˆ
Back
Page 115